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\title{COMBINATORICS, SUPERALGEBRAS, INVARIANT THEORY AND REPRESENTATION THEORY}

\author{A. Brini \\ [18pt]
Dipartimento di Matematica \\ [13pt] ``Alma Mater Studiorum"
Universit\`{a} degli Studi di Bologna}



\date{}

\begin{document}

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Combinatoire \bfs 55 \rms (2007), Article~B55g\hfill}
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\begin{abstract}
We provide an elementary introduction to the (characteristic zero)
theory of {\it Letterplace Superalgebras}, regarded as bimodules
with respect to the {\it superderivation} actions of a pair of
{\it general linear Lie superalgebras}, and discuss some
applications.
\end{abstract}
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%\markboth{\SMALL ANDREA BRINI}{\SMALL COMBINATORICS, SUPERALGEBRAS, INVARIANT THEORY AND REPRESENTATION THEORY}
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\newpage

\section{Introduction}

The purpose of the present work is to provide an elementary
introduction to the (characteristic zero) theory of {\it
Letterplace Superalgebras}, regarded as bimodules with respect to
the {\it superderivation} actions of a pair of {\it general linear
Lie superalgebras}, as well as to show that this theory yields (by
specialization) a simple unified treatment of classical theories.







The idea of Lie superalgebras arises from Physics, since they
implement ``transitions of symmetry" and, more generally, {\it
supersymmetry} (see, e.g., \cite{Iachello}, \cite{FSS},
\cite{Varad}, \cite{BalB}, \cite{CWS}, \cite{DJ}, \cite{KOST}).



General linear Lie superalgebra actions on letterplace superalgebras yield a natural
setting that allows Capelli's method of {\it virtual (auxiliary)
variables} to get its full effectiveness and suppleness. The superalgebraic version
of Capelli's method was introduced by Palareti, Teolis and the present author in 1988 in order to prove
the complete decomposition theorem for letterplace superalgebras \cite{Brini1},  to introduce
the notion of  {\it Young--Capelli symmetrizers}, and to provide a ``natural matrix form"
of Schur superalgebras  \cite{Brini2}.


The theory of letterplace superalgebras, regarded as bimodules
with respect to the action of a pair of general linear Lie
superalgebras, is a fairly general one, and encompasses a variety
of classical theories, by specialization and/or restriction.

We limit ourselves to mention the following:
\begin{itemize}

\item The ordinary representation theory of the symmetric group,
up to the Young natural form of irreducible matrix representations.

\item The classical representation theory
of general linear and symmetric groups over tensor spaces, as well
as its pioneering generalization to the $\BZ_2$-graded case due to
Berele and Regev \cite{Berele1}, \cite{Berele2}.

\item Vector invariant  theory (see, e.g.,  \cite{Weyl}, \cite{drs}, \cite{DKR},
\cite{DEP}).

\item After the work of Grosshans, Rota and Stein \cite{GRS},
letterplace superalgebras provide a unified language for the {\it
symbolic representation} of invariants of symmetric tensors
(Aronhold symbols) and skew-symmetric tensors (Weitzenb{\"o}ck's
Komplex-Symbolik) (see, e.g., Weyl \cite{Weyl},  Brini, Huang and Teolis \cite {BHT}, Grosshans
\cite{Gr1}).

\item The Deruyts theory of {\it covariants of weight zero}
\cite{Deruyts}, 1892.

Deruyts developed a theory of covariants of weight zero that
anticipates by nearly a decade the main results of
Schur's celebrated Dissertation on the irreducible polynomial
representations of $GL_n(\BC)$.

This work of Deruyts has been defined by Green {\it ``a pearl of
nineteenth century algebra"} (\cite[page~249]{Green1}).


\item Letterplace superalgebras  provide a natural setting
  to extend (in an effective and non-trivial way), the theory of {\it transvectants}
  from  binary forms to $n$-ary forms, $n$ an arbitrary positive integer (Brini, Regonati and Teolis \cite{BRT5}).


\end{itemize}




Two  methodological remarks. Thanks to the systematic use of
the superalgebraic version of Capelli's method of {\it virtual
variables}, the theory described below turns out to be a quite
compact one.

As a matter of fact, the whole theory relies upon two basic
results: the (super) Straightening Formula (Grosshans, Rota and Stein \cite{GRS}) and
the Triangularity Theorem for the action of (superstandard)  {\it
Young--Capelli symmetrizers} on the basis of (superstandard) {\it
symmetrized bitableaux} (Brini and Teolis \cite{Brini2}).

The (super) Straightening Formula admits a few lines proof in
terms of virtual variables (see, e.g., \cite{Brini5}, \cite{BRT2}). A quite direct proof
of the Triangularity Theorem has been recently derived from a
handful of combinatorial lemmas on Young tableaux (see, e.g., \cite{BRT4} and Regonati \cite{Regonati}).

I extend my heartfelt thanks to Francesco Regonati and Antonio
G.B. Teolis for their advice, encouragement and invaluable help;
without their collaboration this work would have never been
written.

I also thank the referees for their valuable comments and suggestions.


\section{Synopsis}

The work is organized as follows.

\vskip 0.2cm

In Section~3, we recall some elementary definitions about
associative and Lie superalgebras, and describe some basic
examples.

\vskip 0.2cm

In Section~4, Letterplace Superalgebras, regarded as bimodules
with respect to the  actions of a pair of general linear Lie
superalgebras, are introduced by comparing two  equivalent
languages, namely, the {\it combinatorial} one (``signed
alphabets") and the more traditional language of multilinear
algebra (``$\BZ_2$-graded vector spaces").


\vskip 0.2cm

In Section~5, we summarize a handful of basic definitions and
facts about Young tableaux on signed alphabets.

\vskip 0.2cm



In Section~6, we provide an introduction to the superalgebraic
version of Capelli's method of {\it virtual variables}.




Indeed, the basic  operators one needs to manage are operators
that induce ``symmetries". The starting point of the method is
that these operators, that we call  {\it Capelli-type operators},
can be  defined, in a quite natural and simple way, by appealing
to ``extra" variables (the {\it virtual variables}).

The true meaning of Theorem~\ref{theorem: action of Capelli-type
operator} is that the action of these operators  is the {\it same}
as the action of  operators induced by the action of the
enveloping algebra ${\cal U}(pl(V)$ of the general linear Lie
superalgebra of a $\BZ_2$-graded vector space $V$, and, therefore,
they are indeed of representation-theoretical meaning.

The action of these ``virtual" Capelli-type operators is much
easier to study than the action of their ``non-virtual"
companions, and computations are consequently carried out in the
virtual context.



\vskip 0.2cm

In Sections~7, 8 and 9, by using the method of virtual variables,
we introduce some crucial concepts of the theory, namely, the
concepts of {\it biproducts}, {\it bitableaux}, and {\it left,
right and doubly symmetrized bitableaux}.

Bitableaux provide the natural  generalization of {\it
bideterminants} of a pair of Young tableaux in the sense of
\cite{drs} and \cite{DKR}.

Left, right and doubly symmetrized bitableaux generalize a variety
of classical notions. We mention:

\begin{itemize}
\item Images of highest weight vectors, under the action of the
``negative" {\it root spaces} of $sl_n(\BC)$ (see, e.g., \cite{FH}).
\item Generators of the irreducible symmetry classes
of tensors (see, e.g., \cite{Weyl}, \cite{Berele1}, \cite{Berele2}).
\item Generators of minimal left ideals of the group
algebra $\BK [{\bf S}_n]$, ${\bf S}_n$ the symmetric group on $n$
elements (see, e.g., \cite{JK}).

\end{itemize}

\vskip 0.3cm

We provide an elementary proof of the (super)-Straightening Law of
Grosshans, Rota and Stein \cite{GRS} and exhibit four classes of
``representation-theoretically" relevant bases of the letterplace
superalgebra: the standard basis and three classes  of
Clebsch--Gordan--Capelli bases. The first basis is given by
(super)standard bitableaux, while the others are given by
(super)standard right, left and doubly symmetrized bitableaux,
respectively. (We submit that there are deep relations among these
bases. The Clebsch--Gordan--Capelli bases yield {\it complete
decompositions} of the letterplace superalgebra as a module, while
the standard basis yields an {\it invariant filtration} (this is a
characteristic-free fact);  in this filtration, the irreducible
modules that appear in the decomposition associated to the
Clebsch--Gordan--Capelli bases are complements of each invariant
subspace  to the preceding one).



\vskip 0.2cm

In Section~10, we introduce the basic operators of the theory, the
{\it Young--Capelli symmetrizers}. These operators are defined, via
the method of virtual variables, as special Capelli-type operators
and turn out to be a generalization of the classical Capelli
operators as well as of the Young symmetrizers.

In spite of the fact that they could be represented as (extremely
complicated) ``polynomials" in the (proper) polarization
operators, their virtual definition is quite simple and leads to
the main result of the theory, the {\it Triangularity Theorem}
(Theorem~\ref{theorem: triangularity of Young--Capelli symmetrizers
and symmetrized bitableaux}, Subsection~10.2).



\vskip 0.2cm

In Section~11, Schur and Weyl modules are described as subspaces
spanned by {\it right} and {\it left} symmetrized bitableaux.

Schur and Weyl modules provide the two basic classes of
irreducible submodules of the letterplace superalgebra with
respect to the action of a general linear Lie superalgebra. (In
the special case of a trivial {$\BZ_2$}-graduation, they yield the
two basic constructions of the irreducible representations of the
general linear group; see, e.g., \cite{ABW}, \cite{TOWB}).

\vskip 0.2cm

In Section~12, complete decompositions of the letterplace
superalgebra and of its Schur superalgebra are exhibited.

It is worth noticing that these decomposition results follow at
once from the Triangularity Theorem about the action of (standard)
Young--Capelli symmetrizers on (standard) symmetrized bitableaux.
Furthermore, a fairly general version of the {\it double
centralizer theorem} also follows from the same argument.



\vskip 0.2cm

In Section~13, we derive from the Triangularity Theorem a {\it
matrix form} for the irreducible representations of a general
linear Lie superalgebra over a letterplace superalgebra.

The matrix entries are strictly related (Theorem~\ref{theorem:
natural form of irreducible matrix representation}) to the {\it
symmetry transition coefficients} discussed in Subsection~8.2 that
are, in turn, a generalization of the  D\'esarm\'enien
straightening coefficients \cite{DC}.

Furthermore, these matrix representations yield, as a very special
case, the {\it Young natural form} of the irreducible matrix
representations of the symmetric group ${\bf S}_n$.

\vskip 0.5cm

The third part of the paper is devoted to the discussion of some
applications of the general theory.

\vskip 0.2cm

In Section~14, we derive an explicit decomposition result for
spaces obtained by performing  tensor products of spaces of {\it
symmetric} tensors and {\it skew-symmetric} tensors, regarded as
modules with respect to the classical diagonal action  of the
general linear group.

The present approach turns out to be a nice and transparent
application of the Feyn\-man--Rota idea of the
entangling/disentangling operator in the language of letterplace
superalgebras. Specifically, the components of the tensor product
are {\it faithfully encoded} by  {\it ``places with
multiplicity"}, and, therefore, the decomposition results are
special cases of the general decomposition results for letterplace
algebras. For example, Howe's version of the first fundamental
theorem of the invariant theory of $GL_n$ (Theorem~1.A,
\cite{Howe}) can be obtained from Corollary~\ref{corollary:
decomposition of symmetric and skew-symmetric tensor spaces}.

\vskip 0.2cm

In Section~15, the classical  representation theory of the special
linear Lie algebra $sl_m(\BC)$ in terms of highest weight vectors
and root spaces is derived in a  short and  concrete way.

We exhibit two characterizations of highest weight vectors in
$Super_n[\mathcal{L}|\mathcal{P}]$ as linear combinations of
standard bitableaux and of standard right symmetrized bitableaux
whose left tableau is a fixed tableau of Deruyts type (i.e.,
column-constant in the first symbols of the letter alphabet
$\mathcal{L}$) (Subsection~15.1).

These characterizations allow us to recognize that the action of
the subalgebra $n^-_m$ of strictly lower triangular matrices on
the highest weight vectors just ``replaces" the left Deruyts
tableau by any symmetrized standard tableau (Subsection~15.2).


By combining these two facts with the Clebsch--Gordan--Capelli bases
theorem, we infer that the $b^-_m$-cyclic modules generated by the
highest weight vectors are indeed {\it irreducible}
$sl_m(\BC)$-modules, and provide  a combinatorial description of
two families of bases of these irreducible $sl_m(\BC)$-modules in
terms of left symmetrized bitableaux and of doubly symmetrized
bitableaux (Subsection~15.3).

\vskip 0.2cm

In Section~16, we discuss Deruyts' theory of covariants, following
along the lines of its admirable reconstruction provided by J.A.
Green \cite{Green1} (see also \cite{Curtis}).

In his remarkable but undervalued paper of 1892, Deruyts developed
a theory of covariants of weight zero that yields, in modern
language, an exhaustive description of the irreducible polynomial
representations of $GL_n(\BC)$, as well as a proof of the complete
reducibility of any polynomial representations of $GL_n(\BC)$,
and, therefore, he  anticipates by nearly a decade the main
results of Schur's Dissertation.

{\it His language is the language of invariant theory, and he
makes little use of matrices. But we can now look back on Deruyts'
work and find a wealth of methods which, to our eyes, are pure
representation theory; some of these methods are still unfamiliar
today} (Green \cite[p. 248]{Green1}).

The starting point of Deruyts and Green's approach is the
construction of an algebra isomorphism $\sigma$ from a commutative
letterplace algebra to the algebra of covariants of weight zero
and the fact that, given any such covariant $\varphi$, its $\it
left \ span$ $L(\varphi)$ is a $\BK[GL_n]$-module that turns out
to be equal to the cyclic $\BK[GL_n]$-module generated by the
unique preimage $\gamma$ of $\varphi$ with respect to the
isomorphism $\sigma$ ($\gamma$ is called the {\it source} of the
covariant $\varphi$).

Thanks to this correspondence, one can develop the theory in the
context of covariants of weight zero and, then, translate the main
results in the language of their sources, where the
$GL_n$-representation theoretic meaning of the main results is, in
a sense, more transparent. This is precisely the strategy of
Deruyts and Green.

In order to exploit the full power of the theory developed in Part
II, we reverse this strategy and work directly in the algebra of
sources. The main results (e.g., characterization of
semiinvariants, construction of irreducible representations,
complete reducibility) turn out to be simple applications of
general results (again, the notion of symmetrized bitableau plays
a central role). Then, by using the map $\sigma$, we translate
these results into Deruyts' language of covariants.


\vskip 0.5cm

We believe that letterplace superalgebras and virtual variables
provide also a ``quick and good way" to learn and understand
classical theories. In order to support this assertion, in Sections~17 and 18 we describe  the way to deduce,
just by
specialization, the pioneering theory of tensor representations of
the symmetric group and general linear Lie superalgebras \cite{Berele1}, \cite{SERG1}, \cite{Berele2}
(special case: general linear groups \cite{Weyl}), as well
as the theory of regular representations of symmetric groups
(see, e.g., \cite{JK, Rut}).

\vskip 0.2cm

In Section~17, we consider the special case of a positively
$\BZ_2$-trivially graded alphabet ${\cal L} = \{1, \ldots,n \}$,
while ${\cal P}$ is any finite signed set.

Let $Super_n[\underline {\cal L}|{\cal P}]$ be the subspace of the
letterplace algebra spanned by all the {\it multilinear} monomials
of degree $n$ in the symbols of ${\cal L}$; this subspace is
isomorphic (via the so-called Feynman entangling/disentangling
operators \cite{drs}) to the space $T^n[W_0 \oplus W_1]$ of $n$-tensors over
the $\BZ_2$-graded vector space $W_0 \oplus W_1$ whose basis is
identified with the signed alphabet $\mathcal{P}$. There is a
natural action of the symmetric group $\bf S_n$ on the subspace
$Super_n[\underline {\cal L}|{\cal P}]$; via the linear
isomorphism mentioned above, this action corresponds to the {\it
Berele--Regev action} \cite{Berele1,Berele2} of $\bf S_n$ on the
tensor space $T^n[W_0 \oplus W_1]$.

Here, the point of the specialization argument is the fact that
the operator algebra induced by the action of the symmetric group
$\bf S_n$ admits a simple description in terms of polarization
operators (Proposition~\ref{proposition: operator algebra induced
by action of symmetric group on letter multilinear subspace}).
Therefore, the Berele--Regev theory follows, as a special case,
from the general theory of letterplace superalgebras (we recall
that, as a further special case --- {\it in the case of trivial
positive $\BZ_2$-graduation on $\mathcal{P}$} --- the Berele--Regev
theory reduces to the classical Schur--Weyl tensor representation
theory of  symmetric and general linear groups, see, e.g.,
\cite{Weyl}).

\vskip 0.2cm

In Section~18, we further specialize the theory and derive the
theory of regular representations of symmetric groups.

We consider the subspace $Super_n[\underline{\cal
L}|\underline{\cal P}]$ of the letterplace superalgebra spanned by
doubly multilinear monomials (on the negatively signed set ${\cal
L} = {\cal P} = \{1, \ldots, n \}$). The space
$Super_n[\underline{\cal L}|\underline{\cal P}]$ is a $\BK[\bf
S_n]$-module in an obvious way; here, the point of the
specialization argument is that the operator algebra
$\underline{\underline {\mathcal{B}}}_n$ induced by the action of $\BK[\bf S_n]$
is an isomorphic copy of $\BK[\bf S_n]$ and, again, it admits a
simple description in terms of polarization operators 
(Proposition~\ref{proposition: operator algebra induced by action of symmetric
group on doubly multilinear subspace}).

We recall that the $\BK[\bf S_n]$-modules $Super_n[\underline{\cal
L}|\underline{\cal P}]$ and  $\BK[\bf S_n]$ (regarded as left
regular module) are naturally isomorphic.

Therefore, the decomposition theory of  $\BK[\bf S_n]$ as a left
regular module is ``the same" as the theory of the module $\BK[\bf
S_n] \cdot \ Super_n[\underline{\cal L}|\underline{\cal P}] =
\underline{\underline {\mathcal{B}}}_n \cdot \ Super_n[\underline{\cal
L}|\underline{\cal P}]$ that, in turn, immediately follows by
specializing the general theory.

In the isomorphic structures
$$
\BK[{\bf S_n}] \simeq Super_n[\underline{\cal L}|\underline{\cal
P}] \simeq \underline{\underline {\mathcal{B}}}_n,
$$
{\it Young symmetrizers} in $\BK[\bf S_n]$ correspond, up to a
sign, to ``doubly multilinear" {\it symmetrized bitableaux}  in
$Super_n[\underline{\cal L}|\underline{\cal P}]$ and to ``doubly
multilinear" {\it Young--Capelli symmetrizers} in $\underline{\underline {\mathcal{B}}}_n$.v


As a consequence, the general natural form of irreducible matrix
representations  specializes to the classical Young
natural form of irreducible matrix representations of symmetric
groups (see, e.g., \cite{Rut}, \cite{JK}, \cite{GML}).




\vskip 0.5cm




Following a cogent suggestion of the referees, in the fourth part
of the paper we briefly discuss how to connect the present theory
with the general representation theory of Lie superalgebras (over
the complex field $\BC$) as developed by Kac, Brundan, Penkov,
Serganova, Van der Jeugt and Zhang, to name but a few.

In Section~19 and 20, we  provide a brief outline of the basic
ideas  of Kac's approach to the representation theory of finite
dimensional  Lie superalgebras  (see, e.g., \cite{KAC1}, \cite{KAC2}, \cite{KAC3}),
and describe in detail the case of general linear Lie superalgebras. The representation
theory of these superalgebras is very close to the representation
theory of the so-called {\it basic classical simple Lie
superalgebras of type I} (see, e.g \cite{KAC1}, \cite{BRU},
\cite{BRU1}).

The main constructions  are, in this context, those of the {\it
integral highest weight modules} and of the {\it Kac modules
relative to dominant integral highest weights} (see 
Subsections~20.4 and 20.5). A deep result of Kac (see, e.g., \cite{KAC2},
\cite{KAC3}, \cite{FSS}) states that highest weight modules and
Kac modules coincide if and only if the highest weight $\Lambda$
of the representation is a {\it typical} one (Subsection~20.5,
Proposition~\ref{proposition: typical modules}).


It follows from Theorem~\ref{theorem: decomposition of Super} and
from the results in Section~17 that the irreducible modules that
appear in the theory of letterplace superalgebra representations are
{\it covariant modules}, in the sense of \cite{CW}, \cite{VANDER}.

Covariant modules are finite dimensional highest weight
representations but they are not, in general,  Kac modules
(since their highest weights can be {\it atypical} ones,
see  Remark~\ref{remark: Highest weight in covariant}).

In Subsection~20.6, we provide a detailed combinatorial analysis
of covariant modules as highest weight representations, as well as
a direct description of their highest weights and highest weight
vectors; these results follow at once from the fact that
covariant modules are Schur--Weyl modules (Section~11).

In Subsection~20.7, we show that the theory of letterplace
representations yields --- up to the action of the so-called umbral
operator (see, e.g., \cite{GRS}, \cite{BHT}, \cite{Gr1} ---  the
decomposition theory of the super-symmetric algebra $\textbf S
(\textbf S^2 (V))$  and of the super-antisymmetric algebra
$\bigwedge (\textbf S^2 (V))$, recently rediscovered by Cheng and
Wang (\cite{CW}, \cite{CW1}) and Sergeev (\cite{SERG2},
\cite{SERG3}).

\vskip 0.2cm


We provide a rather extensive bibliography.
Some items  are not mentioned in the text; they are books and papers of general or historical
interest (\cite{BabR}, \cite{BOE}, \cite{DiCarr}, \cite{GY}, \cite{Green}, \cite{Gr2},
\cite{MAC}, \cite{MARCUS}, \cite{Mead}, \cite{Pr2}, \cite{Scheu1}, \cite{TURN}, \cite{Wall},
\cite{Weitz1}, \cite{Weitz2}, \cite{YOUNG}) and a couple of papers that deal with some aspects of
the theory not treated in this work (\cite{Brini4}, \cite{BT}).


\newpage

\part{The General Setting}

Throughout the paper, $\BK$ will denote a field of characteristic
zero, even if a substantial part of the theory below still holds,
modulo suitable normalizations, over fields of arbitrary
characteristic.


\section{Superalgebras}


A {\it superalgebra} $A$ is simply a $\BZ_2$-graded algebra, in
symbols
$$
A = A_0 \oplus A_1,
$$
such that
$$
A_iA_j \subseteq A_{i+j}, \quad i,j \in \BZ_2.
$$
Given a $\BZ_2$-homogeneous element $a \in A$, its $\BZ_2$-degree
is denoted by $|a|.$


\subsection{The supersymmetric superalgebra of a $\BZ_2$-graded vector space}

Given a $\BZ_2$-graded vector space $U = U_0 \oplus U_1,$ its {\it
supersymmetric superalgebra} $Super[U]$ is the superalgebra
$$
Super[U] = Sym[U_0] \otimes \Lambda[U_1],
$$
where
\begin{align*}
Super[U] &= Super[U]_0 \oplus Super[U]_1,\\
Super[U]_0 &= Sym[U_0] \otimes \bigg( \bigoplus_{h \in \mathbb N}
\Lambda ^{2h}[U_1]\bigg),\\
Super[U]_1 &= Sym[U_0] \otimes \bigg(\bigoplus_{h \in \mathbb N}
\Lambda^{2h+1} [U_1]\bigg).
\end{align*}





The supersymmetric algebra $Super[U]$ has a natural structure of a
$\mathbb{Z}_2$-graded bialgebra (see Subsection~3.5).


\subsection{Lie superalgebras}


A {\it Lie superalgebra} is a superalgebra $L = L_0 \oplus L_1$
whose product (Lie superbracket) satisfies the following
identities:
\begin{itemize}
\item$[x,y] = -(-1)^{|x||y|}[y,x]$ \item$(-1)^{|x||z|} [x, [y,z]]
+ (-1)^{|z||y|}   [z, [x,y]] + (-1)^{|y||x|} [y, [z,x]] = 0$
($\BZ_2$-graded Jacobi identity.)
\end{itemize}


\subsection{Basic example: the general linear Lie superalgebra of a $\BZ_2$-graded vector space}


Given a $\BZ_2$-graded vector space $U = U_0 \oplus U_1,$ its {\it
general linear Lie superalgebra} $pl(U)$ is the $\BZ_2$-graded
vector space
\begin{gather*}
End_\mathbb{K}[U] = End_\mathbb{K}[U]_0 \oplus
End_\mathbb{K}[U]_1,\\
End_\mathbb{K}[U]_i = \{\varphi \in End_\mathbb{K}[U]; \varphi
[U_j] \subseteq U_{i+j}, \ j \in \BZ_2 \}, \quad i \in \BZ_2,
\end{gather*}
endowed with the {\it supercommutator}
$$
[\varphi, \psi] = \varphi \psi - (-1)^{|\varphi| |\psi|} \psi
\varphi.
$$
MATRIX VERSION: let $\mathcal{L} = \{x_1, \ldots, x_n \}$ be a
$\BZ_2$-homogeneous basis of $U = U_0 \oplus U_1$. A standard
basis of $pl(U)$ is given by the set of linear endomorphisms
(elementary matrices)
$$
E_{x_i,x_j}, \quad E_{x_i,x_j}(x_k) = \delta_{j,k}x_i, \quad
i,j,k=1, \ldots, n.
$$
The $E_{x_i,x_j}$'s are $\BZ_2$-homogeneous elements of
$End_\mathbb{K}[U]$ of degree $|x_i|+|x_j|$ and satisfy the
identities
$$
[E_{x_i,x_j},E_{x_h,x_k}] = \delta_{j,h}E_{x_i,x_k} -
(-1)^{(|x_i|+|x_j|)(|x_h|+|x_k|)} \delta_{i,k}E_{x_h,x_j}.
$$

\subsection{The supersymmetric superalgebra $Super[\mathcal{A}]$ of a signed set $\mathcal A$}

A {\it signed set} (or, equivalently, a $\BZ_2$-{\it graded set})
is a set $\mathcal A$ endowed with a {\it sign} map $|\ |:
{\mathcal A} \rightarrow {\mathbb Z}_2;$ the sets ${\mathcal A}_0
= \{a \in {\mathcal A}; |a|=0 \}$ and ${\mathcal A}_1 = \{a \in
{\mathcal A}; |a|=1 \}$ are called the subsets of {\it positive}
and {\it negative} symbols, respectively.





The {\it supersymmetric  ${\mathbb K}$-superalgebra}
$Super[\mathcal{A}]$ is the quotient algebra of the free
associative ${\mathbb K}$-algebra with $1$ generated by the signed
set $\mathcal{A}$ modulo the bilateral ideal generated by the
elements of the form:
$$
xy - (-1)^{|x||y|}yx, \qquad x, y,  \in \mathcal{A}.
$$

\begin{remark}\label{remark: grading on algebras}
\begin{enumerate}

\item $Super[\mathcal{A}]$ is a ${\mathbb Z}_2$-graded algebra
$$
Super[\mathcal{A}] = (Super[\mathcal{A}])_0 \oplus
(Super[\mathcal{A}])_1,
$$
where $(Super[\mathcal{A}])_i$ is the subspace of
$Super[\mathcal{A}]$ spanned by the monomials $m$ of ${\mathbb
Z}_2$-degree $|m|=i,$ where, for $m = a_{i_1} \cdots a_{i_n},$
$|m| = |a_{i_1}| + \cdots + |a_{i_n}|.$

With respect to this grading, $Super[\mathcal{A}]$ is
supersymmetric, i.e.:
$$
m m' = (-1)^{|m||m'|} m' m,
$$
for every  ${\mathbb Z}_2$-homogeneous elements $m, m' \in
Super[\mathcal{A}].$

\item $Super[\mathcal{A}]$ is an ${\mathbb N}$-graded algebra
\begin{align*}
Super[\mathcal{A}] &= \bigoplus_{n\in {\mathbb N}}
Super_n[\mathcal{A}]\\
Super_n [\mathcal{A}] &=
\langle a_{i_1} a_{i_2} \cdots a_{i_n};\ a_{i_h}\in \mathcal{A}
\rangle_{\mathbb K};
\end{align*}

\item the ${\mathbb Z}_2$-graduation and the ${\mathbb
N}$-graduation of $Super[\mathcal{A}]$ are {\it coherent}, that is
$$
(Super[\mathcal{A}])_i = \bigoplus_{n\in {\mathbb N}}
(Super_n[\mathcal{A}])_i, \qquad i\in {\mathbb Z}_2.
$$
\end{enumerate}
\end{remark}


Let $\mathcal{A} = \mathcal{A}_0 \cup  \mathcal{A}_1$ be a {\it
finite} signed set, and let $U = U_0 \oplus U_1$, where $ U_0 =
\langle \mathcal{A}_0 \rangle_\BK$, $ U_1 = \langle \mathcal{A}_1
\rangle_\BK$. The superalgebra $Super[\mathcal{A}]$ is isomorphic
to the supersymmetric algebra $Super[U]$ of the $\BZ_2$-graded
vector space $U = U_0 \oplus U_1.$



\subsection{ $\BZ_2$-graded bialgebras. Basic definitions and the Sweedler notation}

Let us consider an algebraic structure $(X, \pi, \Delta, \eta,
\varepsilon)$ where:
\begin{itemize}
\item $X = X_0 \oplus X_1$ is a $\BZ_2$- graded $\BK$-vector
space. \item $\pi$ is an {\it associative product} on $X$, that is
an even linear map
$$
\pi : X \otimes X \rightarrow X
$$
such that
$$
\pi (I \otimes \pi) = \pi ( \pi \otimes I).
$$
\item $\Delta$ is a {\it coassociative coproduct} on $X$, that is
an even linear map
$$
\Delta : X \rightarrow X \otimes X
$$
such that
$$
(\Delta \otimes I) \Delta = (I \otimes \Delta) \Delta.
$$
The {\it Sweedler notation} \cite{SWEE} is a  way to write the
coproduct of an element $x \in X$, namely
$$
\Delta(x) = \sum_{(x)} \ x_{(1)} \otimes x_{(2)}.
$$
For example, in the Sweedler notation the  fact that $\Delta$ is a
coassociative coproduct reads as follows:
$$
\sum_{(x)} \ x_{(1)} \otimes \left( \sum_{(x_{(2)})} \
x_{(2)_{(1)}} \otimes x_{(2)_{(2)}} \right) = \sum_{(x)} \ \left(
\sum_{(x_{(1)})} \ x_{(1)_{(1)}} \otimes x_{(1)_{(2)}} \right)
\otimes x_{(2)}.
$$

\item $\eta$ is a {\it unit map}, that is an even linear map
$$
\eta : \BK \rightarrow X
$$
such that
$$
\eta(k)x = kx, \quad k \in \BK, \quad x \in X.
$$
\item $\varepsilon$ is a {\it counit map}, that is an even linear
map
$$
\varepsilon : X \rightarrow \BK
$$
such that
$$
\sum_{(x)} \ \varepsilon (x_{(1)})x_{(2)} =  \sum_{(x)} \
x_{(1)}\varepsilon (x_{(2)}) = x, \quad x \in X.
$$
\item $(\varepsilon \circ \eta)(1) = 1, \quad 1 \in \BK.$
\end{itemize}

An algebraic structure $(X, \pi, \Delta, \eta, \varepsilon$ is
said to be a  {\it $\BZ_2$-graded  $\BK$-bialgebra} whenever the
following conditions hold:
\begin{itemize}
\item The coproduct  $\Delta$ is an algebra morphism. Notice that
$X \otimes X$ is meant as the {\it tensor product of
$\BZ_2$-graded algebras}, that is $(x \otimes x')(y \otimes y') =
(-1)^{|x'||y|} xy \otimes x'y',$  $x, y, x', y'$
$\BZ_2$-homogeneous elements in $X$. In the Sweedler notation, the
above condition reads as follows:
$$
\Delta(xy) = \sum_{(xy)} \ (xy)_{(1)} \otimes (xy)_{(2)} =
\sum_{(x),(y)} \ (-1)^{|x_{(2)}||y_{(1)}|} \ x_{(1)}y_{(1)}
\otimes x_{(2)}y_{(2)} = \Delta(x)\Delta(y),
$$
$x,y \in X.$ \item The unit map $\eta$ is a coalgebra morphism,
that is
$$
 \Delta(\eta(1)) =  \eta(1) \otimes \eta(1), \quad 1 \in \BK.
$$
\item The counit map $\varepsilon$ is an algebra morphism, that is
$$
\varepsilon(xy) = \varepsilon(x) \varepsilon(y), \quad x, y \in X.
$$
\end{itemize}



\subsection{ The  superalgebra $Super[\mathcal{A}]$ as a $\BZ_2$-graded bialgebra.
Left and right derivations, coderivations and polarization
operators}

Let $\mathcal{A} = \mathcal{A}_0 \cup  \mathcal{A}_1$ be a {\it
finite} signed set, $\mathcal{A}_0 = \{a_1, a_2, \ldots, a_m \}$,
$\mathcal{A}_1 = \{a_{m+1}, a_{m+2}, \ldots,\break a_{m+n} \}$, and let
$U = U_0 \oplus U_1$, where $ U_0 = \langle \mathcal{A}_0
\rangle_\BK$, $ U_1 = \langle \mathcal{A}_1 \rangle_\BK$.



The superalgebra $Super[\mathcal{A}] \cong Super[U]$ is a
$\BZ_2$-graded bialgebra, where the structure maps are defined in
the following way:
\begin{itemize}
\item $\Delta(1) = 1 \otimes 1, \quad \Delta(a_i) = a_i \otimes 1
+ 1 \otimes a_i, \ a_i \in \mathcal{A};$ \item $\eta(1) = 1;$
\item $\varepsilon(1) = 1, \quad \varepsilon(a_i) = 0, \ a_i \in
\mathcal{A}.$
\end{itemize}



A linear map
$$
{\it D}: Super[\mathcal{A}] \rightarrow Super[\mathcal{A}],
$$
homogeneous of degree $d=|{\it D}| \in \BZ_2,$ i.e., such that
${\it D} (Super[\mathcal{A}])_i \subseteq
(Super[\mathcal{A}])_{i+d},$ is a  {\it left superderivation} if
$$
{\it D}(m\ m') = {\it D}(m)\ m' + (-1)^{|{\it D}| |m|} m\ {\it
D}(m'),
$$
for all monomials $m, m' \in Super[\mathcal{A}].$

Let $a_i, a_j \in \mathcal{A}.$ The {\it left  superpolarization
${\it D}_{a_i, a_j}$ of the letter $a_j$ to the letter $a_i$} is
the unique left superderivation of ${\mathbb Z}_2$-degree
$|a_i|+|a_j|,$ such that
$$
{\it D}_{a_i, a_j} (a_h) = \delta_{j, h} \ a_i,
$$
for every $a_h \in \mathcal{A}.$

Here and in the following the Greek letter $\delta$ will denote
the Kronecker symbol.

Any  linear map  ${\it D} : Super[\mathcal{A}] \rightarrow
Super[\mathcal{A}]$ may be extended to an operator
$$
({\it D} \oplus {\it D}) : Super[\mathcal{A}] \otimes
Super[\mathcal{A}] \rightarrow Super[\mathcal{A}] \otimes
Super[\mathcal{A}],
$$
by the rule of ``left superderivation", that is by setting
$$
({\it D} \oplus {\it D}) (m \otimes m') = {\it D}(m) \otimes m' +
(-1)^{|{\it D}||m|} m \otimes {\it D}(m'),
$$
for every $m, m' \in Super[\mathcal{A}].$ If no confusion arises,
we will frequently write ${\it D}$ in place of ${\it D} \oplus
{\it D}$.

We notice that if $({\it D}$ is a left superderivation of
${\mathbb Z}_2$-degree $|{\it D}|,$ then $({\it D} \oplus {\it
D})$ is a left superderivation of ${\mathbb Z}_2$-degree $|{\it
D}|.$


A linear map
$$
{\it D}: Super[\mathcal{A}] \rightarrow Super[\mathcal{A}],
$$
homogeneous of degree $|{\it D}| \in \BZ_2,$  is said to be a {\it
left coderivation} if the following condition holds:
$$
\Delta ({\it D}(m)) = ({\it D} \oplus {\it D})(\Delta(m)),
$$
for every $m \in Super[\mathcal{A}].$ In the Sweedler notation,
the above condition reads as follows:
$$
\Delta({\it D}(m)) = \sum_{(m)} \left[ {\it D}(m_{(1)}) \otimes
m_{(2)} + (-1)^{|{\it D}||m_{(1)}|} m_{(1)} \otimes {\it
D}(m_{(2)}) \right].
$$

\begin{proposition}
Any left superpolarization ${\it D}_{a_i, a_j}$ is a  left
coderivation of the bialgebra $Super[\mathcal{A}].$
\end{proposition}

The next result is one of the basic tools of the $\it method \ of
\ vitual \ variables$ and exploits a deep connection between the
language of superpolarizations and the language of bialgebras.

\begin{corollary}
Let $ m = a_{i_1} a_{i_2} \cdots a_{i_p} \in Super[\mathcal{A}]$
and let $a_j \in \mathcal{A}$, $|a_j| = 0$, such that $a_j \neq
a_{i_h}, \ h = 1, 2, \ldots, p.$ Then
$$
\Delta(m) = {\frac 1 {p!}} {\it D}_{a_{i_1}, a_j}{\it D}_{a_{i_2},
a_j} \cdots {\it D}_{a_{i_p}, a_j} (\Delta((a_j)^p)).
$$
\end{corollary}

\begin{example}
Let $ a_1, a_2, a_3 \in \mathcal{A}$, $|a_1| = 1, |a_2| = 1, |a_3|
= 0 $ and let $a \in \mathcal{A}, \ |a| = 0,$ such that $a \neq
a_{i}, \ i = 1, 2, 3.$ Let $ m  = a_1 a_2 a_3.$ We have the
following identity:
\begin{multline*}
\Delta(m) = \Delta(a_1)\Delta(a_2)\Delta(a_3) = \\
(a_1 \otimes 1 + 1 \otimes a_1)(a_2 \otimes 1 + 1 \otimes a_2)(a_3 \otimes 1 + 1 \otimes a_3) = \\
a_1a_2a_3 \otimes 1 +
a_1a_2  \otimes a_3 +  a_1a_3  \otimes a_2 -  a_2a_ 3 \otimes a_1 + \\
a_1  \otimes a_2a_3 - a_2   \otimes a_1a_3 +  a_3  \otimes  a_1a_2 + 1 \otimes a_1a_2a_3 = \\
{\frac 1 {3!}}{\it D}_{a_{1}, a}{\it D}_{a_{2}, a}{\it D}_{a_{3},
a}(a^3 \otimes 1 + 3a^2 \otimes a + 3a \otimes a^2 + 1 \otimes
a^3) =
 {\it D}_{a_{1}, a}{\it D}_{a_{2}, a}{\it D}_{a_{3}, a}({\frac 1 {3!}}\Delta(a^3)) .
\end{multline*}
\end{example}

In the following sections, we also need the notion of a right
superderivation: a linear map
$$
Super[\mathcal{A}] \leftarrow Super[\mathcal{A}]:\ {\it {\tilde
D}},
$$
homogeneous of degree $|{\it {\tilde D}}|$, is a {\it right
superderivation} if
$$
(-1)^{|{\it {\tilde D}}||m'|} (m){\it {\tilde D}}\ m' + m\
(m'){\it {\tilde D}}  =\ (m\ m'){\it {\tilde D}},
$$
for all monomials $m, m' \in Super[\mathcal{A}].$

Let $a_h, a_k \in \mathcal{A}.$ The {\it right  superpolarization
$_{a_h, a_k}{\it D}$ of the letter $a_h$ to the letter $a_k$} is
the unique right superderivation of ${\mathbb Z}_2$-degree
$|a_h|+|a_k|,$ such that
$$
(a_j) _{a_h, a_k}{\it D}  = \delta_{j, h} \ a_k,
$$
for every $a_j \in \mathcal{A}.$








Consider linear automorphism
$$
R : Super[\mathcal{A}] \rightarrow Super[\mathcal{A}]
$$
such that
$$
R( a_{i_1}a_{i_2} \cdots a_{i_n} ) = a_{i_n}a_{i_{n-1}} \cdots
a_{i_1},
$$
for every $a_{i_1}a_{i_2} \cdots a_{i_n} \in Super[\mathcal{A}]$.

Clearly, $R$ is an involutorial map, that is $R^2 = id$, and $R(m
m') = R(m')R(m)$, for every $m, m' \in Super[\mathcal{A}]$. It
follows that the map
$$
\ {\it D} \mapsto R \circ {\it D} \circ R = \ {\it {\tilde D}}
$$
is an involutorial isomorphism from the vector space of all left
superderivations to the vector space of all right
superderivations.



Notice that the  right superpolarization $_{a_h, a_k}{\it D}$ of
the letter $a_h$ to the letter $a_k$ is the right superderivation
$$
_{a_h, a_k}{\it D} = R \circ {\it D}_{a_k, a_h} \circ R.
$$


The next result follows from the definitions.
\begin{proposition}
\begin{enumerate}
\item $ _{a_h, a_k}{\it D}$ is a  {\it right coderivation} of the
bialgebra $Super[\mathcal{A}],$ that is
\begin{multline*}
\Delta ( (m) _{a_h, a_k}{\it D} )\\ = \sum_{(m)} \ \left(
(-1)^{(|a_h|+|a_k|)|m_{(2)}|} \ (m_{(1)})\! _{a_h, a_k}{\it D} \
\otimes m_{(2)} + m_{(1)}  \otimes  (m_{(2)}) _{a_h, a_k}{\it D}
\right),
\end{multline*}
for every $m \in Super[\mathcal{A}].$ \item Let $ m = a_{i_1}
a_{i_2} \cdots a_{i_p} \in Super[\mathcal{A}]$ and let $a_j \in
\mathcal{A}$, $|a_j| = 0$, such that $a_j \neq a_{i_h}, \ h = 1,
2, \ldots, p.$ Then
$$
\Delta(m) = {\frac 1 {p!}} (\Delta((a_j)^p))) _{a_j, a_{i_1}}{\it
D}_{a_j, a_{i_2}}{\it D} \cdots _{a_j, a_{i_p}}{\it D}.
$$
\end{enumerate}
\end{proposition}








\section{The Letterplace Superalgebra as a Bimodule}


\subsection{Letterplace superalgebras}


In the following, we consider a pair of signed sets $\mathcal{X} =
\mathcal{X}_0 \cup \mathcal{X}_1$ and $\mathcal{Y} = \mathcal{Y}_0
\cup \mathcal{Y}_1$, that we call the {\it letter set} and the
{\it place set}, respectively. The {\it letterplace set}
$$
[\mathcal{X}\,|\,\mathcal{Y}] = \{ (x\,|\,y);\ x\in \mathcal{X},\ y\in
\mathcal{Y}\}
$$
inherits a sign  by setting $|(x\,|\,y)|=|x|+|y|\in {\mathbb Z}_2.$



The {\it letterplace ${\mathbb K}$-superalgebra}
$Super[\mathcal{X}|\mathcal{Y}]$ is the quotient algebra of the
free associative ${\mathbb K}$-algebra with $1$ generated by the
letterplace alphabet $[\mathcal{X}\,|\,\mathcal{Y}]$ modulo the
bilateral ideal generated by the elements of the form:
$$
(x\,|\,y)(z\,|\,t)-(-1)^{(|x|+|y|)(|z|+|t|)}(z\,|\,t)(x\,|\,y), \qquad x,z\in
\mathcal{X},\ y,t\in \mathcal{Y}.
$$

In other words, the  letterplace ${\mathbb K}$-superalgebra
$Super[\mathcal{X}|\mathcal{Y}]$ is the supersymmetric
superalgebra of the $Z_2$-graded set $\mathcal{A} =
[\mathcal{X}\,|\,\mathcal{Y}]$ (see Subsection~3.4).



\begin{remark}\label{remark: grading on letterplace}
\begin{enumerate}

\item $Super[\mathcal{X}|\mathcal{Y}]$ is a ${\mathbb Z}_2$-graded
algebra
$$
Super[\mathcal{X}|\mathcal{Y}] =
(Super[\mathcal{X}|\mathcal{Y}])_0 \oplus
(Super[\mathcal{X}|\mathcal{Y}])_1,
$$
where $(Super[\mathcal{X}|\mathcal{Y}])_i$ is the subspace of
$Super[\mathcal{X}|\mathcal{Y}]$ spanned by the letterplace
monomials $M$ of ${\mathbb Z}_2$-degree $|M|=i,$ where, for $M =
(x_{i_1}\,|\,y_{i_1}) \cdots (x_{i_n}\,|\,y_{i_n}),$ $|M| =
|(x_{i_1}\,|\,y_{i_1})| + \cdots + |(x_{i_n}\,|\,y_{i_n})|.$

With respect to this grading, $Super[\mathcal{X}|\mathcal{Y}]$ is
supersymmetric, i.e.:
$$
M N = (-1)^{|M||N|} N M,
$$
for every  ${\mathbb Z}_2$-homogeneous elements $M, N \in
Super[\mathcal{X}|\mathcal{Y}].$

\item $Super[\mathcal{X}|\mathcal{Y}]$ is an ${\mathbb N}$-graded
algebra
\begin{gather*}
Super[\mathcal{X}|\mathcal{Y}] = \bigoplus_{n\in {\mathbb N}}
Super_n[\mathcal{X}|\mathcal{Y}]\\
Super_n [\mathcal{X}|\mathcal{Y}] = \langle (x_{i_1}\,|\,y_{i_1})
(x_{i_2}\,|\,y_{i_2}) \cdots (x_{i_n}\,|\,y_{i_n}),\ x_{i_h}\in
\mathcal{X},\ y_{j_k}\in \mathcal{Y} \rangle_{\mathbb K};
\end{gather*}

\item the ${\mathbb Z}_2$-graduation and the ${\mathbb
N}$-graduation of $Super[\mathcal{X}|\mathcal{Y}]$ are {\it
coherent}, that is
$$
(Super[\mathcal{X}|\mathcal{Y}])_i = \bigoplus_{n\in {\mathbb N}}
(Super_n[\mathcal{X}|\mathcal{Y}])_i, \qquad i\in {\mathbb Z}_2.
$$
\end{enumerate}
\end{remark}


\subsection{Superpolarization operators}

Let $x', x\in \mathcal{X}.$ The {\it superpolarization ${\mathcal
D}_{x', x}$ of the letter $x$ to the letter $x'$} is the unique
left superderivation
$$
{\mathcal D}_{x', x} : Super[\mathcal{X}|\mathcal{Y}] \rightarrow
Super[\mathcal{X}|\mathcal{Y}]
$$
of ${\mathbb Z}_2$-degree $|x'|+|x|,$ such that
$$
{\mathcal D}_{x', x} (z\,|\,t) = \delta_{x, z} (x'\,|\,t),
$$
for every $(z\,|\,t)\in [\mathcal{X}\,|\,\mathcal{Y}].$



Let $y, y'\in  \mathcal{Y}.$ The {\it superpolarization $_{y,
y'}\! {\mathcal D}$ of the place $y$ to the place $y'$} is the
unique right superderivation
$$
Super[\mathcal{X}|\mathcal{Y}] \leftarrow
Super[\mathcal{X}|\mathcal{Y}]:\ _{y, y'}\! {\mathcal D}
$$
of ${\mathbb Z}_2$-degree $|y|+|y'|,$ such that
$$
(z\,|\,t)\ _{y, y'}\! {\mathcal D} = \delta_{t, y} (z\,|\,y'),
$$
for every $(z\,|\,t)\in [\mathcal{X}\,|\,\mathcal{Y}].$

In passing, we point out that every letter-polarization operator
{\it commutes} with every place-polarization operator.



\subsection{Letterplace superalgebras and supersymmetric algebras: the classical description}


Given a pair of {\it finite} alphabets $\mathcal{L}=\mathcal{L}_0
\cup \mathcal{L}_1$ and $\mathcal{P}=\mathcal{P}_0 \cup
\mathcal{P}_1,$ $\mathcal{L}_0 \subseteq \mathcal{X}_0,$
$\mathcal{L}_1 \subseteq \mathcal{X}_1,$ $\mathcal{P}_0 \subseteq
\mathcal{Y}_0,$ $\mathcal{P}_1 \subseteq \mathcal{Y}_1,$ consider
the $\BZ_2$-graded vector spaces
$$
V = V_0  \oplus V_1 = \langle \mathcal{L}_0 \rangle_\mathbb{K}
\oplus \langle \mathcal{L}_1 \rangle_\mathbb{K}
$$
and
$$
W = W_0  \oplus W_1 = \langle \mathcal{P}_0 \rangle_\mathbb{K}
\oplus \langle \mathcal{P}_1 \rangle_\mathbb{K}.
$$
The tensor product $V \otimes W$ has a natural $\BZ_2$-grading
$$
V \otimes W = [(V_0 \otimes W_0) \oplus (V_1 \otimes W_1)] \oplus
[(V_0 \otimes W_1) \oplus (V_1 \otimes W_0)] ).
$$
The {\it supersymmetric algebra} of the tensor product $V \otimes
W$ is the superalgebra
$$
Super[V \otimes W]= Sym[(V_0 \otimes W_0) \oplus (V_1 \otimes
W_1)] \otimes \Lambda [(V_0 \otimes W_1) \oplus (V_1 \otimes
W_0)].
$$
Clearly, one has a natural isomorphism
$$
Super[\mathcal{L}|\mathcal{P}] \cong Super[V \otimes W].
$$



\subsection{General linear Lie superalgebras, representations and polarization operators}


Given a pair of {\it finite} alphabets $\mathcal{L} =
\mathcal{L}_0 \cup \mathcal{L}_1 $ and $\mathcal{P} =
\mathcal{P}_0 \cup \mathcal{P}_1,$ regard them as homogeneous
bases of the pair of vector spaces $V = V_0 \oplus V_1$ and $W =
W_0 \oplus W_1.$

The  Lie superalgebras $pl(\mathcal{L})$ and $pl(\mathcal{P})$
are, by definition, the general linear Lie superalgebras $pl(V)$
and $pl(W)$ of $V$ and $W$ respectively. Therefore, we have the
standard bases
$$
\{E_{x,x'}; x,x' \in \mathcal{L} \}
$$
and
$$
\{E_{y,y'}; y,y' \in \mathcal{P} \}
$$
of $pl(\mathcal{L})=pl(V)$ and $pl(\mathcal{P})=pl(W)$.

We recall the canonical isomorphism 
$$
Super[\mathcal{L}|\mathcal{P}] \cong Super[V \otimes W].
$$

The (even) mappings
$$
E_{x' x} \mapsto {\mathcal D}_{x',x}, \quad x, x'\in \mathcal{L},
\qquad E_{y' y} \mapsto\ _{y, y'}\! {\mathcal D},\quad y, y'\in
\mathcal{P}
$$
induce Lie superalgebra actions of $pl(\mathcal{L})$ and
$pl(\mathcal{P})$ over any ${\mathbb N}$-{\it homogeneous
component} $Super_n[\mathcal{L}|\mathcal{P}]$ of the letterplace
algebra.

In the following, we will denote by
$$
{\mathcal B}_n,\ _n{\mathcal B}
$$
the (finite dimensional) homomorphic images in $End_{\mathbb
K}(Super_n[\mathcal{L}|\mathcal{P}])$ of the universal enveloping
algebras ${\mathcal U}(pl(\mathcal{L}))$ and ${\mathcal
U}(pl(\mathcal{P})),$ induced by the actions of $pl(\mathcal{L})$
and $pl(\mathcal{P}),$ respectively.

The operator algebras ${\mathcal B}_n,\ _n{\mathcal B}$ are
therefore the algebras generated by the proper letter and place
polarization operators (restricted to
$Super_n[\mathcal{L}|\mathcal{P}]),$ respectively.

Furthermore, by the commutation property,
$Super_n[\mathcal{L}|\mathcal{P}]$ is a bimodule over the
universal enveloping algebras ${\mathcal U}(pl(\mathcal{L})))$ and
${\mathcal U}(pl(\mathcal{P})).$


%\subsection{The actions of the general linear Lie superalgebras pl(V) and pl(W)}


%The general linear Lie superalgebras $pl(V)$ and $pl(W)$ act on
%$Super[V \otimes W$ (and, therefore, on the letterplace superalgebra $Super[L|P]$) by
%{\it left superderivation} and {\it right superderivation},
%respectively. Furthermore,
%$$
%pl(V) \cdot Super[V \otimes W] \cdot pl(W)
%$$
%is a {\it bimodule}, that is,  the actions of $pl(V)$ and $pl(W)$
%{\it COMMUTE}.




\subsection{General linear groups and even polarization operators}


Let $\mathcal{L} = \mathcal{L}_1 = \{x_1, \ldots, x_m \}$ be a
finite alphabet of  letters,  let $\mathcal{P}$ be a finite
alphabet of places; since $\mathcal{L}$ is trivially
$\BZ_2$-graded, the general linear Lie superalgebra
$pl(\mathcal{L})$ reduces to the usual general linear Lie algebra
$gl_m(\BK)$ of all square matrices of order  $m$ over $\BK$, and
the letterplace superalgebra $Super[\mathcal{L}|\mathcal{P}]$ is a
(left) $gl_m(\BK)$-module via the usual action, that is,  a matrix
$S=[s_{ij}]$ acts on a letterplace variable as
$$
S (x_i\,|\,y) = \sum_{j=1}^m (x_j\,|\,y)\ s_{ji}
$$
and is extended as a {\it derivation} (the action of any
elementary matrix $E_{hk}$ is implemented by the polarization
operator ${\mathcal D}_{x_h, x_k}$, $|{\mathcal D}_{x_h, x_k}| = 0
\in \BZ_2$). We denote by $\sigma_n$ the corresponding
representation of the universal enveloping algebra ${\mathcal
U}[gl_m(\BK)]$ over the homogeneous component
$Super_n[\mathcal{L}|\mathcal{P}].$



Let $Gl_m(\BK)$ be the general linear group of nonsingular
matrices of order  $m$ over $\BK;$ the letterplace superalgebra
$Super[\mathcal{L}|\mathcal{P}]$ is a (left) $GL_m(\BK)$-module
via the usual action, that is,  a matrix $S=[s_{ij}]$ acts on a
letterplace variable as
$$
S (x_i\,|\,y) = \sum_{j=1}^m (x_j\,|\,y)\ s_{ji}
$$
and is extended as an {\it algebra automorphism}. We denote by
$\rho_n$ the corresponding representation of the group algebra
$\BK[GL_m(\BK)]$ over the homogeneous component
$Super_n[\mathcal{L}|\mathcal{P}].$

The following standard result will be systematically used in
Sections~15 and 16.


\begin{proposition}\label{proposition: action of general linear group and general linear Lie algebra}
The algebra $\rho_n\ [ \BK[GL_m(\BK)] ]$ generated by the action
of the general linear group coincides with the algebra $\sigma_n\
[ {\mathcal U}[gl_m(\BK)] ]$ generated by the action of the
general linear Lie algebra.

Therefore, the algebra $\rho_n\ [ \BK[GL_m(\BK)] ]$ is the algebra
${\mathcal B}_n$ generated by the letter polarization operators
${\mathcal D}_{x_ix_j}, \ x_i, x_j \in \mathcal{L}$.

\end{proposition}

\begin{proof}
Recall that the group $GL_m(\BK)$ is generated by the {\it
transvections}, namely:
\begin{itemize}
\item $ T_{ij}(\lambda) = I + \lambda\ E_{ij}, $ with $i \neq j;$

\item $ T_{ii}(\lambda) = I + \lambda\ E_{ii}, $ with $\lambda
\neq -1.$
\end{itemize}

The statement now follows  from a standard argument (the so-called
Vandermonde matrix argument):

\begin{itemize}
\item The image under the representation  $\rho_n$ of any
transvection $ T_{ij}(\lambda) = I + \lambda\ E_{ij}, $ with $i
\neq j,$ is a polynomial in the image under the representation
$\sigma_n$ of the elementary matrix $E_{ij}.$ Specifically, we
have:
$$
\rho_n (T_{ij}(\lambda)) = \sum_{h=0}^n  {\lambda^h} \frac
{\sigma_n(E_{ij})^h} {h!};
$$
by evaluating this relation in $n+1$ different values $\lambda=
\lambda_1, \ldots, \lambda_{n+1},$ one gets a system of $n+1$
linear relations that can be solved with respect to the divided
powers of the representation of the elementary matrix $E_{ij}.$

\item The same argument applies to the representation of any
transvection $ T_{ii}(\lambda). $ Specifically, we have:
$$
\rho_n (T_{ii}(\lambda)) = \sum_{h=0}^n  {\lambda^h} \left(
\begin{array}{c}
\sigma_n(E_{ii})
\\
h
\end{array}
\right);
$$
by evaluating this relation in $n+1$ different values $\lambda=
\lambda_1, \ldots, \lambda_{n+1},$ one gets a system of $n+1$
linear relations that can be solved with respect to the formal
binomials of the representation of the elementary matrix $E_{ii}.$

\end{itemize}


\end{proof}

\section{Tableaux}


\subsection{Young tableaux}

We recall that {\it signed set} is a set $\mathcal A$ endowed with
a {\it sign} map $|\ |: {\mathcal A} \rightarrow {\mathbb Z}_2;$
the sets ${\mathcal A}_0 = \{a \in {\mathcal A}; |a|=0 \}$ and
${\mathcal A}_1 = \{a \in {\mathcal A}; |a|=1 \}$ are called the
subsets of {\it positive} and {\it negative} symbols,
respectively.

A {\it signed alphabet} is a linearly ordered signed set.

\vskip 0.2cm A {\it Young tableau} over a signed alphabet
${\mathcal A}$ is a sequence
$$
S=(w_1, w_2, \ldots, w_p)
$$
of words $w_i= a_{i1} a_{i2} \ldots a_{i\lambda_i}, \ a_{ij}\in
{\mathcal A},$ whose lengths form a weakly decreasing sequence,
i.e., a partition
$$
\lambda =(\lambda_1\geq \lambda_2\geq \ldots\geq \lambda_p) =
\sh(S),
$$
called the {\it shape} of $S.$ The concatenation of the words
$w_i$
$$
w =w_1 w_2 \ldots w_p = w(S)
$$
is called the {\it word} of $S.$ If $n$ is the length of $w,$ then
$\lambda$ is a partition of $n$:
$$
\lambda \vdash n.
$$
The {\it content} $c(S)$ of a tableau $S$ is the multiset of the
symbols occurring in $S.$

We will frequently represent tableaux in the array notation:
$$S=(abb,\ bae,\ c)=\begin{array}{ccc} a & b & b
\\ b & a & e
\\ c
\end{array}
$$ 
The set of all the tableaux over ${\mathcal A}$ is denoted by
$Tab({\mathcal A}).$


\subsection{Co-Deruyts and Deruyts tableaux}

A tableau $C$ is said to be of {\it co-Deruyts} type whenever any
two symbols in the same row of $C$ are equal, while any two
symbols in the same column of $C$ are distinct. For example:
$$C=\begin{array}{ccccc} a & a & a & a & a
\\ b & b & b
\\ c
\end{array}$$
A tableau $D$ is said to be of {\it Deruyts} type whenever any two
symbols in the same column of $D$ are equal, while any two symbols
in the same row of $D$ are distinct. For example:
$$D=\begin{array}{ccccc} a & b & c & d & e
\\ a & b & c
\\ a
\end{array}$$

In the following, the symbol $C$ will denote a co-Deruyts tableau
filled with {\it  positive} symbols, and the symbol $D$ will
denote a Deruyts tableau filled with {\it  negative} symbols. In
the formulas below, the shapes of the tableaux $C$ and $D,$ and
the fact that the  symbols were letter or place symbols should be
easily inferred from the context.


\subsection{Standard Young tableaux}


Following Grosshans, Rota and Stein \cite{GRS}, a Young tableau
$S$ over a (linearly ordered)  signed alphabet ${\mathcal A}$ is
called {\it (super)standard} when each row of $S$ is
non-decreasing, with no negative repeated symbols and each column
of $S$ is non-decreasing, with no positive repeated symbols. For
example, if $a<b<c<d$, the tableau
$$
S=\begin{array}{ccccc} a & a & c & c & d
\\ b & c & d
\\ b
\end{array}
\qquad a, c\in {\mathcal A}_0,\ b, d\in {\mathcal A}_1
$$
is a standard tableau. The set of all the standard tableaux over
${\mathcal A}$ is denoted by $Stab({\mathcal A}).$


\subsection{The Berele--Regev hook property}

Assume now that the linearly ordered signed alphabet ${\mathcal
A}$ is finite, with $|{\mathcal A}_0|=r$ and $|{\mathcal A}_1|=s.$
The hook set of ${\mathcal A}$ is
$$
H({\mathcal A}) = \{(\lambda_1\geq\lambda_2\geq\ldots);\quad
\lambda_{r+1}<s+1\}.
$$
\begin{proposition}\label{proposition: Berele--Regev hook property}
There are some standard tableaux of shape $\lambda$ over $A$ if
and only if $\lambda\in H({\mathcal A}).$ Furthermore, the number
$p_{\lambda}({\mathcal A})$ of standard tableaux of any given
shape $\lambda$ over ${\mathcal A}$ is independent of the linear
order defined on ${\mathcal A}.$
\end{proposition}
\begin{proof}
Let ${\mathcal A}_0 = \{x_1, x_2, \ldots, x_m \}$, ${\mathcal A}_1
= \{x_{m+1}, x_{m+2}, \ldots, x_{m+n} \}$. If ${\mathcal A}$ is
endowed with a linear order such that $x_i < x_j$ for every $i =
1, 2, \ldots, m$ and $j = m+1, m+2, \ldots, m+n$, the proof
follows from a straightforward argument (see, e.g., \cite{Berele1},
\cite{Berele2}). If ${\mathcal A}$ is endowed with an arbitrary
linear order, the proof follows from the previous assertion in
combination with the standard basis theorem of Grosshans, Rota and
Stein (see, e.g \cite{GRS} and Proposition~\ref{proposition: Schur
modules}).
\end{proof}


\subsection{Orders on tableaux}


Let $\mathcal{L}$ be a finite signed alphabet. We define a {\it
partial} order on the set of all standard tableaux over
$\mathcal{L}$ which have a given content, and, therefore, have
shapes which are partitions of a given integer $n.$

For every standard tableau $S,$ we consider the sequence $ S^{(p)}$,
$p=1,2, \ldots $, of the subtableaux obtained from $S$ by
considering only the first $p$ symbols of the alphabet, and
consider the family $ \sh(S^{(p)})$, $p=1,2,\ldots $, of the
corresponding shapes. Since the alphabet is assumed to be finite,
this sequence is finite and its last term is $\sh(S).$

Then, for standard tableaux $S,T,$ we set
$$
S \leq T \ \Leftrightarrow \ \sh(S^{(p)}) \unlhd \sh(T^{(p)}), \ p
=1,2,\ldots,
$$
where $\unlhd$ stands for the {\it dominance order} on partitions.
We recall that the dominance order on partitions is defined as
follows: $ \lambda = ( \lambda_1 \geq \lambda_2 \geq \ldots)
\unlhd \mu = ( \mu_1 \geq \mu_2 \geq \ldots) $ if and only if $
\lambda_1+ \cdots +\lambda_i \leq \mu_1+ \cdots +\mu_i, $ for
every $i=1,2, \ldots$

We extend this partial order to the set of all  standard tableaux
on $\mathcal{L}$ which contain a total number (taking into account
multiplicities) of $n$ symbols simply by stating that two tableaux
$S$ and $T$ such that $c(S) \neq c(T)$ are incomparable.

We define a {\it linear} order on the set of all tableaux over
$\mathcal{L}$ which contain a total number (taking into account
multiplicities) of $n$ symbols, by setting $ Q < Q' $ if and only if
$$
\begin{array}{lll}
  & \sh(Q) <_l \sh(Q')&                    \\
or& \sh(Q) = \sh(Q')  & and\ w(Q) >_l w(Q'),
\end{array}
$$
where the shapes and the words are compared in the lexicographic
order.

We remark that this linear order, restricted to standard tableaux,
is a linear extension of the partial order defined above.



\newpage

\part{The General Theory}


\section{The Method of Virtual Variables}

\subsection{The metatheoretic significance of Capelli's idea of
{\it virtual} variables}

Let $\mathcal{L} = \mathcal{L}_0 \cup \mathcal{L}_1 \subset
\mathcal{X}$ and $\mathcal{P} = \mathcal{P}_0 \cup \mathcal{P}_1
\subset \mathcal{Y}$ be {\it finite} signed subsets of the
``universal" signed letter and place alphabets $\mathcal{X}$ and
$\mathcal{Y},$ respectively. The elements $x\in \mathcal{L}$
($y\in \mathcal{P}$) are called {\it proper} letters ({\it proper}
places), and the elements $x\in \mathcal{X} \setminus \mathcal{L}$
($y\in \mathcal{Y} \setminus \mathcal{P}$) are called {\it
virtual} letters (places). Usually we denote virtual symbols by
Greek letters.

The signed subset $[\mathcal{L}\,|\,\mathcal{P}] = \{(x\,|\,y);\ x\in
\mathcal{L}, \ y\in \mathcal{P}\} \subset
[\mathcal{X}\,|\,\mathcal{Y}]$ is called a {\it proper letterplace
alphabet}.


Consider an operator of the form:
\begin{gather*}
{\mathcal D}_{x_{i_1},\alpha_{i_1}} \cdots {\mathcal
D}_{x_{i_n},\alpha_{i_n}} \cdot {\mathcal
D}_{\alpha_{i_1},x_{j_1}} \cdots {\mathcal
D}_{\alpha_{i_n},x_{j_n}}\\
(x_{i_1}, \ldots, x_{i_n}, x_{j_1}, \ldots, x_{j_n} \in \mathcal{L}, 
\text{ i.e., proper letters})
\end{gather*}
that is an operator that {\it creates} some virtual letters
$\alpha_{i_1}, \ldots, \alpha_{i_n}$ (with  prescribed
multiplicities) {\it times} an operator that {\it annihilates} the
{\it same} virtual letters (with the {\it same} prescribed
multiplicities).

Such an operator will be called a (letter) {\it Capelli-type
operator}.

Clearly, the proper letterplace superalgebra
$Super[\mathcal{L}|\mathcal{P}]$ is {\it left invariant} under the
action of a Capelli-type operator.

\begin{theorem}[\cite{Brini2, BRT3}] \label{theorem: action of Capelli-type operator}
The action of a Capelli-type operator over the proper letterplace
superalgebra $Super[\mathcal{L}|\mathcal{P}]$ is the  same as the
action of a ``polynomial" operator in the proper polarizations
${\mathcal D}_{x_{i_h},x_{i_k}}$, $x_{i_h},x_{i_k} \in
\mathcal{L}$, that is an operator that does not involve virtual
variables.
\end{theorem}

Informally speaking, a Capelli-type operator is of {\it
$pl(\mathcal{L})$-representation theoretic meaning}, and, in
general is much more manageable than its ``non-virtual companion".

In the following, we will write $\textbf{T}_1 \cong \textbf{T}_2$
to mean that two operators $\textbf{T}_1 , \textbf{T}_2$ on
$Super[{\mathcal{X}}|\mathcal{Y}]$ are the same when restricted to
the proper letterplace algebra $Super[{\mathcal{L}}|\mathcal{P}]$
and say that the operators $\textbf{T}_1 , \textbf{T}_2$ are
$[{\mathcal{L}}\,|\,\mathcal{P}]-equivalent.$

\begin{example}
Let $x, y\in \mathcal L_1,$ with $x\neq y$, and $\alpha\in
{\mathcal X \setminus \mathcal{L}},$ with $|\alpha|=0.$ Then
\begin{align*}
{\mathcal D}_{y \alpha} {\mathcal D}_{x \alpha} {\mathcal
D}_{\alpha x} {\mathcal D}_{\alpha y}
 &= -{\mathcal D}_{y \alpha} {\mathcal D}_{\alpha x} {\mathcal D}_{x \alpha} {\mathcal D}_{\alpha y}
+{\mathcal D}_{y \alpha} {\mathcal D}_{x x} {\mathcal D}_{\alpha
y} +{\mathcal D}_{y \alpha} {\mathcal D}_{\alpha \alpha} {\mathcal
D}_{\alpha y}
\\ &= +{\mathcal D}_{y \alpha} {\mathcal D}_{\alpha x} {\mathcal D}_{\alpha y} {\mathcal D}_{x \alpha}
-{\mathcal D}_{y \alpha} {\mathcal D}_{\alpha x} {\mathcal D}_{x
y}
\\ &\hphantom{{}=} -{\mathcal D}_{x x} {\mathcal D}_{\alpha y} {\mathcal D}_{y \alpha}
+{\mathcal D}_{x x} {\mathcal D}_{y y} -{\mathcal D}_{x x}
{\mathcal D}_{\alpha \alpha}
\\ &\hphantom{{}=} +{\mathcal D}_{y \alpha} {\mathcal D}_{\alpha y} {\mathcal D}_{\alpha \alpha}
+{\mathcal D}_{y \alpha} {\mathcal D}_{\alpha y}
\\ &\cong  -{\mathcal D}_{y \alpha} {\mathcal D}_{\alpha x} {\mathcal D}_{x y}
+{\mathcal D}_{x x} {\mathcal D}_{y y} + {\mathcal D}_{y \alpha}
{\mathcal D}_{\alpha y}
\\ &\cong  \ldots
\\ &\cong  -{\mathcal D}_{y x} {\mathcal D}_{x y}
+{\mathcal D}_{x x} {\mathcal D}_{y y} +{\mathcal D}_{y y}.
\end{align*}
\end{example}


Here the identities $=$ are obtained by applying the commutator
identity, and the $[{\mathcal{L}}\,|\,\mathcal{P}]$-equivalences
$\cong$ are obtained by applying again the commutator identity and
by deleting the summands which turn out to be the zero operator
when restricted to the proper letterplace algebra
$Super[{\mathcal{L}}|\mathcal{P}].$



\subsection{Tableau polarization monomials}

Let $u'$ and $u$ be words of the same length $m$ on the alphabet
$\mathcal{X}$, say $u' = x'_1x'_2 \cdots x'_m$, $u = x_1x_2 \cdots
x_m$. The {\it letter polarization monomial} of the word $u$ to
the word $u'$ is defined to be the ${\mathbb K}$-linear operator
$$
{\mathcal D_{u'u}} = {\mathcal D}_{x'_1 x_1} {\mathcal D}_{x'_2
x_2} \ldots {\mathcal D}_{x'_m x_m}\in End_{\mathbb K}[
Super_n[\mathcal{X}|\mathcal{Y}]].
$$

Let $S', S\in Tab(\mathcal{X}),$ with $\sh(S')= \sh(S)\vdash n.$
The {\it letter polarization monomial} of the tableau $S$ to the
tableau $S'$ is defined to be the ${\mathbb K}$-linear operator
$$
{\mathcal D_{S'S}} = {\mathcal D}_{x'_1 x_1} {\mathcal D}_{x'_2
x_2} \ldots {\mathcal D}_{x'_n x_n}\in End_{\mathbb K}[
Super_n[\mathcal{X}|\mathcal{Y}]],
$$
where $x'_1 \ldots x'_n = w(S')$ and $x_1 \ldots x_n = w(S).$

In passing, we point out that, if $S'=(u'_1, \ldots ,u'_p)$ and
$S=(u_1, \ldots ,u_p)$, then
$$
{\mathcal D_{S'S}} = {\mathcal D}_{u'_1 u_1} {\mathcal D}_{u'_2
u_2} \ldots {\mathcal D}_{u'_p u_p}.
$$

\begin{example}
Let
$$
S'=\begin{array}{cc} x & y
\\ x & z
\end{array}
\ , \ S=\begin{array}{cc} x & z
\\ z & t
\end{array};
$$
then
$$
{\mathcal D}_{S'S} \ = \ {\mathcal D}_ {\begin{array}{cc} x & y
\\ x & z
\end{array}
\ , \
\begin{array}{cc} x & z
\\ z & t
\end{array}} \ = \
{\mathcal D}_{xx} {\mathcal D}_{yz} {\mathcal D}_{xz} {\mathcal
D}_{zt}
$$
\end{example}


Let $v$ and $v'$ be words of the same length $m$ on the alphabet
$\mathcal{Y}$, say $v = y_1y_2 \cdots y_m$, $v' = y'_1y'_2 \cdots
y'_m$. The {\it place polarization monomial} of the word $v$ to
the word $v'$ is defined to be the ${\mathbb K}$-linear operator
$$
_{vv'} {\mathcal D} =  _{y_1 y'_1}\! {\mathcal D}\ _{ y_2 y'_2}\!
{\mathcal D}\ \ldots\ _{ y_m y'_m}\! {\mathcal D} \in End_{\mathbb
K}[ Super_n[\mathcal{X}|\mathcal{Y}]].
$$




Let $V, V' \in Tab(\mathcal{Y}),$ with $\sh(V) = \sh(V')\vdash n.$
The {\it place polarization monomial} of the tableau $V$ to the
tableau $V'$ is defined to be the ${\mathbb K}$-linear operator
$$
_{VV'} {\mathcal D} =  _{y_1 y'_1}\! {\mathcal D}\ _{ y_2 y'_2}\!
{\mathcal D}\ \ldots\ _{ y_n y'_n}\! {\mathcal D} \in End_{\mathbb
K}[ Super_n[\mathcal{X}|\mathcal{Y}]],
$$
where $y_1 \ldots y_n = w(V)$ and $y'_1 \ldots y'_n = w(V').$

Clearly, if $V=(v_1, \ldots ,v_p)$ and $V'=(v'_1, \ldots ,v'_p)$,
then
$$
_{VV'} {\mathcal D} =  _{v_1 v'_1}\! {\mathcal D}\ _{ v_2 v'_2}\!
{\mathcal D}\ \ldots\ _{ v_p v'_p}\! {\mathcal D}.
$$


\begin{example}
Let
$$
V=\begin{array}{cc} a & c
\\ b & c
\end{array}
\ , \ V'=\begin{array}{cc} b & b
\\ c & d
\end{array};
$$
then
$$
_{VV'} {\mathcal D}\ = \ _{\begin{array}{cc} a & c
\\  b & c
\end{array}
\ , \
\begin{array}{cc}  b & b
\\ c & d
\end{array}} {\mathcal D} \ = \
_{ab} {\mathcal D} _{cb} {\mathcal D}_{bc} {\mathcal D}_{cd}
{\mathcal D}
$$
\end{example}


\subsection{Capelli bitableaux and Capelli rows}


Among Capelli-type operators, a distinguished role is played by
those involving virtual letters of the {\it same} sign.
Specifically, let $U = (u_1, \ldots, u_p)$ and $V =(v_1, \ldots,
v_p)$ be  tableaux on the alphabet of proper letters
$\mathcal{L}$,
 $\sh(U) = \sh(V) = \lambda = (\lambda_1,
\ldots, \lambda_p)$, and let $\alpha_1, \ldots,  \ldots, \alpha_p$
be $p$ {\it distinct} virtual letters of the same sign.

The Capelli-type operator
$$
[U\,|\,V] = \left[
\begin{array}{l}
u_1\\
u_2\\
\vdots\\
u_p
\end{array}
\right| \left.
\begin{array}{l}
v_1\\
v_2\\
\vdots\\
v_p
\end{array}
\right] = {\mathcal D}_{u_1, \alpha_1^{\lambda_1}} \cdots
{\mathcal D}_{u_p, \alpha_p^{\lambda_p}} {\mathcal D}_{
\alpha_1^{\lambda_1}, v_1} \cdots {\mathcal D}_{
\alpha_p^{\lambda_p}, v_p}
$$
will be called a {\it Capelli bitableau}. A Capelli bitableau is
said to be {\it positive} ({\it negative}) if the virtual
letters $\alpha_1, \ldots,  \ldots, \alpha_p$ have positive
(negative) sign.

\begin{example}
Let $U = (u_1,u_2), \ u_1 = xyz, \ u_2 = yz$ and let $V =
(v_1,v_2), \  v_1 = xzz, \  v_2 = xy$. Then,
 given two positive virtual letters $\alpha_1, \alpha_2$, the operator
$$
[U\,|\,V] = \left[
\begin{array}{l}
u_1\\
u_2\\
\end{array}
\right| \left.
\begin{array}{l}
v_1\\
v_2\\
\end{array}
\right] = {\mathcal D}_{u_1, \alpha_1^3} {\mathcal D}_{u_2,
\alpha_2^2} {\mathcal D}_{ \alpha_1^3, v_1} {\mathcal D}_{
\alpha_2^2, v_p},
$$
where
\begin{alignat*}2
{\mathcal D}_{u_1, \alpha_1^3} &= {\mathcal D}_{x
\alpha_1}{\mathcal D}_{y \alpha_1}{\mathcal D}_{z \alpha_1} &\quad
{\mathcal D}_{u_2, \alpha_2^2} &= {\mathcal D}_{y
\alpha_2}{\mathcal D}_{z \alpha_2}\\
{\mathcal D}_{\alpha_1^3, v_1} &= {\mathcal D}_{\alpha_1
x}{\mathcal D}_{\alpha_1 z}{\mathcal D}_{\alpha_1 z} &\quad
{\mathcal D}_{\alpha_2^2, v_2} &= {\mathcal D}_{\alpha_2
x}{\mathcal D}_{\alpha_2 y},
\end{alignat*}
is a positive Capelli bitableau.

\end{example}

If  $U, V$ are tableaux with just one row, the Capelli bitableau
$[U\,|\,V]$ is said to be a {\it Capelli row}. Note that a Capelli
bitableau, is not, in general, a product of Capelli rows.


The following result is a more detailed reformulation of
Proposition~4 of \cite{Brini3}; it  provides the basic identity
that relates Capelli bitableaux and products of Capelli rows.


\begin{proposition}\label{proposition: Capelli bitableaux and Capelli rows}
Let
$$
u_{1}, u_{2}, \ldots u_{p}, u_{p+1}; \ v_{1}, v_{2}, \ldots v_{p},
v_{p+1}
$$
be   words over the proper letter alphabet $\mathcal{L}$, where
$v_i$ has the same length as $u_i.$

We have the following $[{\mathcal{L}}\,|\,\mathcal{P}]$-equivalence
involving positive (negative) Capelli bi\-tabl\-eaux:
$$
\left[
\begin{array}{l}
u_1\\
u_2\\
\vdots\\
u_p
\end{array}
\right| \left.
\begin{array}{l}
v_1\\
v_2\\
\vdots\\
v_p
\end{array}
\right] \ \left[ u_{p+1} \,|\, v_{p+1} \right] \cong \sum c_{*} \left[
\begin{array}{l}
u_1\\
u_2\\
\vdots\\
u_p\\
u'_{p+1}
\end{array}
\right| \left.
\begin{array}{l}
v'_1\  v_{p+1,(1)}\\
v'_2\  v_{p+1,(2)}\\
\vdots\\
v'_p\  v_{p+1,(p)}\\
v_{p+1,(p+1)}
\end{array}
\right]
$$
where the $c_*$'s are rational coefficients and the sum is taken
over all the $(p+1)$-tuples of  subwords
$$
u'_{p+1}, v'_1, v'_2, \ldots v'_p
$$
of the words $ u_{p+1}, v_1, v_2, \ldots v_p $ such that the
(multiset) difference of contents
$$
c(u_{p+1}) - c(u'_{p+1})
$$
equals the sum of (multiset) differences of contents
$$ \sum_{i=1}^{p} \left( c(v_{i}) -
c(v'_{i}) \right),
$$
and over all the splittings
$$
v_{p+1, (1)}, v_{p+1, (2)}, \ldots, v_{p+1, (p+1)}
$$
of the word $v_{p+1},$ such that
$$
l(v'_i\  v_{p+1,(i)}) = l(u_i), \qquad i=1, \ldots, p.
$$
\end{proposition}

It is a simple fact  that, in the above summation, the Capelli
tableau
$$
\left[
\begin{array}{l}
u_1\\
u_2\\
\vdots\\
u_p\\
u_{p+1}
\end{array}
\right| \left.
\begin{array}{l}
v_1\\
v_2\\
\vdots\\
v_p\\
v_{p+1}
\end{array}
\right]
$$
appears  with coefficient $\pm 1.$

Therefore, the identity in the previous proposition can be
rewritten as
\begin{equation*}
\pm \left[
\begin{array}{l}
u_1\\
u_2\\
\vdots\\
u_p\\
u_{p+1}
\end{array}
\right| \left.
\begin{array}{l}
v_1\\
v_2\\
\vdots\\
v_p\\
v_{p+1}
\end{array}
\right] \cong \left[
\begin{array}{l}
u_1\\
u_2\\
\vdots\\
u_p
\end{array}
\right| \left.
\begin{array}{l}
v_1\\
v_2\\
\vdots\\
v_p
\end{array}
\right] \ \left[ u_{p+1} \,|\, v_{p+1} \right] -
\sum c_{*} \left[
\begin{array}{l}
u_1\\
u_2\\
\vdots\\
u_p\\
u'_{p+1}
\end{array}
\right| \left.
\begin{array}{l}
v'_1\  v_{p+1,(1)}\\
v'_2\  v_{p+1,(2)}\\
\vdots\\
v'_p\  v_{p+1,(p)}\\
v_{p+1,(p+1)}
\end{array}
\right],
\end{equation*}
where the sum is taken under the previous conditions, and
$u'_{p+1}$ ranges among the strict subwords of $u_{p+1}.$

By iterating this argument, we obtain the following theorem.

\begin{theorem}\label{theorem: Capelli bitableaux and products of Capelli Rows}

\begin{itemize}

\item Every Capelli bitableau is
$[{\mathcal{L}}\,|\,\mathcal{P}]$-equivalent
 to a linear combination of products of Capelli
rows, with rational coefficients. \item Every product of Capelli
rows is $[{\mathcal{L}}\,|\,\mathcal{P}]$-equivalent to a linear
combination of Capelli bitableaux, with rational coefficients.
\end{itemize}
\end{theorem}

\subsection{Devirtualization of Capelli rows  and Laplace expansion type identities}

If we consider the restricted action of a Capelli operator on the
proper superalgebra $Super\mathcal[L|\mathcal{P}]$, we get the
following Laplace expansion type identities.

\begin{theorem}[\cite{Brini3, BRT3}]\label{theorem: Laplace expansion type identities} 
Let $\alpha \in \mathcal X\setminus \mathcal{L}$ be a virtual
letter, $|\alpha|=0;$ then
\begin{multline*}
 {\mathcal D}_{y_1 \alpha} {\mathcal D}_{y_2 \alpha} \ldots
{\mathcal D}_{y_n \alpha} \cdot {\mathcal D}_{\alpha z_1}
{\mathcal D}_{\alpha z_2} \ldots {\mathcal D}_{\alpha z_n}
\\ \cong\sum_{i=1}^n \pm ({\mathcal D}_{y_i
z_1} -(-1)^{|y_i||z_1|} (n-1)\delta_{y_i z_1}I) {\mathcal D}_{y_1
\alpha} \ldots \widehat{{\mathcal D}_{y_i \alpha}} \ldots
{\mathcal D}_{y_n \alpha} \cdot {\mathcal D}_{\alpha z_2} \ldots
{\mathcal D}_{\alpha z_n}
\end{multline*}
where $\pm$ is the sign associated to the pair of words 
$$y_1
\ldots y_i \ldots y_n z_1 \ldots z_n\quad \text{and}\quad 
y_i z_1 y_1 \ldots
\widehat{y_i} \ldots y_n z_2 \ldots z_n,$$ 
that is
$$
(-1)^{|z_1|(|y_1|+\ldots+\widehat{|y_i|}+\ldots+|y_n|)+|y_i|(|y_1|+\ldots+|y_{i-1}|)}.
$$
\noindent A similar result holds in the case $|\alpha|=1.$
\end{theorem}



By Theorem~\ref{theorem: Capelli bitableaux and products of
Capelli Rows} and iterating the identity of Theorem~\ref{theorem:
Laplace expansion type identities}, one can eliminate all the
virtual variables $\alpha_i,$  in any Capelli bitableau, therefore
obtaining a devirtualization of the associated  operator. The
crucial point of the virtual method is that, in the study of the
actions, the virtual form is much more preferable than a
devirtualized form.

\subsection{Basic examples}
\begin{itemize}
\item Let  $x\neq y$ be proper letters, $\alpha$ a virtual letter,
$|\alpha|=|x|=|y|=0;$ then
$${\mathcal D}_{x \alpha}^n {\mathcal D}_{\alpha y }^n \cong n! {\mathcal
D}_{x y}^n.$$ 
\item Let  $x$ be a proper letter, $\alpha$ a
virtual letter, $|\alpha|=|x|=0;$ then
$${\mathcal D}_{x \alpha}^n {\mathcal D}_{\alpha x }^n \cong n! {\mathcal
D}_{x x} ({\mathcal D}_{x x}-I) \ldots ({\mathcal D}_{x
x}-(n-1)I)= (n!)^2\binom {{\mathcal D}_{x x}}  n.$$
\item Let $y_{i_1}, \ldots, y_{i_n}, x_{j_1}, \ldots, x_{j_n}$ be
two $n$-tuples of proper letters of the same sign, say $|y_{i_h}|=
|x_{j_k}|= 0,$ for every $h, k=1, \ldots, n.$ Assume that the two
$n$-tuples above have no letters in common. Then
\begin{equation*}
{\mathcal D}_{y_{i_1}\alpha} \ldots {\mathcal D}_{y_{i_n}\alpha}
{\mathcal D}_{\alpha x_{j_n}} \ldots {\mathcal D}_{\alpha x_{j_n}}
=
\begin{cases} \per[{\mathcal D}_{y_{i_h} x_{j_k}}]_{h,k} &
\text{if }|\alpha|= 0,
\\ \det[{\mathcal D}_{y_{i_h} x_{j_k}}]_{h,k} &\text{if } |\alpha|= 1.
\end{cases}
\end{equation*}
\item Let ${\mathcal L}= {\mathcal L}^- =\{x_1, \ldots x_m\}$ be a
linearly ordered set of (distinct) negative proper letters; then,
we get the classical Capelli operator:
\begin{multline*}
{\mathcal D}_{x_m \alpha} \ldots {\mathcal D}_{x_1 \alpha} \cdot
{\mathcal D}_{\alpha x_1} \ldots {\mathcal D}_{\alpha x_m} 
\\ =\det\left[
\begin{array}{llll}
{\mathcal D}_{x_1 x_1} + (m-1)I & {\mathcal D}_{x_1 x_2} & \ldots
& {\mathcal D}_{x_1 x_m}
\\{\mathcal D}_{x_2 x_1} & {\mathcal D}_{x_2 x_2}+(m-2)I & \ldots & {\mathcal D}_{x_2 x_m}
\\ \vdots & \vdots &  \ddots & \vdots
\\ {\mathcal D}_{x_m x_1} & {\mathcal D}_{x_m x_2} & \ldots & {\mathcal D}_{x_m x_m}
\end{array}\right]= H_m,
\end{multline*}
where the expansion of the ``determinant" is by column from left
to right (Weyl \cite{Weyl}).
\end{itemize}






\section{Biproducts and Bitableaux in $Super[{\mathcal L}|{\mathcal \mathcal{P}}]$}


\subsection{Capelli rows and supersymmetries in
$Super[{\mathcal L}|{\mathcal \mathcal{P}}]$}

Let $x_1, \ldots, x_n\in {\mathcal L}$ and $y_1, \ldots, y_n\in
{\mathcal P}$ be (not necessarily distinct) proper letters and
places, respectively. Let $\alpha\in \mathcal X\setminus
\mathcal{L}$ be a virtual letter. By applying the Capelli operator
${\mathcal D}_{x_1 \alpha} {\mathcal D}_{x_2 \alpha} \ldots
{\mathcal D}_{x_n \alpha} \cdot {\mathcal D}_{\alpha x_n} \ldots
{\mathcal D}_{\alpha x_2} {\mathcal D}_{\alpha x_1}$ to the
monomial $(x_1\,|\,y_1) (x_2\,|\,y_2) \ldots (x_n\,|\,y_n)\in
Super_n\mathcal[\mathcal{L}|\mathcal{P}],$ we get
\begin{multline*}{\mathcal D}_{x_1 \alpha}
{\mathcal D}_{x_2 \alpha} \ldots {\mathcal D}_{x_n \alpha} \cdot
{\mathcal D}_{\alpha x_n} \ldots {\mathcal D}_{\alpha x_2}
{\mathcal D}_{\alpha x_1} (x_1\,|\,y_1) (x_2\,|\,y_2) \ldots (x_n\,|\,y_n) 
\\= k \cdot {\mathcal D}_{x_1
\alpha} {\mathcal D}_{x_2 \alpha} \ldots {\mathcal D}_{x_n \alpha}
(\alpha\,|\,y_1) (\alpha\,|\,y_2) \ldots (\alpha\,|\,y_n), \quad k \in
{\mathbb{Z}}
\end{multline*}
which is
\begin{itemize}
\item supersymmetric in the $x$'s and the $y$'s if $|\alpha|=0$
(that is equals zero whenever the word $x_1 \cdots x_n$ or the
word $y_1 \cdots y_n$ contain repeated negatively signed symbols);


\item ``dual" supersymmetric in the $x$'s and the $y$'s if
$|\alpha|=1$ (that is equals zero whenever the word $x_1 \cdots
x_n$ or the word $y_1 \cdots y_n$ contain repeated positively
signed symbols).
\end{itemize}

Informally speaking, Capelli rows are supersymmetrization
operators in disguise.

\subsection{Biproducts as basic symmetrized elements in $Super[{\mathcal L}|{\mathcal P}]$}

The argument of the previous subsection leads naturally to a
virtual definition of the basic supersymmetric and of the basic
dually supersymmetric objects in
$Super_n[\mathcal{L}|\mathcal{P}],$ both associated to pairs of
sequences of the same length in $\mathcal L$ and $\mathcal P.$
These objects, here presented in their three different virtual
forms \cite{Brini1}, are called {\it biproducts} and {\it
*--biproducts}, respectively; in particular, the biproducts
coincide, in characteristic $0,$ with the Grosshans--Rota--Stein
biproducts \cite{GRS}.

Let $x_1, \ldots, x_n\in {\mathcal L}$ and $y_1, \ldots, y_n\in
{\mathcal P}$ be proper letters and places, respectively. Let
$\alpha\in \mathcal X\setminus \mathcal{L}$ be a virtual letter
and $\beta\in \mathcal Y\setminus \mathcal{P}$ be a virtual place,
with $|\alpha|=|\beta|.$

The element of $Super_n[\mathcal{L}|\mathcal{P}]$
\begin{align*}
{\mathcal D}_{x_1 \alpha} {\mathcal D}_{x_2 \alpha} \ldots
{\mathcal D}_{x_n \alpha}& (\alpha\,|\,y_1) (\alpha\,|\,y_2) \ldots
(\alpha\,|\,y_n)
\\ &= {\mathcal D}_{x_1 \alpha} {\mathcal D}_{x_2 \alpha} \ldots {\mathcal D}_{x_n
\alpha} (\frac {(\alpha\,|\,\beta)^n} {n!})\ _{\beta y_1}{\mathcal D}
_{\beta y_2}{\mathcal D} \ldots _{\beta y_n}{\mathcal D}
\\ &= (x_1\,|\,\beta) (x_2\,|\,\beta) \ldots (x_n\,|\,\beta) _{\beta y_1}{\mathcal D}
_{\beta y_2}{\mathcal D} \ldots _{\beta y_n}{\mathcal D}
\end{align*}
is called, for $|\alpha|= |\beta|= 0,$ the {\it biproduct} of $x_1
\ldots x_n$ and $y_1 \ldots y_n,$  denoted by
\begin{equation}
(x_1 x_2 \ldots x_n\,|\,y_1 y_2 \ldots y_n),
\end{equation}
and, for $|\alpha|= |\beta|= 1,$ the {\it *-biproduct} of $x_1
\ldots x_n$ and $y_1 \ldots y_n,$ denoted by
\begin{equation}
(x_1 x_2 \ldots x_n\,|\,y_1 y_2 \ldots y_n)^*.
\end{equation}




\begin{remark}\label{remark: laplace expansion}
The Laplace expansions of biproducts and *--biproducts correspond
to the Leibniz rule for superderivations.
\end{remark}

\begin{example}
\begin{enumerate}
\item Let $x_1, \ldots, x_n\in {\mathcal L_0= \mathcal{L}}$ and
$y_1, \ldots, y_n\in {\mathcal P_0 = \mathcal{P}}$ be proper
letters and places, respectively; then
$Super_n[\mathcal{L}|\mathcal{P}] =
Sym_n[\mathcal{L}|\mathcal{P}].$ We have
\begin{align*}
(x_1 x_2 \ldots x_n\,|\,y_1 y_2 \ldots y_n) &= \per
\left((x_i\,|\,y_j)\right)_{i, j=1, \ldots, n}
\\ (x_1 x_2 \ldots
x_n\,|\,y_1 y_2 \ldots y_n)^* &= (-1)^{(^n_2)} \det
\left((x_i\,|\,y_j)\right)_{i, j=1, \ldots, n}
\end{align*}
\item Let $x_1, x_2 \in \mathcal L_0=\mathcal{L},$ $y_1, y_2 \in
\mathcal P_1=\mathcal{P};$ then
$Super[\mathcal{L}_0|\mathcal{P}_1]=\Lambda[\mathcal{L}|\mathcal{P}]$.
We have
\begin{align*}
(x_1x_2\,|\,y_1y_2) & = {\mathcal D}_{x_1 \alpha} {\mathcal D}_{x_2
\alpha} \left((\alpha\,|\,y_1)(\alpha\,|\,y_2)\right)
\\ & = (x_2\,|\,y_1)(x_1\,|\,y_2)+(x_1\,|\,y_1)(x_2\,|\,y_2);
\end{align*}
on the other hand,
\begin{align*}
(x_1x_2\,|\,y_1y_2) & = \left((x_1\,|\,\beta)(x_2\,|\,\beta)\right)\ _{\beta
y_1}{\mathcal D}\ _{\beta y_2}{\mathcal D}
\\ & = -(x_1\,|\,y_2)(x_2\,|\,y_1)+(x_1\,|\,y_1)(x_2\,|\,y_2).
\end{align*}
Note that $(x_1 x_2\,|\,y_1 y_2)$ is symmetric in the $x'$s and
skew-symmetric in the $y'$s.
\end{enumerate}
\end{example}



The notion of biproduct   is extended to a {\it bilinear map}
$$
Super[{\mathcal L}] \times Super[{\mathcal P}] \rightarrow
Super[\mathcal{L}|\mathcal{P}]
$$
by setting
\begin{itemize}
\item If $\omega = x_{i_1}x_{i_2} \cdots  x_{i_p}$ is a monomial
of $\BZ$-degree $p$ in $Super[\mathcal L]$, and $\varpi =
y_{j_1}y_{j_2} \cdots  y_{_q}$ is a monomial of $\BZ$-degree $q$
in $Super[\mathcal P]$, with $p \neq q$, then $(\omega\,|\,\varpi) =
0.$ \item $\big(\sum_h \ c_h \omega_h\,|\,\sum_k \ d_k \varpi_k\big) =
\sum_{h,k} \ c_h d_k \ (\omega_h\,|\,\varpi_k).$
\end{itemize}


The actions of  superpolarizations on  biproducts can be computed
in a quite direct way.



\begin{proposition} \label{proposition: polarization of biproduct}
Let $x_i, x_j \in \mathcal L$, $y_h, y_k \in \mathcal P$, $\omega
\in Super[\mathcal L]$, $\varpi \in Super[\mathcal P]$. We have
the following identities:
\begin{align*}
{\mathcal D}_{x_i x_j}(\omega\,|\,\varpi)_{y_h y_k}{\mathcal D} & =
({\it D}_{x_i x_j}(\omega)\,|\,\varpi)_{y_h y_k}{\mathcal D} \\ &  =
{\mathcal D}_{x_i x_j}(\omega\,|\,(\varpi)_{y_h y_k}{\it D}) \\ &  =
({\it D}_{x_i x_j}(\omega)\,|\,(\varpi)_{y_h y_k}{\it D}),
\end{align*}
where ${\it D}_{x_i x_j}$ and $_{y_h y_k}{\it D}$ are left and
right superpolarizations on $Super[\mathcal L]$ and
$Super[\mathcal P]$, respectively (see Subsection~3.6).
\end{proposition}


\begin{example} Let $x_1, x_2, x_3 \in \mathcal L$, $|x_1| = |x_3| =1, |x_2| = 0$,
$y_1, y_2, y_3 \in \mathcal L$, $|y_1| = |y_2| =1, |y_3| = 0$. We
have the following identities:
\begin{align*}
{\mathcal D}_{x_3 x_2}(x_1x_2\,|\,y_1y_2)_{y_1 y_3}{\mathcal D} & =
({\it D}_{x_3 x_2}(x_1x_2)\,|\,y_1y_2)_{y_1 y_3}{\mathcal D} \\ & = -
(x_1x_3\,|\,y_1y_2)_{y_1 y_3}{\mathcal D} \\ & = -
(x_1x_3\,|\,(y_1y_2)_{y_1 y_3}{\it D}) \\ & = (x_1x_3\,|\,y_3y_2).
\end{align*}
\end{example}

Proposition~\ref{proposition: polarization of biproduct} may be
rephrased as a ``representation-theoretical" result.

Recall that $Super[\mathcal L]$ is $pl(\mathcal L)$-module and
$Super[\mathcal P]$ is $pl(\mathcal P)$-module, where the actions
of $pl(\mathcal L)$ and $pl(\mathcal P)$ are implemented by left
and right superpolarizations, respectively. Thus, the tensor
product $Super[\mathcal L] \otimes Super[\mathcal P]$ is a
$(pl(\mathcal L), pl(\mathcal P))$-bimodule
$$
pl(\mathcal L) \cdot Super[\mathcal L] \otimes Super[\mathcal P]
\cdot pl(\mathcal P),
$$
since the actions of $pl(\mathcal L)$ and $pl(\mathcal P)$ clearly
commute.




\begin{corollary}
The biproduct induces a $(pl(\mathcal(L),
pl(\mathcal(P))$-equivariant  linear map
\begin{align*}
Super[\mathcal L] \otimes Super[\mathcal P] &\rightarrow
Super[\mathcal{L}|\mathcal{P}],\\
\omega \otimes \varpi &\mapsto (\omega\,|\,\varpi),
\end{align*}
$\omega \in Super[\mathcal L]$, $\varpi \in Super[\mathcal P]$.
\end{corollary}





\subsection{Bitableau monomials}


For any words $x' = x'_1 x'_2 \cdots x'_n$ and $x = x_1 x_2 \cdots
x_n$ on the letter alphabet $\mathcal{X},$ any words $y = y_1 y_2
\cdots y_n$ and $y' = y'_1 y'_2 \cdots y'_n$ on the place alphabet
$\mathcal{Y},$ all of the same length, we set
\begin{align*}
\mathcal{D}_{x' x} &= \mathcal{D}_{x_1' x_1}\ \mathcal{D}_{x_2'
x_2} \cdots \mathcal{D}_{x_n' x_n},
\\
\langle x\,|\, y \rangle &= (x_1\,|\, y_1)\ (x_2\,|\, y_2) \cdots (x_n\,|\, y_n),
\\
_{y y'}\mathcal{D} &=\ _{y_1 y_1'}\mathcal{D}\ _{y_2
y_2'}\mathcal{D}\ \cdots _{y_n y_n'}\mathcal{D}.
\end{align*}
For any pair of tableaux $S'$ and $S$ on the letter alphabet
$\mathcal{X},$ and any pair of tableaux $T$ and $T'$ on the place
alphabet $\mathcal{Y},$ all of the same shape, we set
\begin{align*}
\mathcal{D}_{S' S} &= \mathcal{D}_{x' x}, \qquad x' = w(S'),\ x
=w(S),
\\
\langle S\,|\, T\rangle &= \langle x\,|\, y\rangle, \qquad x = w(S),\ y
=w(T),
\\
_{T T'}\mathcal{D} &=\ _{y y'}\mathcal{D}, \qquad y = w(T),\ y'
=w(T'),
\end{align*}
where $x', x, y, y'$ are the row words of the tableaux $S', S, T,
T',$ respectively. We recall that the row word $w(U)$ of a tableau
$U$ is the word obtained by reading the entries of $U$ row by row,
from left to right and from top to bottom.

We will often use the short forms
\begin{align*}
S'S\quad & for\quad \mathcal{D}_{S' S},
\\
\underline{ST}\quad & for\quad \langle S\,|\, T\rangle,
\\
TT'\quad & for\quad _{T T'}\mathcal{D}.
\end{align*}

The definition of biproduct can also be written in the following
form:
$$
\mathcal{D}_{x \alpha^n}\ \langle \alpha^n \,|\, y \rangle =
\mathcal{D}_{x \alpha^n}\ \frac {\langle \alpha^n \,|\, \beta^n
\rangle}{n!}\ _{\beta^n y}\mathcal{D} = \langle x \,|\, \beta^n
\rangle\ _{\beta^n y}\mathcal{D},
$$
or in an even shorter form:
$$
x\alpha^n\ \underline{\alpha^n y} = \frac 1 {n!}\ x\alpha^n\
\underline{\alpha^n \beta^n}\ \beta^n y = \underline{x \beta^n}\
\beta^n y;
$$
in these formulas $x$ and $y$ denote words over the proper
alphabets $\mathcal{L}$ and $\mathcal{P},$ while $\alpha$ and
$\beta$ denote constant words of positive virtual symbols.







\subsection{Bitableaux in $Super[\mathcal{L}|\mathcal{P}]$}


For every partition $\lambda = (\lambda_1, \lambda_2, \ldots,
\lambda_p) \vdash n,$ and every pair of tableaux $T \in
Tab(\mathcal{L}),$ $U \in Tab(\mathcal{P}),$ with $\sh(T) =
\lambda = \sh(U),$ we define the {\it bitableau} \cite{GRS,
Brini1}
$$
(T\,|\,U) \in Super_n[\mathcal{L}|\mathcal{P}]
$$
as the common value of the expressions
$$
TC_1\ \underline{C_1U} = \frac 1 {\lambda !}\ TC_1\
\underline{C_1C_2}\ C_2U, = \underline{TC_2}\ C_2U,
$$
where $C_1$ is any virtual letter tableau of co-Deruyts type, $C_2$
is any virtual place tableau of co-Deruyts type, all of shape
$\lambda,$ and $\lambda! = \prod_i \lambda_i!.$

If $\sh(T) \neq \sh(U),$ the bitableau $(T\,|\,U)$ is set to be zero.

\begin{example}
In the following, let $|\alpha_i| = 0$ and $|\beta_i| = 0$ be any
virtual positive symbols.
\begin{align*}
\left(
\begin{array}{cc} x & y
\\ x & z
\\ y
\end{array}
\right. \left|
\begin{array}{cc}
a & b\\
a & c\\
c
\end{array} \
\right)  &= {\mathcal D}_ {
\begin{array}{cc}
x & y\\
x & z\\
y
\end{array}
\begin{array}{cc}
\alpha_1 & \alpha_1\\
\alpha_2 & \alpha_2\\
\alpha_3
\end{array}
} \left<
\begin{array}{cc}
\alpha_1 & \alpha_1
\\ \alpha_2 & \alpha_2
\\ \alpha_3
\end{array}
\right. \left|
\begin{array}{cc}
a & b\\
a & c\\
c
\end{array} \
\right> 
\\
&\kern-5pt=
\frac 1 {2! 2! 1!}\ {\mathcal D}_ {
\begin{array}{cc}
x & y\\
x & z\\
y
\end{array}
\begin{array}{cc}
\alpha_1 & \alpha_1\\
\alpha_2 & \alpha_2\\
\alpha_3
\end{array}
} \left<
\begin{array}{cc}
\alpha_1 & \alpha_1
\\ \alpha_2 & \alpha_2
\\ \alpha_3
\end{array}
\right. \left|
\begin{array}{cc}
\beta_1 & \beta_1\\
\beta_2 & \beta_2\\
\beta_3
\end{array} \
\right> \ _{
\begin{array}{cc}
\beta_1 & \beta_1\\
\beta_2 & \beta_2\\
\beta_3
\end{array}
\begin{array}{cc}
a & b\\
a & c\\
c
\end{array}
} {\mathcal D} 
\\
&\kern-5pt=
\left<
\begin{array}{cc}
x & y\\
x & z\\
y
\end{array}
\right. \left|
\begin{array}{cc}
\beta_1 & \beta_1\\
\beta_2 & \beta_2\\
\beta_3
\end{array} \
\right> \ _{
\begin{array}{cc}
\beta_1 & \beta_1\\
\beta_2 & \beta_2\\
\beta_3
\end{array}
\begin{array}{cc}
a & b\\
a & c\\
c
\end{array}
} {\mathcal D}.
\end{align*}
\end{example}

\begin{proposition}\label{proposition: bitableaux and biproducts}
Let $T = (\omega_1, \ldots, \omega_p)$, $U = (\omega'_1, \ldots,
\omega'_p)$ be tableaux of the same shape. Then the bitableau
$(T\,|\,U)$ equals, up to a sign, the product of the biproducts
$(\omega_1\,|\,\omega'_1), \ldots, (\omega_p\,|\,\omega'_p)$. In symbols
$$
(T\,|\,U) = \left(
\begin{array}{l}
\omega_1\\ \omega_2\\ \vdots\\ \omega_p
\end{array}
\right| \left.
\begin{array}{l}
\omega'_1\\ \omega'_2\\ \vdots\\ \omega'_p
\end{array}
\right) = (-1)^{|\omega_2||\omega'_1|+ \cdots +
|\omega_p|(|\omega'_1|+ \cdots + |\omega'_{p-1}|)}
(\omega_1|\omega'_1) \cdots (\omega_p|\omega'_p).
$$
\end{proposition}


\begin{example}
$$
\left(
\begin{array}{cc} x & y
\\ x & z
\\ y
\end{array}
\right. \left|
\begin{array}{cc}
a & b\\
a & c\\
c
\end{array} \
\right) \ = \pm (xy\,|\,ab)\ (xz\,|\,ac)\ (y\,|\,c)
$$
where the sign is given by the parity of $ (|xz|)(|ab|) +
|y|(|ab|+|ac|) .$
\end{example}

\begin{remark}\label{remark: column tableaux}
The bitableaux of shape $(1,1, \ldots, 1),$ i.e., {\it
column-bitableaux}, are monomials and, conversely,  monomials can
be written, up to a sign, as  column bitableaux. Bitableaux of
shape $(n),$ i.e., {\it row-bitableaux}, are biproducts.
\end{remark}


The actions of  superpolarizations on  bitableaux can be computed
in a quite direct way.



\begin{proposition} \label{proposition: polarization of bitableaux}
Let $x_i, x_j \in \mathcal L$, $y_h, y_k \in \mathcal P$, and let
$S = (\omega_1, \ldots, \omega_p) \in Tab(\mathcal L)$, $T =
(\omega'_1, \ldots, \omega'_p) \in Tab(\mathcal P)$. We have the
following identity:
\begin{align*}
{\mathcal D}_{x_i x_j}(T\,|\,U) _{y_h y_k}{\mathcal D} \ & = \
{\mathcal D}_{x_i x_j} \ \left(
\begin{array}{c}
\omega_1\\ \omega_2\\ \vdots\\ \omega_p
\end{array}
\right| \left.
\begin{array}{c}
\omega'_1\\ \omega'_2\\ \vdots\\  \omega'_p
\end{array}
\right) \ _{y_h y_k}{\mathcal D} \\ &  = \ \sum_{s,t=1}^p  \
(-1)^{(|x_i| + |x_j|)\epsilon_s} (-1)^{(|y_h| + |y_k|)\epsilon'_t}
\ \left(
\begin{array}{c}
\omega_1\\ \omega_2\\ \vdots\\ \vdots\\ {\it D}_{x_i
x_j}(\omega_s)\\ \vdots \\ \omega_p
\end{array}
\right| \left.
\begin{array}{c}
\omega'_1\\ \omega'_2\\ \vdots\\ (\omega'_t)_{y_h y_k}{\it D}\\
\vdots \\ \vdots\\ \omega'_p
\end{array}
\right),
\end{align*}
where
\begin{alignat*}2
\epsilon_s &= |\omega_1| + \cdots + |\omega_{s-1}|, &\quad s &= 2,
\ldots, p,\\
\epsilon'_t &= |\omega'_{t+1}| + \cdots + |\omega'_p|, &\quad t &= 1,
2, \ldots, p-1,
\end{alignat*}
and ${\it D}_{x_i x_j}$, $_{y_h y_k}{\it D}$ are left and right
superpolarizations on $Super[\mathcal L]$ and $Super[\mathcal P]$,
respectively (see Subsection~3.6).
\end{proposition}



Proposition~\ref{proposition: polarization of bitableaux} may be
rephrased as a ``representation-theoretical" result.

Let $\lambda = (\lambda_1, \lambda_2, \ldots, \lambda_p) \vdash n$
be a partition. Set
$$
Super^\lambda[\mathcal L] = Super_{\lambda_1}[\mathcal L] \otimes
Super_{\lambda_2}[\mathcal L] \otimes \cdots \otimes
Super_{\lambda_p}[\mathcal L],
$$
and
$$
Super^\lambda[\mathcal P] = Super_{\lambda_1}[\mathcal P] \otimes
Super_{\lambda_2}[\mathcal P] \otimes \cdots \otimes
Super_{\lambda_p}[\mathcal P].
$$




The tensor product $Super^\lambda[\mathcal L]$ is $pl(\mathcal
L)$-module and the tensor product $Super^\lambda[\mathcal P]$ is
$pl(\mathcal P)$-module, where the actions of $pl(\mathcal L)$ and
$pl(\mathcal P)$ are implemented by left and right
superpolarizations and extended as left and right superderivations
, respectively. Thus, the tensor product $Super^\lambda[\mathcal
L] \otimes Super^\lambda[\mathcal P]$ is a $(pl(\mathcal L),
pl(\mathcal P))$-bimodule
$$
pl(\mathcal L) \cdot Super^\lambda[\mathcal L] \otimes
Super^\lambda[\mathcal P] \cdot pl(\mathcal P),
$$
since the actions of $pl(\mathcal L)$ and $pl(\mathcal P)$ clearly
commute.




\begin{corollary}
The map
$$
\omega_1 \otimes \omega_2 \otimes \cdots \otimes \omega_p \otimes
\omega'_1 \otimes \omega'_2 \otimes \cdots \otimes \omega'_p
\mapsto \left(
\begin{array}{l}
\omega_1\\ \omega_2\\ \vdots\\ \omega_p
\end{array}
\right| \left.
\begin{array}{l}
\omega'_1\\ \omega'_2\\ \vdots\\ \omega'_p
\end{array}
\right),
$$
induces a $(pl(\mathcal(L), pl(\mathcal(P))$-equivariant  linear
map
$$
Super^\lambda[\mathcal L] \otimes Super^\lambda[\mathcal P]
\rightarrow Super_n[\mathcal{L}|\mathcal{P}],
$$
$(\omega_1, \omega_2, \cdots, \omega_p) \in
Tab_\lambda(\mathcal(L)$, $(\omega'_1, \omega'_2, \cdots,
\omega'_p) \in Tab_\lambda(\mathcal(P)$.
\end{corollary}













\section{The Standard Basis}



\subsection{The Straightening Law of Grosshans, Rota and Stein.}


\begin{theorem}[\sc Straightening Law, general form \cite{GRS}]\label{theorem: Straightening Law} 
For all monomials $u, v, w \in Super(\mathcal{L}),\ x,y \in
Super(\mathcal{P})$,
$$
\sum_{(v)} \left(
\begin{array} {c}
u v_{(1)} \\
v_{(2)} w
\end{array}
\right. \left|
\begin{array} {c}
 x \\
 y
\end{array}
\right) = (-1)^{|u||v|} \sum_{{(u)}\ {(y)}} (-1)^{l(u_{(2)})}
\left(
\begin{array} {c}
v u_{(1)}  \\
u_{(2)} w
\end{array}
\right. \left|
\begin{array} {l}
 x y_{(1)} \\
 y_{(2)}
\end{array}
\right).
$$
\end{theorem}







In the statement above, the summations are meant with respect to
the {\it coproducts} of the supersymmetric algebras
$Super(\mathcal{L})$ and $Super(\mathcal{P})$, regarded as
$\BZ_2$-bialgebras (Subsection~3.6).

In the following, for any positive symbol $\xi$ and any natural
number $n,$ we set $ \xi^{(n)} = \frac {\xi^{n}} {n!}; $ notice
that
$$
\xi^{(n)} \xi^{(m)} = \binom {n+m}n \xi^{(n+m)}, \qquad
\Delta\left(\xi^{(n)}\right) = \sum_{k=0}^n \xi^{(k)}\otimes
\xi^{(n-k)}.
$$
\begin{proof} \cite{Brini5}
First we prove the identity in the case in which the words $u, v,
w, x, y$ are powers of positive symbols: let $\alpha, \beta,
\gamma, \delta, \epsilon$ be positive virtual symbols, and $a, b,
d, e$ natural numbers such that $b\geq d;$ then
\begin{equation} 
 \left(
\begin{array}{l}
\alpha^{(a)}  \beta^{(b-d)}\\
\beta^{(d)}   \epsilon^{(e)}
\end{array}
\right| \left.
\begin{array}{l}
\gamma^{(a+b-d)}\\
\delta^{(d+e)}\\
\end{array}
\right) = \sum_{k=0}^{\min(a, d)} (-1)^k \left(
\begin{array}{l}
\beta^{(b)}  \alpha^{(a-k)} \\
\alpha^{(k)} \epsilon^{(e)} \\
\end{array}
\right| \left.
\begin{array}{l}
\gamma^{(a+b-d)} \delta^{(d-k)}\\
\delta^{(k+e)}                 \\
\end{array}
\right).
\tag*{($*$)}
\end{equation}
Indeed, starting from the right-hand side
$$
\sum_{k=0}^{\min(a, d)} (-1)^k \left( \beta^{(b)} \alpha^{(a-k)} \,|\,
\gamma^{(a+b-d)} \delta^{(d-k)} \right) \left( \alpha^{(k)}
\epsilon^{(e)} \,|\, \delta^{(k+e)} \right)
$$
we get
\begin{align*}
& \sum_{k=0}^{\min(a, d)} (-1)^k \sum_{s=0}^{\min(a-k,d-k)}
(\beta\,|\,\gamma)^{(b-d+k+s)} (\beta\,|\,\delta)^{(d-k-s)}
(\alpha\,|\,\gamma)^{(a-k-s)} (\alpha\,|\,\delta)^{(s)}
(\alpha\,|\,\delta)^{(k)} (\epsilon\,|\,\delta)^{(e)}
\\
&= \sum_{k=0}^{\min(a, d)} (-1)^k \sum_{s=0}^{\min(a-k,d-k)}
\binom {s+k}k
(\beta\,|\,\gamma)^{(b-d+k+s)} (\beta\,|\,\delta)^{(d-k-s)}\\
&\kern7cm\cdot
(\alpha\,|\,\gamma)^{(a-k-s)} (\alpha\,|\,\delta)^{(s+k)}
(\epsilon\,|\,\delta)^{(e)}
\\
&= \sum_{k=0}^{\min(a, d)} (-1)^k \sum_{t=k}^{\min(a,d)} 
\binom tk
(\beta\,|\,\gamma)^{(b-d+t)} (\beta\,|\,\delta)^{(d-t)}
(\alpha\,|\,\gamma)^{(a-t)} (\alpha\,|\,\delta)^{(t)}
(\epsilon\,|\,\delta)^{(e)}
\\
&= \sum_{t=0}^{\min(a, d)} \sum_{k=0}^{t} (-1)^k 
\binom tk
(\beta\,|\,\gamma)^{(b-d+t)} (\beta\,|\,\delta)^{(d-t)}
(\alpha\,|\,\gamma)^{(a-t)} (\alpha\,|\,\delta)^{(t)}
(\epsilon\,|\,\delta)^{(e)}
\\
&= (\beta\,|\,\gamma)^{(b-d)} (\beta\,|\,\delta)^{(d)}
(\alpha\,|\,\gamma)^{(a)} (\epsilon\,|\,\delta)^{(e)},
\end{align*}
which equals the left-hand side
$$
\left(
\begin{array}{l}
\alpha^{(a)}  \beta^{(b-d)}\\
\beta^{(d)}   \epsilon^{(e)}
\end{array}
\right| \left.
\begin{array}{l}
\gamma^{(a+b-d)}\\
\delta^{(d+e)}\\
\end{array}
\right) = \left( \alpha^{(a)} \beta^{(b-d)} \,|\, \gamma^{(a+b-d)}
\right) \left( \beta^{(d)} \epsilon^{(e)} \,|\, \delta^{(d+e)}
\right).
$$
Consider the letter polarization monomial
$$
{\mathcal D}_{u \alpha^{(a)}} {\mathcal D}_{v \beta^{(b)}}
{\mathcal D}_{w \epsilon^{(e)}} = (-1)^{|u||v|} {\mathcal D}_{v
\beta^{(b)}} {\mathcal D}_{u \alpha^{(a)}} {\mathcal D}_{w
\epsilon^{(e)}}
$$
and the place polarization monomial
$$
_{\gamma^{(a+b-d)} x}{\mathcal D} _{\delta^{(d+e)} y}{\mathcal D}.
$$
and apply these operators on both sides of the positive identity
$(*)$.




Since letter and place polarizations operators are {\it
coderivations} (see Subsection~3.6, and, for further details,
\cite{BRT1} and \cite{BRT2}), we get:
\begin{align*}
\left(
\begin{array} {c}
u v_{(1)} \\
v_{(2)} w
\end{array}
\right| \left.
\begin{array} {c}
 x \\
 y
\end{array}
\right) &=
{\mathcal D}_{u \alpha^{(a)}} {\mathcal D}_{v \beta^{(b)}}
{\mathcal D}_{w \epsilon^{(e)}} \left(
\begin{array}{l}
\alpha^{(a)}  \beta^{(b-d)}  \\
\beta^{(d)}   \epsilon^{(e)} \\
\end{array}
\right|
 \left.
\begin{array}{l}
\gamma^{(a+b-d)}\\
\delta^{(d+e)}\\
\end{array}
\right) \ _{\gamma^{(a+b-d)} x}{\mathcal D}\ _{\delta^{(d+e)}
y}{\mathcal D} \\
&=
(-1)^{|u||v|} {\mathcal D}_{v \beta^{(b)}} {\mathcal D}_{u
\alpha^{(a)}} {\mathcal D}_{w \epsilon^{(e)}} \\
&\kern.8cm\times
\sum_{k=0}^{\min(a,
d)} (-1)^k \left(
\begin{array}{l}
\beta^{(b)}  \alpha^{(a-k)} \\
\alpha^{(k)} \epsilon^{(e)}
\end{array}
\right| \left.
\begin{array}{l}
\gamma^{(a+b-d)} \delta^{(d-k)}\\
\delta^{(k+e)}                 \\
\end{array}
\right) \ _{\gamma^{(a+b-d)} x}{\mathcal D}\ _{\delta^{(d+e)}
y}{\mathcal D} \\
&=
 (-1)^{|u||v|} \sum_{{(u)}\ {(y)}} (-1)^{l(u_{(2)})} \left(
\begin{array} {l}
v u_{(1)} \\
u_{(2)} w
\end{array}
\right| \left.
\begin{array} {l}
x y_{(1)} \\
y_{(2)}
\end{array}
\right).
\end{align*}
\end{proof}


\subsection{Triangularity and nondegeneracy results}


For the sake of readability, we recall the partial order already
defined in Subsection~5.5. This order is defined on the set of all
standard tableaux over $\mathcal{L}$ of a given content; note that
the shapes of these tableaux are partitions of the same integer.

For every standard tableau $S,$ we consider the sequence $S^{(p)}
\ p=1,2, \ldots,$ of the subtableaux obtained from $S$ by
considering only the first $p$ symbols of the alphabet, and
consider the family $\sh(S^{(p)}), \ p=1,2,\ldots$ of the
corresponding shapes. Since the alphabet is assumed to be finite,
this sequence is finite and its last term is $\sh(S).$

Then, for standard tableaux $S,T,$ we set
$$
S \leq T \ \Leftrightarrow \ \sh(S^{(p)}) \unlhd \sh(T^{(p)}), \ p
=1,2,\ldots,
$$
where $\unlhd$ stands for the {\it dominance order} on partitions.
We recall that the dominance order on partitions is defined as
follows: $ \lambda = ( \lambda_1 \geq \lambda_2 \geq \ldots) \leq
\mu = ( \mu_1 \geq \mu_2 \geq \ldots) $ if and only if
$$
\lambda_1+ \cdots +\lambda_i \leq \mu_1+ \cdots +\mu_i,
$$
for every $i=1,2, \ldots$.

\begin{lemma}[\cite{BRT4}]\label{lemma: -+ triangulariy} 
Let $S, T$ be standard letter tableaux of the same content, $D$ a
virtual Deruyts letter tableau, $C$ a co-Deruyts virtual  place
tableau, with $\sh(D)=\sh(S)$ and $\sh(C)=\sh(T).$
\begin{itemize}
\item $DS\ \underline{TC} = 0,$ for $S\not\geq T;$ \item $DS\
\underline{TC} = \theta^{-+}_{ST}\ \underline{DC},$ for
$\sh(S)=\sh(T).$
\end{itemize}
Notice that $\theta^{-+}_{ST}$ are uniquely determined integer
coefficient, since $\underline{DC} \neq 0.$
\end{lemma}
For the proof see \cite{Regonati}.

The coefficients $\theta^{-+}_{ST}$ are called {\it  symmetry
transition coefficients} and turn out to be a generalization of
the `` D\' esarm\' enien coefficients" \cite{DC}.

\begin{lemma}[\cite{BRT4}]\label{lemma: symmetry coefficients}
For every pair of standard letter tableaux $S, T$ of the same
content, with $\sh(S)=\sh(T),$ we have
$$
\theta^{-+}_{ST} 
\begin{cases}
   = 0& \text{if }  S\not\geq T,\\
\neq 0& \text{if }   S =       T.\\
\end{cases}
$$
Moreover, each diagonal coefficient $\theta^{-+}_{SS}$ is, up to a
sign, the product of the factorials of the multiplicities of
positive symbols in each row and of negative symbols in each
column of the tableau $S.$
\end{lemma}
For the proof see \cite{Regonati}.

We have also the following nondegeneracy result.

\begin{lemma}[\cite{BRT4}]\label{lemma: nondegeneracy} 
Let $C, C_1$ be co-Deruyts tableaux and $D_1, D$ be Deruyts
tableaux, all of the same shape $\lambda \vdash n.$ Then
$$
CD_1\ \underline{D_1C_1}\ C_1D = (-1)^{(^n_2)} h_\lambda\
\underline{CD},
$$
where $h_\lambda$ is a positive coefficient; indeed, it equals the
product of the hook lengths of the shape $\lambda.$
\end{lemma}
For the proof, see Subsection~18.3.

All these results have many variations.

For example, interchanging the role of $C$ and $D,$ we have the
following analogue of Lemma~\ref{lemma: -+ triangulariy}.

\begin{lemma}\label{lemma: +- triangularity}
Let $S, T$ be standard letter tableaux of the same content, $C$ a
virtual co-Deruyts letter tableau, $D$ a virtual Deruyts place
tableau, with $\sh(C)=\sh(S)$ and $\sh(D)=\sh(T).$
\begin{itemize}
\item $CS\ \underline{TD} = 0,$ for $S\not\leq T;$ \item $CS\
\underline{TD} = \theta^{+-}_{ST}\ \underline{CD},$ for
$\sh(S)=\sh(T).$
\end{itemize}
\end{lemma}

Notice that the two types of symmetry transition coefficients are
related by
$$
\theta^{+-}_{ST} = \theta^{-+}_{TS}.
$$

As another example, we have the following operator analogue of
Lemma~\ref{lemma: -+ triangulariy}.

\begin{lemma}[\cite{BRT4}]\label{lemma: -+ operator triangulariy} 
Let $S, T$ be standard letter tableaux of the same content, $D$ a
virtual Deruyts letter tableau, $C$ a virtual co-Deruyts letter
tableau, with $\sh(D)=\sh(S)$ and $\sh(C)=\sh(T).$
\begin{itemize}
\item $DS\ TC = 0,$ for $S\not\geq T;$ \item $DS\ TC =
\theta^{-+}_{ST}\ DC,$ for $\sh(S)=\sh(T).$
\end{itemize}
The coefficients $\theta^{-+}_{ST}$ are indeed the symmetry
transition coefficients, as before.
\end{lemma}



\subsection{The standard basis}


On the set of all the letter tableaux of a given content, we
consider the linear order in which
$$
P \leq P' \quad \Leftrightarrow \quad
\begin{array}{c}
\sh(P) \leq_l \sh(P')\\
w(P) \geq_l w(P')
\end{array},
$$
where the shapes and the words are compared in the lexicographic
order. We recall that this order, restricted to standard tableaux,
is a linear extension of the partial order defined above (see
Subsection~5.5). We consider an analogous order on the set of all
the place tableaux of a given content.

On the set of all the pairs $(P, Q),$ where $P$ ranges over the
letter tableaux of a given content, with sum of multiplicities
$n,$ where $Q$ ranges over the place tableaux of a given content,
with sum of multiplicities $n,$ and $\sh(P) = \sh(Q),$ we consider
the partial order
$$
(P, Q) \leq (P', Q') \quad \Leftrightarrow \quad
\begin{array}{c}
\sh(P) = \sh(Q) \leq_l \sh(P') = \sh(Q')\\
w(P) \geq_l w(P')\\
w(Q) \geq_l w(Q')
\end{array}.
$$

\begin{theorem}[\cite{GRS}]\label{theorem: standard expansion of bitableaux} 
Every bitableau $(P\,|\,Q)\in Super_n[\mathcal{L}|\mathcal{P}]$ can be
written as a linear combination, with rational coefficients,
$$
(P\,|\,Q) = \sum_{S,T\ \text{standard}} c_{S,T}\ (S\,|\,T)
$$
of standard bitableaux $(S\,|\,T)\in
Super_n[\mathcal{L}|\mathcal{P}],$ where the standard tableaux $S$
have the same content as the content of the tableau $P,$ the
standard tableaux $T$ have the same content of the tableau $Q,$
and $(P,Q) \leq (S,T).$
\end{theorem}

\begin{proof}(Sketch)
We consider the following rewriting rules on a bitableau $(A\,|\,B)$:
\begin{enumerate}
\item if the bitableau $(A\,|\,B)$ is not row-ordered, rewrite it as a
row-ordered bitableau $\pm (A'\,|\,B');$ \item if the row-ordered
bitableau $(A\,|\,B)$ has a column violation in the tableau $A,$
perform the following process. Notice that in this case $(A\,|\,B)$
has a factor of the form
\begin{equation}
\left(
\begin{array}{lllllll}
u_1& \ldots& u_p& v_{p+1}  & v_{p+2}& \ldots& v_{m}\\
v_1& \ldots& v_p& v^*_{p+1}& w_{p+2}& \ldots& w_{n}
\end{array}
\right| \left.
\begin{array}{lll}
x_1& \ldots & x_m\\
y_1& \ldots & y_n
\end{array}
\right), 
\tag*{$(*)$}
\end{equation}
where
$$
v_1 \leq \ldots \leq v_p \leq v^*_{p+1} \leq v_{p+1} \leq v_{p+2}
\leq \ldots v_{m},
$$
and $v^*_{p+1} < v_{p+1}$ or $v^*_{p+1} = v_{p+1},$ $v^*_{p+1}$ a
positive letter. Consider the straightening law
$$
\sum_{(v)} \left(
\begin{array} {l}
u v_{(1)} \\
v_{(2)} w
\end{array}
\right| \left.
\begin{array} {l}
x \\
y
\end{array}
\right) = (-1)^{|u||v|} \sum_{{(u)}\ {(y)}} (-1)^{l(u_{(2)})}
\left(
\begin{array} {l}
v u_{(1)} \\
u_{(2)} w
\end{array}
\right| \left.
\begin{array} {l}
x y_{(1)} \\
y_{(2)}
\end{array}
\right),
$$
where
\begin{align*}
u &= u_1 \cdots u_p, \\
v &= v_1 \cdots v_p v^*_{p+1} v_{p+1} v_{p+2} \cdots v_{m}, \\
w &= w_{p+2} \cdots w_{n},\\
x &= x_1 \cdots x_m, \\
y &= y_1 \cdots y_n; 
\end{align*}
Notice that the bitableau $(*)$ appears in the left-hand side with
a nonzero integral coefficient, thus it can be rewritten as a
linear combination, with rational coefficients, of the other
bitableaux in the identity; notice that these are strictly bigger
than $(*)$ in the order of pairs of tableaux. Then rewrite the
bitableau $(A\,|\,B)$ by replacing its factor $(*)$ by this linear
combination. \item if the row-ordered bitableau $(A\,|\,B)$ has a
column violation in the tableau $B,$ perform a process analogous
to the previous one. Use the variation of the straightening law
obtained by interchanging the roles of letters and places.
\end{enumerate}
Notice that each rule replaces a nonstandard bitableau $(A\,|\,B)$
with a linear combination, with rational coefficients, of
bitableaux $(A'\,|\,B')$ where $A'$ has the same content as the
content of $A,$ $B'$ has the same content as the content of $B,$
and $(A,B) < (A',B').$ Starting with a bitableau $(P\,|\,Q),$ the
iteration of these rewriting rules gives, in a finite number of
steps, a rewriting of $(P\,|\,Q)$ as a linear combination, with
rational coefficients, of standard bitableaux $(S\,|\,T),$ where $S$
has the same content as the content of $P,$ $T$ has the same
content as the content of $Q,$ and $(P,Q) \leq (S,T').$
\end{proof}


\vskip 0.5cm

\begin{theorem}[\sc The Grosshans, Rota, Stein standard basis theorem for
{$Super_n[\mathcal{L}|\mathcal{P}]$}, \cite{GRS}]
\label{theorem: standard basis}

The following set is a basis of
$Super_n[\mathcal{L}|\mathcal{P}]$:
$$
\{ (S\,|\,T);\ S\in Stab_{\lambda}(\mathcal{L}),\ T\in
Stab_{\lambda}(\mathcal{P}),\ \lambda \vdash n, \ \lambda \in
H(\mathcal{L})\cap H(\mathcal{P}) \},
$$
where $H(\mathcal{L})$ and $H(\mathcal{P})$ are the hook sets
defined in Subsection~5.4.
\end{theorem}

\begin{proof}
The standard bitableaux whose shape are partitions of $n$ span
$Super_n[\mathcal{L}|\mathcal{P}]$ since they span a subspace
containing all the bitableaux whose  shapes are partition of $n,$;
among these tableaux there are
 all the bitableaux of shape $1,1,
\ldots,1$ which are the monomials of order $n.$

The standard bitableaux whose shape is a  partition of $n$ are
linearly independent. Assume, for the purpose of contradiction,
that there is a nontrivial linear relation
$$
\sum c_{ST}\ (S\,|\,T) = 0
$$
among standard bitableaux. Let $S_0$ be minimal among the letter
tableaux $S$ such that $c_{ST} \neq 0$ for some place tableaux
$T,$ and let $T_0$ be minimal among the place tableaux $T$ such
that $c_{S_0T} \neq 0.$ Denote by $\lambda_0$ the shape of $S_0$
and $T_0,$ let $D_1$ be a letter Deruyts tableau of shape
$\lambda_0,$ and let $D_2$ be a place Deruyts tableau of shape
$\lambda_0.$

By applying the polarization monomials $ D_1S_0 $ and $ T_0D_2 $
on both sides of the nontrivial relation, we get the contradiction
\begin{align*}
0 &= D_1S_0\ \left( \sum_{S,T} c_{ST}\ (S\,|\,T) \right)\ T_0D_2
\\
&=\sum_{S,T} c_{ST}\ D_1S_0\ \underline{SC}\ CT\ T_0D_2 \\
&= \sum_{S
\leq S_0,\ T \leq T_0} c_{ST}\ D_1S_0\ \underline{SC}\ CT\ T_0D_2
\\&=
\theta^{-+}_{S_0S_0}\ c_{S_0T_0}\ \theta^{+-}_{T_0T_0}\ D_1C\
\underline{CD_2} = \theta^{-+}_{S_0S_0}\ c_{S_0T_0}\
\theta^{+-}_{T_0T_0}\ (D_1\,|\,D_2)\\
& \neq 0.
\end{align*}
\end{proof}

An alternative proof of the linear independence of
standard bitableaux follows from the superalgebraic
version of the Robinson--Schensted correspondence
\cite{BSV}, \cite{LNS}.


\subsection{An invariant filtration associated to the standard basis}


Consider the  linear order --- defined in Subsection~5.5 --- on the
place standard tableaux whose shapes are  partitions of $n.$

We recall that $ Q < Q' $ if and only if
$$
\begin{array}{lll}
  & \sh(Q) <_l \sh(Q')&                    \\
or& \sh(Q) = \sh(Q')  & and\ w(Q) >_l w(Q').
\end{array}
$$
Let
$$
T_1, T_2, \ldots, T_f
$$
be the list of the standard place tableaux, whose shapes are
partitions of $n,$ in ascending order, and let
$$
Super_n[\mathcal{L}|\mathcal{P}] = V_0 \supset V_1 \ldots \supset
V_f = (0)
$$
be the chain of the subspaces
$$
V_i = \langle (S\,|\,T);\ S\ \text{standard},\ T >T_i \rangle.
$$
Each subspace $V_i$ is a $pl(\mathcal{L})$-submodule; indeed for
any letter polarization ${\mathcal D},$ and any basis element
$(S\,|\,T),$ with $S,T$ standard and $T >T_i,$ we have
$$
{\mathcal D}\ (S\,|\,T) = \sum_{P} (P\,|\,T) = \sum_{P}\ \sum_{U,V\
\text{standard}} c_{UV}^P (U\,|\,V),
$$
where, by the standard expansion theorem, $ V \geq T >T_i. $

Thus, the chain $ Super_n[\mathcal{L}|\mathcal{P}] = V_0 \supset
V_1 \ldots \supset V_f = (0) $ is indeed a
$pl(\mathcal{L})$-invariant filtration of
$Super_n[\mathcal{L}|\mathcal{P}].$ In the associated graded
module
$$
 {V_0}/{V_1} \oplus  {V_1}/{V_2} \oplus \cdots \oplus
{V_{f-1}}/{V_f}
$$
each summand $V_{i-1}/V_i$ has a basis $ \{(S\,|\,T_i) + V_i; S \
\text{standard}, \sh(S) = \sh(T_i)\}$.

An analogous argument gives a $pl(\mathcal{P})$-invariant
filtration of $Super_n[\mathcal{L}|\mathcal{P}].$



\section{Clebsch--Gordan--Capelli Expansions}


\subsection{Right symmetrized bitableaux}


For every $\lambda \vdash n$ and every $T \in Tab(\mathcal{L}),$
$U \in Tab(\mathcal{P}),$ with $\sh(T)=\lambda = \sh(U),$ we
define the {\it right symmetrized bitableau} $\big(T\,\big|\,\fbox{$U$}\big) \in
Super_n[\mathcal{L}|\mathcal{P}]$ by setting
\begin{align*}
\big(T\,\big|\,\fbox{$U$}\big) &= {\mathcal D}_{TC} \ \langle C\,|\, D\rangle\
_{DU}{\mathcal D}
\\
&= TC\ \underline{CD}\ DU
\end{align*}
where $C$ is any virtual tableau of co-Deruyts type and $D$ is any
virtual tableau of Deruyts type, all of shape $\lambda.$


A quite useful, but not trivial, fact is that right symmetrized
bitableaux admit different equivalent definitions.

\begin{proposition}[\cite{Brini1, BRT3, BRT4}]\label{proposition: threefold definition of right symmetrized bitableaux} 
For every $\lambda \vdash n$ and every $T \in Tab(\mathcal{L}),$
$U \in Tab(\mathcal{P}),$ with $\sh(T)=\lambda = \sh(U),$ we have
\begin{align*}
\big(T\,\big|\,\fbox{$U$}\big) &= \underline{TC_1}\ C_1D\ DU
\\
&= TC\ \underline{CD}\ DU
\\
&= TC\ CD_1\ \underline{D_1U},
\end{align*}
where $C, C_1$ are any virtual tableaux of co-Deruyts type and $D,
D_1$ are any virtual tableaux of Deruyts type, all of shape
$\lambda.$
\end{proposition}


\begin{example}
In the following, let $|\alpha_i| = |\alpha'_i| = 0$ and
$|\beta_i| = |\beta'_i| = 1$ be virtual symbols.
\begin{align*}
&\left(
\begin{array}{cc}
x & y\\
x & z\\
y
\end{array}
\right. \left| \fbox{$
\begin{array}{cc}
a & b\\
a & c\\
c
\end{array}
$} \ \right) \\
&\kern2.5cm
=
\left<
\begin{array}{cc}
x & y\\
x & z\\
y
\end{array}
\right. \left|
\begin{array}{cc}
\alpha'_1 & \alpha'_1\\
\alpha'_2 & \alpha'_2\\
\alpha'_3
\end{array}
\right>\ _{
\begin{array}{cc}
\alpha'_1 & \alpha'_1\\
\alpha'_2 & \alpha'_2\\
\alpha'_3
\end{array}
\begin{array}{cc}
\beta_1 & \beta_2\\
\beta_1 & \beta_2\\
\beta_1
\end{array}
} \mathcal{D}\ _{
\begin{array}{cc}
\beta_1 & \beta_2\\
\beta_1 & \beta_2\\
\beta_1
\end{array}
\begin{array}{cc}
a & b\\
a & c\\
c
\end{array}
} {\mathcal D} \\
&\kern2.5cm =
{\mathcal D}_ {
\begin{array}{cc}
x & y\\
x & z\\
y
\end{array}
\begin{array}{cc}
\alpha_1 & \alpha_1\\
\alpha_2 & \alpha_2\\
\alpha_3
\end{array}
} \left<
\begin{array}{cc}
\alpha_1 & \alpha_1
\\ \alpha_2 & \alpha_2
\\ \alpha_3
\end{array}
\right. \left|
\begin{array}{cc}
\beta_1 & \beta_2\\
\beta_1 & \beta_2\\
\beta_1
\end{array} \
\right> \ _{
\begin{array}{cc}
\beta_1 & \beta_2\\
\beta_1 & \beta_2\\
\beta_1
\end{array}
\begin{array}{cc}
a & b\\
a & c\\
c
\end{array}
} {\mathcal D} \\
&\kern2.5cm
=
{\mathcal D}_ {
\begin{array}{cc}
x & y\\
x & z\\
y
\end{array}
\begin{array}{cc}
\alpha_1 & \alpha_1\\
\alpha_2 & \alpha_2\\
\alpha_3
\end{array}
}\ {\mathcal D}_ {
\begin{array}{cc}
\alpha_1 & \alpha_1
\\ \alpha_2 & \alpha_2
\\ \alpha_3
\end{array}
\begin{array}{cc}
\beta'_1 & \beta'_2\\
\beta'_1 & \beta'_2\\
\beta'_1
\end{array} \
}\ \left<
\begin{array}{cc}
\beta'_1 & \beta'_2\\
\beta'_1 & \beta'_2\\
\beta'_1
\end{array}
\right. \left|
\begin{array}{cc}
a & b\\
a & c\\
c
\end{array}
\right>.
\end{align*}
\end{example}

Informally speaking, the right symmetrized bitableau
$\big(T\,\big|\,\fbox{$U$}\big)$ is supersymmetric in the rows of $T$ and dual
supersymmetric in the columns of $U$: as a matter of fact,
$\big(T\,\big|\,\fbox{$U$}\big)$ is zero if $T$ has a row repetition of negative
letters or $U$ has a column repetition of positive places.

Notice that the right symmetrized bitableau $\big(T\,\big|\,\fbox{$U$}\big)$ can
be also regarded as the result of applying a place polarization
operator to a bitableau: more specifically,
$$
\Big(T\,\Big|\,\fbox{$U$}\Big) = (T\,|\,D) _{DU} {\mathcal D},
$$
$D$ any virtual place tableau of Deruyts type, $\sh(U) = \sh(D).$

Therefore, a right symmetrized bitableau is a linear combination
of bitableaux, all of the same shape.


\begin{example}
In the following all the proper symbols are negative and
$|\beta_i| = 1.$
\begin{align*}
&\left(
\begin{array}{cc} x & y
\\ x & z
\\ y
\end{array}
\right. \left| \fbox{$
\begin{array}{cc}
a & b\\
a & c\\
c
\end{array}
$} \ \right) \ = \left(
\begin{array}{cc} x & y
\\ x & z
\\ y
\end{array}
\right. \left|
\begin{array}{cc} \beta_1 & \beta_2
\\ \beta_1 & \beta_2
\\ \beta_1
\end{array}
\right) \ _{\begin{array}{cc} \beta_1 & \beta_2
\\ \beta_1 & \beta_2
\\ \beta_1
\end{array} ,
\begin{array}{cc} a & b
\\ a & c
\\ c
\end{array}}{\mathcal D} \\
&\kern.5cm
= 2 \left( \
\begin{array}{cc} x & y
\\ x & z
\\ y
\end{array}
\right. \left|
\begin{array}{cc} a & b
\\ a & c
\\ c
\end{array}
\ \right) \ +\ 2 \left( \
\begin{array}{cc} x & y
\\ x & z
\\ y
\end{array}
\right. \left|
\begin{array}{cc} a & b
\\ c & c
\\ a
\end{array}
\ \right) \ +\ 2 \left( \
\begin{array}{cc} x & y
\\ x & z
\\ y
\end{array}
\right. \left|
\begin{array}{cc} c & b
\\ a & c
\\ a
\end{array}
\ \right) 
\\
&\kern1cm
+ \ 2 \left( \
\begin{array}{cc} x & y
\\ x & z
\\ y
\end{array}
\right. \left|
\begin{array}{cc} a & c
\\ a & b
\\ c
\end{array}
\ \right) \ +\ 2 \left( \
\begin{array}{cc} x & y
\\ x & z
\\ y
\end{array}
\right. \left|
\begin{array}{cc} a & c
\\ c & b
\\ a
\end{array}
\ \right) \ +\ 2 \left( \
\begin{array}{cc} x & y
\\ x & z
\\ y
\end{array}
\right. \left|
\begin{array}{cc} c & c
\\ a & b
\\ a
\end{array}
 \ \right).
\end{align*}
Indeed, in the case of negatively signed alphabets, our definition
coincides with the classical one \cite{Clausen1}.
\end{example}

\begin{remark}\label{remark: symmetrized Deruyts}
Let $D$ be a {\it Deruyts tableau} of shape $\lambda$ on the
proper place alphabet ${\mathcal P}$.

The {\it right} symmetrized bitableau $\big(T\,\big|\,\fbox{$D$}\big)$ satisfies
the identity:
$$
\Big(T\,\Big|\,\fbox{$D$}\Big) = \tilde \lambda_1! \cdots \tilde \lambda_q! \
(T\,|\,D),
$$
for every $T \in Tab({\mathcal L}).$
\end{remark}


\subsection{Left symmetrized bitableaux}


For every $\lambda \vdash n$ and every $T \in Tab(\mathcal{L}),$
$U \in Tab(\mathcal{P}),$ with $\sh(T)=\lambda = \sh(U),$ we
define the {\it left symmetrized bitableau} $\big(\fbox{$T$}\,\big|\,U\big) \in
Super_n[\mathcal{L}|\mathcal{P}]$ by setting
\begin{align*}
\Big(\fbox{$T$}\,\Big|\,U\Big) &= {\mathcal D}_{TD}\ \langle D\,|\, C\rangle\
_{CU}{\mathcal D}
\\
&= TD\ \underline{DC}\ CU,
\end{align*}
where $D$ is any virtual tableau of Deruyts type and $C$ is any
virtual tableau of co-Deruyts type, all of shape $\lambda.$


A quite useful, but not trivial, fact is that left symmetrized
bitableaux admit different equivalent definitions.


\begin{proposition}[\cite{Brini1, BRT3, BRT4}]
\label{proposition: threefold definition of left symmetrized bitableaux} 
For every $\lambda \vdash n$ and every $T \in Tab(\mathcal{L}),$
$U \in Tab(\mathcal{P}),$ with $\sh(T)=\lambda = \sh(U),$ we have
\begin{align*}
\Big(\fbox{$T$}\,\Big|\,U\Big) &= \underline{TD_1}\ D_1C\ CU
\\
&= TD\ \underline{DC}\ CU
\\
&= TD\ DC_1\ \underline{C_1U}
\end{align*}
where $D, D_1$ are  virtual tableaux of Deruyts type and $C, C_1$ are
virtual tableaux of co-Deruyts type, all of shape $\lambda.$
\end{proposition}




Informally speaking, the left symmetrized bitableau
$\big(\fbox{$T$}\,\big|\,U\big)$ is  dual supersymmetric in the columns of $T$ and
supersymmetric in the rows of $U$: as a matter of fact,
$\big(\fbox{$T$}\,\big|\,U\big)$ is zero if $T$ has a column repetition of
positive letters or $U$ has a row repetition of negative places.

Notice that the left symmetrized bitableau $\big(\fbox{$T$}\,\big|\,U\big))$ can
be also regarded as the result of applying a letter polarization
operator to a bitableau: more specifically,
$$
\Big(\fbox{$T$}\,\Big|\,U\Big) = {\mathcal D}_{TD} (D\,|\,U),
$$
$D$ any virtual letter tableau of Deruyts type, $\sh(T) = \sh(D).$

Therefore, a left symmetrized bitableau is a linear combination of
bitableaux, all of the same shape.


\begin{remark}\label{remark: left symmetrizea Deruyts}
Let $D$ be a {\it Deruyts tableau} of shape $\lambda$ on the
proper letter alphabet ${\mathcal L}$.

The {\it left} symmetrized bitableau $\big(\fbox{$D$}\,\big|\,T\big)$ satisfies the
identity:
$$
\Big(\fbox{$D$}\,\Big|\,T\Big) = \tilde \lambda_1! \cdots \tilde \lambda_q! \
(D\,|\,T),
$$
for every $T \in Tab({\mathcal P}).$
\end{remark}





\subsection{Doubly symmetrized bitableaux}


For every $\lambda \vdash n$ and every $T \in Tab(\mathcal{L}),$
$U \in Tab(\mathcal{P}),$ with $\sh(T)=\lambda = \sh(U),$ we
define the {\it doubly symmetrized bitableau} $\big(\fbox{$T$}\,\big|\,\fbox{$U$}\big)
\in Super_n[\mathcal{L}|\mathcal{P}]$ by setting
$$
\Big(\fbox{$T$}\,\Big|\,\fbox{$U$}\Big)= {\mathcal D}_{TD_1} \ (D_1\,|\,D_2) \ _{D_2U}
{\mathcal D}
$$
where $D_1$ is any virtual (letter) tableau of Deruyts type, $D_2$
is any virtual (place) tableau of Deruyts type, both of shape
$\lambda.$


\subsection{Clebsch--Gordan--Capelli bases of $Super_n[\mathcal{L}|\mathcal{P}]$}


\begin{theorem}[\cite{Brini1, BRT3, BRT4}]
\label{theorem: Clebsch--Gordan--Capelli basis} 
The following sets are bases for
$Super_n[\mathcal{L}|\mathcal{P}]$:
\begin{enumerate}
\item $\big\{ \big(S\,\big|\,\fbox{$T$}\big);\ S\in Stab_{\lambda}(\mathcal{L}),\ T\in
Stab_{\lambda}(\mathcal{P}),\ \lambda \vdash n,\ \lambda \in
H(\mathcal{L})\cap H(\mathcal{P}) \big\};$ 
\item $\big\{ \big(\fbox{$S$}\,\big|\,T\big);\
S\in Stab_{\lambda}(\mathcal{L}),\ T\in
Stab_{\lambda}(\mathcal{P}),\ \lambda \vdash n,\ \lambda \in
H(\mathcal{L})\cap H(\mathcal{P}) \big\};$ 
\item $\big\{
\big(\fbox{$S$}\,\big|\,\fbox{$T$}\big);\ S\in Stab_{\lambda}(\mathcal{L}),\ T\in
Stab_{\lambda}(\mathcal{P}),\ \lambda \vdash n,\ \lambda \in
H(\mathcal{L})\cap H(\mathcal{P}) \big\};$
\end{enumerate}
\end{theorem}

\begin{proof}
We limit ourselves to the set of standard right symmetrized
bitableaux.

This set is linearly independent. Assume, for the purpose of
contradiction, that there is a nontrivial linear relation
$$
\sum c_{ST}\ \Big(S\,\Big|\,\fbox{$T$}\Big) = 0
$$
among standard right symmetrized bitableaux. Let $S_0$ be minimal
among the letter tableaux $S$ such that $c_{ST} \neq 0$ for some
place tableau $T,$ and let $T_0$ be maximal among the place
tableaux $T$ such that $c_{S_0T} \neq 0.$ Denote by $\lambda_0$
the shape of $S_0$ and $T_0,$ let $D_0$ be a letter Deruyts
tableau of shape $\lambda_0,$ and let $C_0$ be a place co-Deruyts
tableau of shape $\lambda_0.$

By applying the polarization monomials $ D_0S_0 $ and $ T_0C_0 $
on both sides of the nontrivial relation, we get the contradiction
\begin{align*}
0 &= D_0S_0\ \left( \sum_{S,T} c_{ST}\ \Big(S\,\Big|\,\fbox{$T$}\Big) \right)\
T_0C_0 \\
&=
\sum_{S,T} c_{ST}\ D_0S_0\ SC\ \underline{CD}\ DT\ T_0C_0\\
& =
\sum_{S \leq S_0,\ T \geq T_0} c_{ST}\ D_0S_0\ SC\ \underline{CD}\
DT\ T_0C_0 \\
&=
\theta^{-+}_{S_0S_0}\ c_{S_0T_0}\ \theta^{-+}_{T_0T_0}\ D_0C\
\underline{CD}\ DC_0 \\
&= (-1)^{(^n_2)} h_{\lambda_0}\
\theta^{-+}_{T_0T_0}\ c_{S_0T_0}\ \theta^{-+}_{S_0S_0}\
\underline{D_0C_0}\\
& \neq 0.
\end{align*}

The linearly independent set of standard right symmetrized
bitableaux spans\break $Super_n[\mathcal{L}|\mathcal{P}],$ since it has
the same cardinality as the basis of standard bitableaux.
\end{proof}




\section{Young--Capelli Symmetrizers and Orthonormal Generators}


\subsection{Young--Capelli symmetrizers}


Let $\lambda \vdash n$ and let $S', S \in Tab(\mathcal{L}),$ with
$ \sh(S')=\lambda = \sh(S).$ The product of letter bitableau
polarization monomials ${\mathcal D}_{S'C} \cdot\ {\mathcal
D}_{CD} \cdot\ {\mathcal D}_{DS}$  defines, by restriction, a
linear operator
$$
\gamma_n\Big(S',\fbox{$S$}\Big) =  {\mathcal D}_{S'C} \cdot\ {\mathcal
D}_{CD} \cdot\ {\mathcal D}_{DS}\ \in End_{\mathbb K}[
Super_n[\mathcal{L}|\mathcal{P}]],
$$
which is independent of the choice of the virtual tableau $C$ of
co-Deruyts type and of the virtual tableau $D$ of Deruyts type,
both of shape $\lambda.$ The operator $\gamma_n\big(S',\fbox{$S$}\big)$ is
called a {\it right Young--Capelli symmetrizer} \cite{Brini2,
BRT3, BRT4}.

\begin{example}
Let $|\alpha_i| = 0$ and $|\beta_i| = 1.$
\begin{multline*}
\gamma_n \left(
\begin{array}{cc} x & y
\\ x & z
\\ y
\end{array}
,\ \fbox{$
\begin{array}{cc} x & y
\\ y & z
\\ z
\end{array} $} \ \right)\\
=
{\mathcal D}_ {\begin{array}{cc} x & y
\\ x & z
\\ y
\end{array}
\ , \
\begin{array}{cc} \alpha_1 & \alpha_1
\\  \alpha_2 & \alpha_2
\\ \alpha_3
\end{array}}
\quad \cdot \quad {\mathcal D}_ {\begin{array}{cc} \alpha_1 &
\alpha_1
\\  \alpha_2 & \alpha_2
\\ \alpha_3
\end{array}
\ , \
\begin{array}{cc} \beta_1 & \beta_2
\\  \beta_1& \beta_2
\\ \beta_1
\end{array}}
\quad \cdot \quad {\mathcal D}_ {\begin{array}{cc} \beta_1 &
\beta_2
\\  \beta_1& \beta_2
\\ \beta_1
\end{array}
\ , \
\begin{array}{cc} x & y
\\ y & z
\\ z
\end{array}}
\end{multline*}

\end{example}

By the metatheoretic significance of the method of virtual
variables (see, e.g., Subsection~6.1), the crucial fact is that
$\gamma_n\big(S',\fbox{$S$}\big)$ belongs to the subalgebra ${\mathcal
B}_n,$ the algebra generated by proper letter polarizations. In
plain words, even though the operator $\gamma_n\big(S',\fbox{$S$}\big)$ is
defined by using virtual variables, it admits presentations
involving only superpolarizations between proper letters.

A right Young--Capelli symmetrizer is called {\it standard} when
both its tableaux are standard.

In an analogous way, we define the {\it left Young--Capelli
symmetrizers}. Let $\lambda \vdash n$ and let $S', S \in
Tab(\mathcal{L}),$ with $\sh(S')=\lambda = \sh(S).$ The product of
letter bitableau polarization monomials ${\mathcal D}_{S'D} \cdot\
{\mathcal D}_{DC} \cdot\ {\mathcal D}_{CS}$ defines, by
restriction, a linear operator
$$
\gamma_n\Big(\fbox{$S'$}, S\Big) = {\mathcal D}_{S'D} \cdot\ {\mathcal
D}_{DC} \cdot\ {\mathcal D}_{CS}\ \in End_{\mathbb K}[
Super_n[\mathcal{L}|\mathcal{P}]],
$$
which is independent of the choice of the virtual tableau $D$ of
Deruyts type and of the virtual tableau $C$ of co-Deruyts type,
both of shape $\lambda.$



\subsection{The Triangularity Theorem}


\begin{theorem}[\cite{Brini2, BRT3, BRT4}]\label{theorem: triangularity of Young--Capelli symmetrizers and symmetrized bitableaux}
The action of standard right
Young--Capelli symmetrizers on standard right symmetrized
bitableaux is given by
$$
\gamma_n\Big(S',\fbox{$S$}\Big) \Big(T\,\Big|\,\fbox{$U$}\Big) = 
\begin{cases}
(-1)^{(^n_2)} h_\lambda \theta^{-+}_{ST} \Big(S'\,\Big|\,\fbox{$U$}\Big),\quad &
\sh(S)= \sh(T) = \lambda, \\ 0 & otherwise,
\end{cases}
$$
where $\theta^{-+}_{ST}$ are the symmetry transition coefficients from
Lemma~\ref{lemma: -+ triangulariy}, thus integers satisfying the
triangularity conditions $\theta^{-+}_{ST}=0$ unless $S \geq T,$
$\theta^{-+}_{ST} \neq 0$ for $S=T,$ and $h_\lambda$ is a positive
integer which depends on the shape $\lambda.$ Furthermore, the
integer $h_\lambda$ equals the product of the hook lengths of the
shape $\lambda$.
\end{theorem}

\begin{proof} (Sketch) \cite{BRT4}
First of all, by definition, we have
$$
\gamma_n\Big(S',\fbox{$S$}\Big) \Big(T\,\Big|\,\fbox{$U$}\Big) = S'C_{1}\ \left(
C_{1}D_{1}\ \left( D_{1}S\quad \underline {TC_{2}}\ \right)
C_{2}D_{2} \right)\ D_{2}U.
$$
We note that:
\begin{align*}
\text{if }D_{1}S\ \underline{TC_{2}} \neq 0 &\text{ then } \sh(D_{1}) \geq
\sh(C_{2});\\
\text{if } C_{1}D_{1}\ D_{1}S\ \underline {TC_{2}}\
C_{2}D_{2} \neq 0 &\text{ then } \sh(C_{1}) \leq \sh(D_{2}).
\end{align*}
Thus, the whole expression is nonzero only if $S$ and $T$ have the
same shape, say $ \sh(S)=\sh(T)=\lambda. $ Under this condition,
we have, using Lemmas~\ref{lemma: -+ triangulariy}, \ref{lemma:
symmetry coefficients} and \ref{lemma: nondegeneracy} from
Subsection~8.2,
\begin{align*}
\gamma_n\Big(S',\fbox{$S$}\Big) \Big(T\,\Big|\,\fbox{$U$}\Big) &= S'C_{1}\ C_{1}D_{1}\ \left(D_{1}S\ \underline{TC_{2}}\right)\ C_{2}D_{2}\ D_{2}U \\
&= \theta^{-+}_{ST} S'C_{1}\ (C_{1}D_{1}\ \underline{D_{1}C_{2}}\ C_{2}D_{2})\ D_{2}U \\
&= (-1)^{(^n_2)} h_\lambda \theta^{-+}_{ST} S'C_{1}\ \underline{C_{1}D_{2}}\ D_{2}U \\
&= (-1)^{(^n_2)} h_\lambda \theta^{-+}_{ST} \Big(S'\,\Big|\,\fbox{$U$}\Big).
\end{align*}
The coefficients $\theta^{-+}_{ST}$ and $h_\lambda$ satisfy the
triangularity and nondegeneracy conditions by Lemmas~\ref{lemma:
symmetry coefficients} and \ref{lemma: nondegeneracy}.

The last assertion will be proved in Subsection~18.3.
\end{proof}

Given any linear extension of the partial order defined above, the
matrix $[\theta^{-+}_{ S,T}]$ is lower triangular with nonzero
integral diagonal entries; the matrix
$$
[\varrho^{-+}_{ S,T}]= [\theta^{-+}_{S,T}]^{-1}
$$
is called the {\it Rutherford matrix}.

\begin{remark}\label{remark: action of Young--Capelli}
The action of standard left Young--Capelli symmetrizers on
standard left symmetrized bitableaux is given by
$$
\gamma_n\Big(\fbox{$S'$}, S\Big) \Big(\fbox{$T$}\,\Big|\, U\Big) = 
\begin{cases}
(-1)^{(^n_2)} h_\lambda \theta^{+-}_{ST} \Big(\fbox{$S'$}\,\Big|\, U\Big),\quad &
\sh(S)= \sh(T) = \lambda, \\ 0 & otherwise,
\end{cases}
$$
where $\theta^{+-}_{ST}$ are the symmetry transition coefficients
defined in Subsection~8.2, thus integers satisfying the
triangularity conditions $\theta^{+-}_{ST}=0$ unless $S \leq T,$
$\theta^{+-}_{ST} \neq 0$ for $S=T,$ and $h_\lambda$ is the
product of the hook lengths of the shape $\lambda$.
\end{remark}


\subsection{Orthonormal generators}



For every $S',S \in Stab(\mathcal{L}),$ with
$\sh(S')=\sh(S)=\lambda\vdash n,$ we define the {\it right
orthonormal generator} $Y_n\big(S',\fbox{$S$}\big) \in {\mathcal B}_n$ by
setting \cite{Brini2}
$$
Y_n\Big(S',\fbox{$S$}\Big) = \frac {(-1)^{(^n_2)}} {h_\lambda} \sum_{ T\in
Stab(\mathcal{L})} \varrho^{-+}_{ST}\ \gamma_n\Big(S',\fbox{$T$}\Big).
$$
From Theorem~\ref{theorem: triangularity of Young--Capelli
symmetrizers and symmetrized bitableaux} and the definitions
above, the next result follows immediately.


\begin{theorem}\label{theorem: action of orthorormal generators on symmetrized bitableaux}
The action of the right orthonormal generators on the standard
right symmetrized bitableaux is given by
$$
Y_n\Big(S',\fbox{$S$}\Big) \Big(T\,\Big|\,\fbox{$U$}\Big) = \delta_{S,T} \Big(S'\,\Big|\,\fbox{$U$}\Big).
$$
Therefore, the orthonormal generators
$$
Y_n\Big(S',\fbox{$S$}\Big) \quad S, \ S' \in Stab_\lambda(\mathcal{L}),
\quad \lambda \in  H(\mathcal{L}) \cap H(\mathcal{P})
$$
form a ${\mathbb K}$-linear basis of the algebra ${\mathcal B}_n.$
\end{theorem}


\begin{remark}\label{remark: left orthonormal}
In an analogous way, we can define the {\it left orthonormal
generators} $ Y_n\big(\fbox{$S'$}, S\big), $ which act orthogonally on
left symmetrized bitableaux.
\end{remark}


\subsection{Factorization properties}


From the operator triangularity in Lemma~\ref{lemma: -+ operator
triangulariy}, we have the following result.

\begin{proposition}[\sc Factorization theorem, \cite{Brini2, BRT3, Regonati}]
\label{proposition: factorization of Young--Capelli symmetrizers}
For any
standard tableau $T,$ $\sh(T)= \lambda,$ with no letter in common
with the tableau $C$ and $D,$ the Young--Capelli symmetrizer
$\gamma_n\big(S',\fbox{$S$}\big)$ can be factorized as follows:
$$
\gamma_n\Big(S',\fbox{$S$}\Big) = \theta^{-+}_{TT}\ S'C\ CT\ TD\ DS.
$$
\end{proposition}

\begin{remark}\label{remark: Capelli operators}
Note that, if the tableau $T$ has no letter in common with the
tableaux $S'$ and $S,$ then the operator $S'C\ CT$ can be
expressed as the product of the Capelli operators relative to
pairs of corresponding rows of $S'$ and $T;$ analogously the
operator $TD\ DS$ can be expressed as the product of the Capelli
operators relative to pairs of corresponding columns of $T$ and
$S.$
\end{remark}


\subsection{Place operators}

Interchanging the roles of letters and places, we can define the
place Young--Capelli symmetrizers
$$
\Big(T, \fbox{$T'$}\Big) _n\gamma, \qquad \Big(\fbox{$T$}, T'\Big) _n\gamma,
$$
and the corresponding place orthonormal generators
$$
\Big(T, \fbox{$T'$}\Big) _nY, \qquad \Big(\fbox{$T$}, T'\Big) _nY,
$$
which act orthogonally on the corresponding symmetrized bitableaux
$$
\Big(U, \fbox{$V$}\Big), \qquad \Big(\fbox{$U$}, V\Big).
$$



\section{Schur and Weyl Modules}


From the results of Sections~9 and 10, one easily infers the
following results.

\subsection{Schur modules}

Let $\lambda \vdash n$ be a partition of $n,$ $\lambda \in
H(\mathcal{L})\cap H(\mathcal{P}).$ Given a standard tableau $T\in
Stab_{\lambda}(\mathcal{P})$, the subspace $\mathcal{S}_{\lambda
T}$ generated by the set
$$
\Big\{\Big(U\,\Big|\,\fbox{$T$}\Big);\ U\in Tab_{\lambda}(\mathcal{L})\Big\}
$$
is called the (letter-){\it Schur module} parametrized by the
place-tableau $T.$

\begin{proposition}\label{proposition: Schur modules}
The Schur modules have the following properties.
\begin{enumerate}
\item $\mathcal{S}_{\lambda T}$ is  a $pl(\mathcal{L})$-invariant
subspace of $Super_n[\mathcal{L}|\mathcal{P}];$ \item the set
$\big\{\big(S\,\big|\,\fbox{$T$}\big);\ S\in Stab_{\lambda}(\mathcal{L})\big\}$ is a
$\mathbb{K}$-linear basis of $\mathcal{S}_{\lambda T};$ \item the
set $\big\{\big(\fbox{$S$}\,\big|\,\fbox{$T$}\big);\ S\in Stab_{\lambda}(\mathcal{L})\big\}$
is a $\mathbb{K}$-linear basis of $\mathcal{S}_{\lambda T};$ \item
$\mathcal{S}_{\lambda T}$ is an irreducible submodule; \item
$\mathcal{S}_{\lambda T}$ and $\mathcal{S}_{\lambda' T'}$ are
isomorphic $pl(\mathcal{L})$-modules if and only if $\lambda =
\lambda'.$
\end{enumerate}
\end{proposition}


\subsection{Weyl modules}

Let $\lambda \vdash n$ be a partition of $n$, $\lambda \in
H(\mathcal{L})\cap H(\mathcal{P})$. Given a standard tableau $T\in
Stab_{\lambda}(\mathcal{P})$, the subspace $\mathcal{W}_{\lambda
T}$ generated by the set
$$
\Big\{\Big(\fbox{$U$}\,\Big|\,T\Big);\ U\in Tab_{\lambda}(\mathcal{L})\Big\}
$$
is called the (letter-){\it Weyl module} parametrized by the
place-tableau $T$.

\begin{proposition}\label{proposition: Weyl modules}
The Weyl modules have the following properties.
\begin{enumerate}
\item $\mathcal{W}_{\lambda T}$ is  a $pl(\mathcal{L})$-invariant
subspace of $Super_n[\mathcal{L}|\mathcal{P}];$ \item the set
$\big\{\big(\fbox{$S$}\,\big|\,T\big);\ S\in Stab_{\lambda}(\mathcal{L})\big\}$ is a
$\mathbb{K}$-linear basis of $\mathcal{W}_{\lambda T};$ \item
$\mathcal{W}_{\lambda T}$ is an irreducible submodule; \item
$\mathcal{W}_{\lambda T}$ and $\mathcal{W}_{\lambda' T'}$ are
isomorphic $pl(\mathcal{L})$-modules if and only if $\lambda =
\lambda'.$
\end{enumerate}
\end{proposition}



\subsection{The Schur--Weyl correspondence}


\begin{proposition}\label{proposition: Weyl-Schur correspondence}
Let $T_1, T_2 \in Stab_{\lambda}(\mathcal{P})$ and consider the
(negative) place Capelli operator
$$
\mathcal{S}_{\lambda T_2} \leftarrow \mathcal{W}_{\lambda T_1}\ :
T_1D\ DT_2.
$$
\begin{itemize}
\item $\big(\fbox{$U$}\,\big|\,T_1\big)\ T_1D\ DT_2 = K_{T_1}
\big(\fbox{$U$}\,\big|\,\fbox{$T_2$}\big),\quad K_{T_1} \in \BK^*,\ 
\text{for all } U\in
Tab_{\lambda}(\mathcal{L});$ \item $T_1D\ DT_2$ is a
$pl(\mathcal{L})$-equivariant isomorphism.
\end{itemize}
\end{proposition}


\begin{proposition}\label{proposition: Schur--Weyl correspondence}
Let $T_1, T_2 \in Stab_{\lambda}(\mathcal{P})$ and consider the
(positive) place Capelli operator
$$
\mathcal{W}_{\lambda T_1} \leftarrow \mathcal{S}_{\lambda T_2}\ :
T_1C\ CT_2.
$$
\begin{itemize}
\item $\big(\fbox{$U$}\,\big|\,\fbox{$T_2$}\big)\ T_2C\ CT_1 =
K_{T_2}\big(\fbox{$U$}\,\big|\,T_1\big),\quad K_{T_2} \in \BK^*,\ 
\text{for all } U\in
Tab_{\lambda}(\mathcal{L});$ \item $T_2C\ CT_1$ is a
$pl(\mathcal{L})$-equivariant isomorphism.
\end{itemize}
\end{proposition}




\section{Decomposition Theorems}


\begin{theorem}[\cite{Brini1, BRT3}]\label{theorem: decomposition of Super} 
We have the following complete decompositions of the semisimple
$pl({\mathcal L})$-module $Super_n[{\mathcal L}|{\mathcal
\mathcal{P}}]$:
\begin{align*}
Super_n[{\mathcal L}|{\mathcal \mathcal{P}}]& =
\bigoplus_{\stackrel{\scriptstyle{\lambda \in H({\mathcal L})\cap
H({\mathcal P})}}{\lambda \vdash n}} \quad
\bigoplus_{\stackrel{\scriptstyle{T \in Stab({\mathcal P})}}
{\sh(T)=\lambda}} S_{\lambda T}
\\
&= \bigoplus_{\stackrel{\scriptstyle{\lambda \in H({\mathcal
L})\cap H({\mathcal P})}}{\lambda \vdash n}} \quad
\bigoplus_{\stackrel{\scriptstyle{T \in Stab({\mathcal P})}}
{\sh(T)=\lambda}} \Big\langle \Big(S\,\Big|\,\fbox{$T$}\Big);\ \ S\in Stab({\mathcal L})
\Big\rangle
\\
&= \bigoplus_{\stackrel{\scriptstyle{\lambda \in H({\mathcal
L})\cap H({\mathcal P})}}{\lambda \vdash n}} \quad
\bigoplus_{\stackrel{\scriptstyle{T \in Stab({\mathcal P})}}
{\sh(T)=\lambda}} \Big\langle \Big(\fbox{$S$}\,\Big|\,\fbox{$T$}\Big);\ \ S\in
Stab({\mathcal L}) \Big\rangle;
\end{align*}
\begin{align*}
Super_n[{\mathcal L}|{\mathcal P}] &=
\bigoplus_{\stackrel{\scriptstyle{\lambda \in H({\mathcal L})\cap
H({\mathcal P})}}{\lambda \vdash n}} \quad
\bigoplus_{\stackrel{\scriptstyle{T \in Stab({\mathcal P})}}
{\sh(T)=\lambda}} W_{\lambda T}
\\
&= \bigoplus_{\stackrel{\scriptstyle{\lambda \in H({\mathcal
L})\cap H({\mathcal P})}}{\lambda \vdash n}} \quad
\bigoplus_{\stackrel{\scriptstyle{T \in Stab({\mathcal P})}}
{\sh(T)=\lambda}} \Big\langle \Big(\fbox{$S$}\,\Big|\,T\Big);\ \ S\in Stab({\mathcal L})
\Big\rangle;
\end{align*}
where the outer sum indicates the isotypic decomposition of the
semisimple module, and the inner sum describes a complete
decomposition of each isotypic component into irreducible
submodules.
\end{theorem}

\begin{remark}\label{remark: invariant filtration}
We recall (see Subsection~8.4) that the basis of standard
bitableaux gives rise to a $pl(\mathcal{L})$-invariant filtration
$$
Super_n[\mathcal{L}|\mathcal{P}] = V_0 \supset V_1 \ldots \supset
V_f = (0),
$$
where $ V_i = \langle (S\,|\,T);\ S\ \text{standard},\ T >T_i \rangle. $
Notice that each term $V_i$ in the invariant filtration admits the
Weyl module $\mathcal{W}_{\lambda T_i}$, $\sh(T_i) = \lambda$, as
a complementary invariant subspace in the preceding term
$V_{i-1}$:
$$
V_{i-1} = V_{i} \oplus \mathcal{W}_{\lambda T_i}.
$$
Therefore, the standard basis theorem must be regarded as a ``weak
form" of the Clebsch--Gordan--Capelli basis theorem.
\end{remark}


\begin{theorem}[\cite{Brini2, BRT3}]\label{theorem: decomposition of B} 
We have the following complete decomposition for the operator
algebra ${\mathcal B}_n$ generated by the letter polarization
operators acting over $Super_n[{\mathcal L}|{\mathcal P}]$:
$$
{\mathcal B}_n= \bigoplus_{\stackrel{\scriptstyle{\lambda \in
H({\mathcal L})\cap H({\mathcal P})}}{\lambda \vdash n}} \quad
\bigoplus_{\stackrel{\scriptstyle{S \in Stab({\mathcal L})}}
{\sh(S)=\lambda}} \Big\langle Y_n\Big(S',\fbox{$S$}\Big);\ \ S'\in
Stab({\mathcal L}) \Big\rangle,
$$
where the outer sum indicates the isotypic decomposition of the
semisimple algebra, and the inner sum describes a complete
decomposition of each simple subalgebra into minimal left ideals.
\end{theorem}

\begin{corollary}[\sc Structure Theorem]\label{corollary: structure of B} 
$$
{\mathcal B}_n \cong \bigoplus_{\stackrel{\scriptstyle{\lambda \in
H({\mathcal L})\cap H({\mathcal P})}}{\lambda \vdash n}}
M_{f_{\lambda}({\mathcal L})}(\BK),
$$
where $M_{f_{\lambda}({\mathcal L})}(\BK)$ is the  full matrix
algebra of square matrices of order $f_{\lambda}({\mathcal L}).$
\end{corollary}






A completely parallel theory holds for the operator algebra
$_n{\mathcal B}$ generated by the proper place polarization
operators acting over $Super_n[{\mathcal L}|{\mathcal P}].$

It should be clear from the preceding discussion how to define
{\it letter left } orthonormal generators $Y\big(\fbox{$S'$}, S\big)$ and
{\it place left} and {\it right } orthonormal generators
$\big(\fbox{$T'$}, T\big)Y$ and $\big(T',\fbox{$T$}\big)Y$.

We have the following ``symmetric" version of 
Theorem~\ref{theorem: action of orthorormal generators on symmetrized
bitableaux}.

\begin{proposition}\label{proposition: letter and place orthonormal generators}
Let $S', S, U \in Stab({\mathcal L})$ and $V, T, T' \in
Stab({\mathcal P})$. Then:
\begin{align*}
Y\Big(S', \fbox{$S$}\Big) \Big(U, \fbox{$V$}\Big) 
\Big(T, \fbox{$T'$}\Big)Y &= \delta_{SU}
\delta_{VT}\Big(S', \fbox{$T'$}\Big),
\\
Y\Big(\fbox{$S'$}, S\Big) \Big(\fbox{$U$}, V\Big) 
\Big(\fbox{$T$}, T'\Big)Y &= \delta_{SU}
\delta_{VT}\Big(\fbox{$S'$}, T'\Big).
\end{align*}
\end{proposition}


As a consequence, we get the double centralizer theorem.
\begin{theorem}[\sc The Double Centralizer Theorem~\cite{Brini2}]
\label{theorem: double centralizer}
The algebras
$$
{\mathcal B}_n,\ _n{\mathcal B} \subset
End_{\BK}(Super_n[{\mathcal L}|{\mathcal P}])
$$
are the centralizers of each other.
\end{theorem}



\section{The Natural Form of Irreducible Matrix Representations of Schur Superalgebras}


Let $\mathcal{L}, \mathcal{P}$ be arbitrary finite signed
alphabets, and $\lambda \vdash n$ a partition of $n.$ In the
following we will write $S_1, S_2, \ldots, S_{p_\lambda}$ to mean
the list of all standard tableaux of shape $\lambda$ over
$\mathcal{L},$ sorted with respect to the linear order defined in
Subsection~5.5. For sake of simplicity, we will write
$$
\theta_{ij}^\lambda \quad \text{in place of} \quad \theta^{-+}_{S_i
S_j},\qquad \text{and} \quad \varrho_{ij}^\lambda \quad \text{in place of} \quad
\varrho^{-+}_{S_i S_j}.
$$


For every operator $G \in {\mathcal B}_n = \oplus_{\lambda \vdash
n}{\mathcal B}_{\lambda},$ we will denote by $G_\lambda$ its
component in the simple subalgebra $B_\lambda$ 
(Theorem~\ref{theorem: decomposition of B}).

On the one hand, we have
$$
Y_n\Big(S_h, \fbox{$S_h$}\Big)\ G\ Y_n\Big(S_k, \fbox{$S_k$}\Big) =
d_{hk}^\lambda(G)\ Y_n\Big(S_h, \fbox{$S_k$}\Big), \qquad
d_{hk}^\lambda(G) \in {\mathbb K};
$$
notice that the coefficients $d_{hk}^\lambda(G)$ are precisely the
coefficients that appear in the expansion of $G_\lambda$ with
respect to the basis $\big\{ Y_n\big(S_h, \fbox{$S_k$}\big);\ h,k =1, 2,
\ldots \, p_\lambda\big\}$ of ${\mathcal B}_\lambda.$

On the other hand, given any standard tableau $T$ of shape
$\lambda$ over $\mathcal{P},$ we have
$$
G \Big(S_k\,\Big|\,\fbox{$T$}\Big) = G_{\lambda} \Big(S_k\,\Big|\,\fbox{$T$}\Big) = \sum_{h}
c_{hk}^T(G)\ \Big(S_h\,\Big|\,\fbox{$T$}\Big), \qquad c_{hk}^T(G) \in {\mathbb K}.
$$
Clearly, we have:
\begin{proposition}\label{proposition: twofold approach to irreducible matrix representations}
For every standard tableau $T$ over $\mathcal{P}$ with
$\sh(T)=\lambda,$ we have
$$
d_{hk}^\lambda(G_\lambda) = d_{hk}^\lambda(G) = c_{hk}^T(G)=
c_{hk}^T(G_\lambda),
$$
for every $G \in {\mathcal B}_n,$ and for every $h,k =1, 2, \ldots
\, p_\lambda.$
\end{proposition}

In the following, we write $c_{hk}^\lambda$ in place of
$c_{hk}^T.$

We remark that, for every  operator $G \in {\mathcal B}_n,$ and
for every
 tableau $S_j \in Stab(\mathcal{L})$ of shape $\lambda$, the following identity holds:
$$
DS_h\ G\ \underline{S_j C} =\ \theta_{hj}^\lambda(G)\ \underline{D
C},
$$
where $\theta_{hj}^\lambda(G)$ is a uniquely determined scalar
coefficient (here, as usual, $D$ denotes any virtual letter
Deruyts tableau of shape $\lambda$ and $C$ denotes any virtual
place co-Deruyts tableau of shape $\lambda$).

Notice that
$$
[\theta_{ij}^\lambda] = [\theta^\lambda_{ij}(I)], \qquad
[\varrho_{ij}^\lambda] = [\theta^\lambda_{ij}(I)]^{-1},
$$
where $I$ denotes the identity in ${\mathcal B}_n.$

\begin{theorem}[\cite{BRT4}]\label{theorem: natural form of irreducible matrix representation} 
We have:
$$
c_{ij}^\lambda(G) = \sum_{h} \varrho_{ih}^\lambda\
\theta_{hj}^\lambda(G),
$$
or, in matrix form:
$$
C^\lambda(G) = \Theta^\lambda(I)^{-1} \times \Theta^\lambda(G).
$$
\end{theorem}

\begin{proof} (Sketch)
We start from the definition of the coefficients
$c^{\lambda}_{hk}(G)$:
$$
G \Big(S_k\,\Big|\,\fbox{$T$}\Big) = \sum_{h=1}^{p_\lambda} c_{hk}^\lambda (G)\
\Big(S_h\,\Big|\,\fbox{$T$}\Big);
$$
we apply the Young--Capelli symmetrizer
$\gamma_n\big(S_l\,\big|\,\fbox{$S_l$}\big)$ to the left-hand side:
\begin{align*}
\gamma_n\Big(S_l\,\Big|\,\fbox{$S_l$}\Big)\ G\ \Big(S_k\,\Big|\,\fbox{$T$}\Big) &= S_lC\ CD\ DS_l\
G\ \underline{S_kC}\ CD\ DT
\\
&= \theta^{\lambda}_{lk}(G)\ S_lC\ CD\ \underline{DC}\ CD\ DT
\\
&= (-1)^{(^n_2)} h_\lambda\ \theta^{\lambda}_{lk}(G)\ S_lC\
\underline{CD}\ DT
\\
&= (-1)^{(^n_2)} h_\lambda\ \theta^{\lambda}_{lk}(G)\
\Big(S_l\,\Big|\,\fbox{$T$}\Big);
\end{align*}
we apply the Young--Capelli symmetrizer
$\gamma_n\big(S_l\,\big|\,\fbox{$S_l$}\big)$ to the right-hand side:
\begin{align*}
\gamma_n\Big(S_l\,\Big|\,\fbox{$S_l$}\Big)\ \sum_{h=1}^{p_\lambda} c_{hk}^\lambda
(G)\ \Big(S_h\,\Big|\,\fbox{$T$}\Big) &= \sum_{h=1}^{p_\lambda} c_{hk}^\lambda
(G)\ \gamma_n\Big(S_l\,\Big|\,\fbox{$S_l$}\Big)\ \Big(S_h\,\Big|\,\fbox{$T$}\Big)
\\
&= (-1)^{(^n_2)} h_\lambda\ \sum_{h=1}^{p_\lambda}
\theta^{\lambda}_{lh}\ c_{hk}^\lambda (G)\ \Big(S_l\,\Big|\,\fbox{$T$}\Big).
\end{align*}
In the end, we have
$$
\theta^{\lambda}_{lk}(G) = \sum_{h=1}^{p_\lambda}
\theta^{\lambda}_{lh}\ c_{hk}^\lambda (G).
$$
\end{proof}

For every $\lambda \vdash n$ such that $\lambda \in H(\mathcal{L})
\cap H(\mathcal{P})$ and every standard place tableau $T$ of shape
$\lambda,$ the module structure ${\mathcal U}(pl(\mathcal{L}))
\cdot {\mathcal S}_T$ on the irreducible Schur module
$\mathcal{S}_T$ induces a surjective algebra morphism
$$
\nu_T :\ {\mathcal U}(pl(\mathcal{L})) \rightarrow End_{\mathbb
K}[{\mathcal S}_T];
$$
by choosing the basis of the standard symmetrized bitableaux
$\big(S_i\,\big|\,\fbox{$T$}\big)$ in ${\mathcal S}_T,$ the morphism $\nu_T$
induces an irreducible matrix representation
$$
\overline \nu_T :\ {\mathcal U}(pl(\mathcal{L})) \rightarrow
M_{p_\lambda}
$$
where, for every ${\mathcal G} \in {\mathcal U}(pl(\mathcal{L})),$
$$
\overline \nu_T ({\mathcal G}) = [ c_{ij}^\lambda (\nu_T
({\mathcal G})) ] = [ c_{ij}^\lambda (G) ],
$$
where $G = \nu_T({\mathcal G})$. Therefore, this irreducible
representation has the matrix form described in the preceding
theorem.

\newpage

\part{Applications}


\section{Decomposition of Tensor Products of Spaces of Symmetric and Skew-Symmetric Tensors}


Let $V$ be a vector space of dimension $m,$ and let $\{x_1,
\ldots, x_m \}$ be a basis of $V.$ Consider a pair of multiindices
$$
I = (i_1, \ldots, i_s) \in {\BZ^+}^s, \quad J = (j_1, \ldots, j_t)
\in {\BZ^+}^t,
$$
and set $ |I| = i_1 + \cdots + i_s, $ $ |J| = j_1 + \cdots + j_t.
$ The space
$$
W^{I, J} = \Lambda^{i_1}(V) \otimes \cdots \otimes
\Lambda^{i_s}(V) \otimes Sym^{j_1}(V) \otimes \cdots \otimes
Sym^{j_t}(V)
$$
is a $gl(m)$-module as well as a $GL(m)$-module, and, by a
standard argument, the operator algebras induced by the action of
$gl(m)$ and $GL(m)$ are the same.

By way of application, we will derive an explicit complete
decomposition result for the  $GL(m)$-module $W^{I, J}.$

Consider the negatively signed letter alphabet $\mathcal{L} =
\mathcal{L}_1 = \{x_1, \ldots, x_m \}$ and the signed place
alphabet $\mathcal{P} =\mathcal{P}_0 \cup \mathcal{P}_1,$ where
$$
\mathcal{P}_0 = \{\varphi_1, \ldots, \varphi_s \}, \quad
\mathcal{P}_1 = \{\psi_1, \ldots, \psi_t \},
$$
and the order is given by
$$
\varphi_1 < \ldots < \varphi_s < \psi_1 < \ldots < \psi_t.
$$

Let $Super^{I,J}[\mathcal{L}|\mathcal{P}]$ be the ``homogeneous"
subspace of $Super[\mathcal{L}|\mathcal{P}]$ spanned by all the
monomials of content $(I,J),$ that is, of content $i_h$ in each
positive place $\varphi_h,$ for every $h = 1, \ldots, s,$ and of
content $j_{k'}$ in each negative place $\psi_k,$ for every $k =
1, \ldots, t.$

Clearly, $Super^{I,J}[\mathcal{L}|\mathcal{P}]$ is  a
$gl(m)$-module as well as a $GL(m)$-module, and again, by a
standard argument ( see, e.g., Subsection~4.5 ), the operator
algebras induced by the action of $gl(m)$ and $GL(m)$ are the
same.

Let
$$
F : W^{I,J} \rightarrow Super^{I,J}[\mathcal{L}|\mathcal{P}]
$$
be the map such that
\begin{multline*}
F(x_{p_{11}} \wedge \cdots \wedge x_{p_{1i_{1}}} \otimes \cdots
\otimes x_{p_{s1}} \wedge \cdots \wedge x_{p_{si_{s}}} \otimes
x_{q_{11}}  \cdots  x_{q_{1j_{1}}} \otimes \cdots \otimes
x_{q_{t1}}  \cdots x_{q_{tj_{t}}}) \\=
(x_{p_{11}} \cdots  x_{p_{1i_{1}}}\,|\,\varphi_1^{i_1})  \cdots
(x_{p_{s1}} \cdots  x_{p_{si_{s}}}\,|\,\varphi_s^{i_s})
(x_{q_{11}}\,|\,\psi_1)  \cdots (x_{q_{1j_{1}}}\,|\,\psi_1) 
\\
 \cdots
(x_{q_{t1}}\,|\,\psi_t)  \cdots (x_{q_{tj_{t}}}\,|\,\psi_t).
\end{multline*}

\begin{proposition}\label{proposition: entangling/disentangling}
The map $F : W^{I,J} \rightarrow
Super^{I,J}[\mathcal{L}|\mathcal{P}]$ is a $GL(m)$-equivariant
isomorphism.
\end{proposition}

By specializing Theorem~\ref{theorem: decomposition of Super}, we
get:

\begin{corollary}\label{corollary: decomposition of symmetric and skew-symmetric tensor spaces}
We have the following complete $GL(m)$-module decomposition
$$
W^{I,J} = \bigoplus_{\stackrel{\scriptstyle{\lambda \vdash
|I|+|J|}} {\lambda \in H(\mathcal{L}) \cap H(\mathcal{P})} } \quad
\bigoplus_{T\in Stab_{\lambda}^{I,J}(\mathcal{P})} F^{-1}[
S_{\lambda T} ],
$$
where the outer sum indicates the isotypic decomposition of the
semisimple module, and the inner sum, which describe a complete
decomposition of each isotypic component into $GL(m)$-irreducible
Schur modules, is over the set $Stab_{\lambda}^{I,J}(\mathcal{P})$
of all standard place tableaux of content $(I, J).$
\end{corollary}

We remark that a weaker version of the preceding result --- just up
to isomorphism --- could be derived by iterated applications of
Pieri's rule.


\section{Letterplace Algebras, Highest Weight Vectors and $sl_m(\BC)$-Irreducible Modules}

\subsection{Representations of $sl_m(\BC)$: basic definitions and results}

Let $sl_m(\BC)$ be the special linear Lie algebra of $m \times m$
{\it traceless} matrices with complex entries. Let $\textbf{h}
\subseteq sl_m(\BC)$ be the {\it Cartan subalgebra} of all
diagonal matrices.

For every $i = 1, \ldots, m$, let $\varepsilon_i$ be the linear
function
$$
\varepsilon_i : gl_m(\BC) \rightarrow \BC
$$
such that
$$
\varepsilon_i(M) = m_{ii},
$$
for every matrix $M = (m_{ij})_{i,j = 1, \ldots, m} \ \in
gl_m(\BC)$.


Thus, $\textbf{h}^{*} = \BC\{\varepsilon_1, \varepsilon_2, \ldots,
\varepsilon_m \} / \langle \varepsilon_1 + \varepsilon_2 + \ldots
+ \varepsilon_m  \rangle$. We often write $\varepsilon_i$ for the
image of $\varepsilon_i$ in $\textbf{h}^{*}$.

Consider the Cartan decomposition
$$
sl_m(\BC) = \textbf{h} \oplus \langle E_{i,j}; \quad i \neq j
\rangle;
$$
the one-dimensional spaces $\langle E_{i,j} \rangle$ are invariant
subspaces of $sl_m(\BC)$ with respect to the adjoint action of the
subalgebra $\textbf{h}$, since $ [H,E_{i,j}] = (\varepsilon_i -
\varepsilon_j)(H) E_{i,j}$ for every $H \in \textbf{h}.$ The
spaces $\langle E_{i,j} \rangle$ are called the {\it root spaces}
and the elements $\varepsilon_i - \varepsilon_j \in
\textbf{h}^{*}$ are called the {\it roots} of the Lie algebra
$sl_m(\BC)$.

As usual, let
$$
\{\varepsilon_i - \epsilon_j; i < j \}
$$
be the set of {\it positive} roots of $sl_m(\BC)$; thus,    $
\{\langle E_{i,j} \rangle; i < j \}$ is the set of positive root
spaces, and $ \{\langle E_{i,j} \rangle; i > j \}$ is the set of
negative root spaces.


Let ${\cal V}$ be a finite-dimensional representation of
$sl_m(\BC)$ and $\varphi$ an element of $\textbf{h}^{*}$;  a
non-trivial subspace ${\cal V}(\varphi) = \{v \in {\cal V}; H(v) =
\varphi(H)v, \ for \ every \ H \in \textbf{h} \}$ is called a {\it
weight subspace} of the representation ${\cal V}$, and $\varphi$
is called the {\it weight} of ${\cal V}(\varphi)$.

It is easy to see that ${\cal V}$ decomposes into the (finite)
direct sum of its weight spaces, in symbols ${\cal V} =
\oplus_\varphi \ {\cal V}(\varphi)$.

A {\it weight vector} is a vector $v \in {\cal V}$ which belongs
to a weight subspace. A {\it highest weight vector} is a weight
vector which is annihilated by the action of any positive root
space.


\begin{proposition}[\rm see, e.g., \cite{FH}]
\begin{enumerate}
\item Every finite-dimensional representation ${\cal V}$ of
$sl_m(\BC)$ possesses a highest weight vector. \item The subspace
${\cal W}$ generated by the images of a highest weight vector $v$
under successive applications of negative root spaces is an
irreducible subrepresentation. \item An irreducible representation
possesses a unique highest weight vector, up to scalar factors.
\item Two irreducible representations ${\cal W}$ and ${\cal W}'$
of $sl_m(\BC)$ are isomorphic if and only if they have the same
highest weight (as an element of $\textbf{h}^{*}$).
\end{enumerate}
\end{proposition}

It follows from assertion 3) of the above proposition that the
{\it highest weight} of an irreducible $sl_m(\BC)$-representation
may be unambiguously defined as the weight of its highest weight
vector.

\subsection{$sl_m(\BC)$-irreducible modules and $gl_m(\BC)$-irreducible modules}

We recall that $gl_m(\BC) = sl_m(\BC) \oplus {\BC}I_m,$ $I_m$ the
identity $m \times m$ matrix (this is an instance of the so-called
{\it Levi decomposition theorem}). The subalgebra ${\BC}I_m$ is
the {\it radical ideal} of $gl_m(\BC)$ and $sl_m(\BC)$ is its
semisimple part. As a matter of fact, $sl_m(\BC)$ is a {\it
simple} Lie algebra and it is also an ideal of $gl_m(\BC)$;
furthermore, the non-trivial ideals of $gl_m(\BC)$ are precisely
${\BC}I_m$ and $sl_m(\BC)$.

The following assertions are special instances of general results
(see, e.g.,  \cite{FH}, \cite{GW}, \cite{Varad}).

\begin{proposition}[\rm see, e.g., \cite{FH}] 
\label{proposition: $gl$ and $sl$ representation}
\begin{enumerate}
\item Every irreducible representation ${\cal V}$ of $gl_m(\BC)$
is of the form $ {\cal V} = {\cal V}_0 \otimes L$, where ${\cal
V}_0$ is an irreducible representation of $sl_m(\BC)$ (i.e., a
representation of $gl_m(\BC)$ that is trivial on the radical
${\BC}I_m$), and $L$ is a one-dimensional representation of
$gl_m(\BC)$. \item Since the subalgebra $sl_m(\BC)$ is a simple
one, it acts on $L$ in the trivial way, and, then,   each element
of $gl_m(\BC)$ acts on $L$ just by multiplying  by a complex
coefficient. \item It follows from the preceding item that any
irreducible representation of $gl_m(\BC)$ restricts to an
irreducible representation of $sl_m(\BC)$, and any irreducible
representation of $sl_m(\BC)$ extends to an irreducible
representation of $gl_m(\BC)$.
\end{enumerate}
\end{proposition}


\begin{claim}
From the preceding proposition, it follows that, if ${\cal W}$ and
${\cal W}'$ are isomorphic $sl_m(\BC)$-modules, it is not in
general true that ${\cal W}$ and ${\cal W}'$ are isomorphic
$gl_m(\BC)$-modules.
\end{claim}











\subsection{Letterplace algebras and representations of $sl_m(\BC)$}
Let $\mathcal{L} = \mathcal{L}_1 = \{x_1, \ldots, x_m \}$ be a
negatively signed alphabet of  letters, and let $\mathcal{P}$ be
any finite $\BZ_2$-graded alphabet of places.

Set $\BK = \BC.$


In the notation of Subsection~4.3,  we have
$$Super[\mathcal{L}|\mathcal{P}] \simeq Sym[V \otimes W_1] \otimes \Lambda[V \otimes W_0],$$
and
$$pl(\mathcal{L}) = pl(V) = gl_m(\BC),$$
the usual Lie algebra of $m \times m$ matrices.

We will describe the connection between our combinatorial approach
and the classical theory (highest weight vectors) of the Lie
module
$$
sl_m(\BC) \cdot Super[\mathcal{L}|\mathcal{P}] \simeq sl_m(\BC)
\cdot Sym[V \otimes W_1] \otimes \Lambda[V \otimes W_0].
$$

Recall that any homogeneous component
$Super_n[\mathcal{L}|\mathcal{P}]$ is a $gl_m(\BC)$-submodule, and
that the action of $gl_m(\BC)$ is implemented by letter
polarization operators; since $\mathcal{L}$ is a trivially
$\mathbb{\BZ}_2$-graded alphabet, these polarization operators are
derivations in the usual sense (Subsection~4.4).



\begin{claim}
The identity matrix $I_m$ acts on the homogeneous component
$Super_n[\mathcal{L}|\mathcal{P}]$  as the polarization operator
$$
{\mathcal D}_ {x_1,x_1} + {\mathcal D}_ {x_2,x_2} + \cdots +
{\mathcal D}_ {x_m,x_m},
$$
and, then, by multiplying each element by the integer $n$.

Thus, every $sl_m(\BC)$-submodule of
$Super_n[\mathcal{L}|\mathcal{P}]$ is a $gl_m(\BC)$-submodule.
\end{claim}




\begin{corollary}
Every irreducible $gl_m(\BC)$-submodule of
$Super_n[\mathcal{L}|\mathcal{P}]$ is an irreducible
$sl_m(\BC)$-submodule.
\end{corollary}


In the case of irreducible $gl_m(\BC)$-submodules of
$Super_n[\mathcal{L}|\mathcal{P}]$, assertion 1) of 
Proposition~\ref{proposition: $gl$ and $sl$ representation} reduces to  a
quite simple construction.
\begin{corollary}
Every irreducible $gl_m(\BC)$-submodule ${\cal V}$ of
$Super_n[\mathcal{L}|\mathcal{P}]$ is of the form ${\cal V} =
{\cal V}_0 \otimes \BC$, where:
\begin{itemize}
\item ${\cal V}_0$ is the irreducible $sl_m(\BC)$-submodule
obtained from ${\cal V}$ by restriction of the action of
$gl_m(\BC)$ to $sl_m(\BC)$. \item $\BC$ is the one-dimensional
$gl_m(\BC)$-module such that the action of $sl_m(\BC)$ is trivial
and $I_m \cdot {\textbf 1} = n{\textbf 1}$.



\end{itemize}
\end{corollary}



\begin{claim}
Let $\underline{\alpha} = (\alpha_1, \alpha_2, \ldots, \alpha_m)
\in \BN^m$ be an $m$-multiindex such that $\alpha_1 + \alpha_2 +
\ldots + \alpha_m = n$. Given a monomial
$$
M = (x_{i_1}\,|\,y_{j_1})(x_{i_2}\,|\,y_{j_2}) \cdots (x_{i_n}\,|\,y_{j_n})
\in Super_n[\mathcal{L}|\mathcal{P}],
$$
we say that $M$ has {\it letter content} $\underline{\alpha}$, and
write $c(M) = \underline{\alpha}$, if
$$
\# \{ x_{i_h} = x_k; h = 1, 2, \ldots, n \}  = \alpha_k,
$$
for every $k = 1, 2, \ldots, m$.

Let
$$
Super_{\underline{\alpha},n}[\mathcal{L}|\mathcal{P}]
$$
be the subspace of $Super_n[\mathcal{L}|\mathcal{P}]$ spanned by
the set
$$
\{ M = (x_{i_1}\,|\,y_{j_1})(x_{i_2}\,|\,y_{j_2}) \cdots
(x_{i_n}\,|\,y_{j_n}); \ c(M) = \underline{\alpha} = (\alpha_1,
\alpha_2, \ldots, \alpha_m) \}.
$$
Note that $Super_{\underline{\alpha},n}[\mathcal{L}|\mathcal{P}]$
may be trivial.

A weight space the $sl_m(\BC)$-module
$Super_n[\mathcal{L}|\mathcal{P}]$ is a non-trivial subspace of
the form $Super_{\underline{\alpha},n}[\mathcal{L}|\mathcal{P}]$,
and its weight is the element
$$
\alpha_1 \varepsilon_1 + \alpha_2 \varepsilon_2 + \cdots +
\alpha_m \varepsilon_m \in \textbf{h}^{*}.
$$

Thus, the weight space decomposition of
$Super_n[\mathcal{L}|\mathcal{P}]$ is given by the following
formula:
$$
Super_n[\mathcal{L}|\mathcal{P}] = \bigoplus_{\underline{\alpha}}
\ Super_{\underline{\alpha},n}[\mathcal{L}|\mathcal{P}],
$$
where the direct sum is over the set of all multiindices
$\underline{\alpha} = (\alpha_1, \alpha_2, \ldots, \alpha_m) \in
\BN^m$ such that the subspaces
$Super_{\underline{\alpha},n}[\mathcal{L}|\mathcal{P}]$ are
non-trivial.




Furthermore, every weight space
$Super_{\underline{\alpha},n}[\mathcal{L}|\mathcal{P}]$ possesses
four classes of bases associated to pairs of standard Young
tableaux. By specializing 
Theorem~\ref{theorem: Clebsch--Gordan--Capelli basis}, 
we infer that the sets:
\begin{itemize}
\item $\{ (S\,|\,T); \ S \in Stab_\lambda(\mathcal{L}),  T \in
Stab_\lambda(\mathcal{P}), \lambda \vdash n, c(S) =
\underline{\alpha} \}, $ 
\item $\big\{ \big(S\,\big|\,\fbox{$T$}\big); \ S \in
Stab_\lambda(\mathcal{L}),  T \in Stab_\lambda(\mathcal{P}),
\lambda \vdash n, c(S) = \underline{\alpha} \big\}, $ 
\item $\big\{
\big(\fbox{$S$}\,\big|\,T\big); \ S \in Stab_\lambda(\mathcal{L}),  T \in
Stab_\lambda(\mathcal{P}), \lambda \vdash n, c(S) =
\underline{\alpha} \big\}, $ 
\item $\big\{ \big(\fbox{$S$}\,\big|\,\fbox{$T$}\big); \ S
\in Stab_\lambda(\mathcal{L}),  T \in Stab_\lambda(\mathcal{P}),
\lambda \vdash n, c(S) = \underline{\alpha} \big\}, $
\end{itemize}
are $\BK$-bases of
$Super_{\underline{\alpha},n}[\mathcal{L}|\mathcal{P}]$, where
$c(S)$ denotes the content of  $S \in Stab_\lambda(\mathcal{L})$
(see Subsection~5.1).
\end{claim}
















\subsection{The action of upper polarizations}
We recall that the finite set of all the standard tableaux over
$\mathcal{L}$ with total content the fixed integer $n$ is meant to
be the partially ordered with respect to the order defined in
Subsection~5.5.

The polarizations ${\mathcal D}_{x_i x_j}$ with $1\leq i \leq j
\leq m$ will be called {\it upper polarizations}; if $i<j$, the
operator ${\mathcal D}_{x_i x_j}$ will be said to be a {\it
strictly upper polarization}.

In the following, we will often use the following identity. For
any Deruyts tableau of shape $\lambda$, we have
$$
\big(\fbox{$D$}\,\big\vert\, T\big) = \bigg(\prod_i \tilde \lambda_i !\bigg)\ (D\,|\,T).
$$



First of all, we establish a triangularity result.
\begin{proposition}\label{proposition: triangularity of negative Capelli bitableaux on bitableaux}
The action of the standard Capelli bitableaux of the type
$$
[\tilde D_\lambda\,|\,\tilde S]_- = \pm D_\lambda D\ DS
$$
with $\sh(D) = \sh(S) \vdash n,$ on the standard bitableaux
$(T\,|\,U),$ with $\sh(T) = \sh(U) \vdash n,$ satisfies the
nondegenerate triangularity conditions
$$
[\tilde D\,|\,\tilde S]_-\ (T\,|\,U) = 
\begin{cases}
c_{S}\ (D\,|\,U)& \text{for}\ S=T,
\\
0 & \text{for}\ S \not\geq T,
\end{cases}
$$
where $c_S$ are nonzero integers.
\end{proposition}

\begin{proof}
By definition, we have
$$
[\tilde D\,|\,\tilde S]_-\ (T\,|\,U) = \pm DD_1\ D_1S\ \underline{TC}\ CU,
$$
for some virtual tableau $D_1$ (Deruyts) and $C$ (co-Deruyts). By
the triangularity and nondegeneracy results of Subsection~8.2,
this expression vanishes for $S \not\geq T,$ while for $S=T$
becomes
$$
DD_1\ D_1S\ \underline{SC}\ CU = \theta^{-+}_{SS}\ DD_1\
\underline{D_1C}\ CU = \pm (\tilde \lambda)!\ \theta^{-+}_{SS}\
\underline{DC}\ CU = c_{S}\ (D\,|\,U),
$$
where $c_S \neq 0.$
\end{proof}

In the following, given a shape $ \lambda = (\lambda_1, \ldots,
\lambda_p) \vdash n, \ \lambda_1 \leq m $, we denote by
$D_\lambda$ the Deruyts tableau of shape $\lambda$ filled, in
order, with the first $\lambda_1$ symbols of the alphabet
$\mathcal{L}$: for example
$$
D_{322} =
\begin{array}{ccc} x_1 & x_2 & x_3
\\  x_1 & x_2
\\  x_1 & x_2
\end{array}.
$$

\begin{proposition}\label{proposition: Capelli bitableaux and upper polarizations}
Let $\lambda \vdash n$ and $S$ be any standard tableau of shape
$\lambda$. Then the action of the Capelli bitableau
$$
[\tilde D_\lambda\,|\,\tilde S]_- = \pm D_\lambda D\ DS
$$
on $Super_n[\mathcal{L}|\mathcal{P}]$ is the same as  the action
of a linear combination of products of upper polarizations.
Furthermore, if $S \neq D_\lambda,$ each term of this linear
combination can be written as a product of upper polarizations,
with a strictly upper polarization as right-most factor.
\end{proposition}

\begin{proof}(Sketch)
The proof is based on three steps.


Step 1. By Proposition~\ref{proposition: Capelli bitableaux and
Capelli rows}, we infer:
\begin{multline*}
[\tilde D_\lambda\,|\,\tilde S]_- = \left[
\begin{array}{l|l}
x_1^{\tilde \lambda_1} & \tilde w_1 \\
x_2^{\tilde \lambda_2} & \tilde w_2 \\
\vdots & \vdots\\
x_{p-1}^{\tilde \lambda_{p-1}} & \tilde w_{p-1} \\
x_p^{\tilde \lambda_p} & \tilde w_p
\end{array}
\right]_- \cong \left[
\begin{array}{l|l}
x_1^{\tilde \lambda_1} & \tilde w_1\\
x_2^{\tilde \lambda_2} & \tilde w_2\\
\vdots & \vdots\\
x_{p-1}^{\tilde \lambda_{p-1}} & \tilde w_{p-1}
\end{array}
\right]_- \left[ x_p^{\tilde \lambda_p} \,|\, \tilde w_p \right]     
\\
- \sum c_{*} \left[
\begin{array}{l|l}
x_1^{\tilde \lambda_1} & \tilde w'_1\  \tilde w_{p,(1)}\\
x_2^{\tilde \lambda_2} & \tilde w'_2\  \tilde w_{p,(2)}\\
\vdots & \vdots\\
x_{p-1}^{\tilde \lambda_{p-1}} & \tilde w'_{p-1}\ \tilde w_{p,(p-1)}\\
x_p^{m} & \tilde w_{p,(p)}\\
\end{array}
 \right]_-,
\end{multline*}
where the sum is taken under the conditions of the same
proposition, and $m < \tilde\lambda_p.$

Since $S$ is a standard tableau, each column word $\tilde w_i$ of
$\tilde S$ contains only letters $x_j$, with $j \geq i.$
Furthermore, if $S \neq D_\lambda,$ there is at least one word
$\tilde w_i$ of $\tilde S$ that  contains a letter $x_j$ with
$j>i.$ Notice that every summand in the preceding identity has the
same properties.

By iterating this identity, one recognizes that $[\tilde
D_\lambda\,|\,\tilde S]$ is $[{\mathcal{L}}\,|\,\mathcal{P}]$-equivalent
to a linear combination  of products of Capelli rows of the form
$$
[x_i \ldots x_i\,|\,w];
$$
if $S \neq D_\lambda,$ in each of these products there is at least
one factor whose right-hand word $w$ contains a letter $x_j$ with
$j>i.$

Step 2. By Theorem~\ref{theorem: Laplace expansion type
identities}, every Capelli row $[x_i \ldots x_i\,|\,w]$ produced by
Step~1 is $[{\mathcal{L}}\,|\,\mathcal{P}]$-equivalent to a linear
combination  of products of upper polarizations. Thus $[\tilde
D_\lambda\,|\,\tilde S]$ is $[{\mathcal{L}}\,|\,\mathcal{P}]$-equivalent
to a linear combination of products of upper polarizations.
Furthermore, if $S \neq D_\lambda,$ each summand in this linear
combination is a product of upper polarizations, and contains at
least a strictly upper polarization as a factor.

Step 3. If $S \neq D_\lambda,$ by iterating the commutator
identity, each of the above products can be turned into a linear
combination of products of upper polarizations, with a {\it
strictly} upper polarization as  right-most factor.
\end{proof}

The elements of $Super_n[\mathcal{L}|\mathcal{P}]$ which are
annihilated by every strictly upper polarization operators are
characterized by the following theorem.

\begin{theorem}\label{theorem: characterization of elements annihilated by strictly upper polarizations}
For any element $F$ of $Super_n[\mathcal{L}|\mathcal{P}],$ the
following statements are equivalent.
\begin{enumerate}
\item $F$ is annihilated by every strictly upper polarization.
\item The expansion of $F$ as a linear combination of standard
bitableaux is of the form
$$
F = \sum_{\lambda \vdash n}\ \sum_{U \in
Stab_\lambda(\mathcal{P})}\ c_{U}\ (D_\lambda\,|\,U).
$$
\item The expansion of $F$ as a linear combination of standard
right symmetrized bi\-tabl\-eaux is of the form
$$
F = \sum_{\lambda \vdash n}\ \sum_{U \in
Stab_\lambda(\mathcal{P})}\ d_{U}\ \Big(D_\lambda\,\Big|\,\fbox {$U$}\Big).
$$
\end{enumerate}
\end{theorem}

\begin{proof}
It is clear that 2) implies 1) and that 3) implies 1). We
limit ourselves to proving
 that 1) implies 2), since the implication $1) \Rightarrow 3)$ follows from a similar argument.
 By way of contradiction, we assume
that there is an element $F$ of $Super_n[\mathcal{L}|\mathcal{P}]$
which is annihilated by all the upper polarizations and has a
standard expansion of the form
$$
F = \sum_{\lambda \vdash n}\  \sum_U\ c_{U}\ (D_\lambda\,|\,U) +
\sum_{T,U}\ c_{TU}\ (T\,|\,U),
$$
where $T$ ranges in a set ${\mathcal T}$ of standard tableaux
different from any tableau $D_\lambda$ and all the coefficients
$c_{TU}$ are nonzero. Let $S$ be any minimal tableau in ${\mathcal
T},$ of shape $\mu,$ say, and consider the Capelli bitableau
$$
[\tilde D_\mu\,|\,\tilde S]_- = \pm D_\mu D\ DS.
$$
By the preceding proposition, $[\tilde D_\mu\,|\,\tilde S]_-$ can be
expressed as a linear combination of products of upper
polarizations, where each product has a strictly upper
polarization as right-most factor. We have
\begin{align*}
0 &= [\tilde D_\mu\,|\,\tilde S]_- \left( \sum_{\lambda \vdash m}\
\sum_U\ c_{U}\ (D_\lambda\,|\,U) + \sum_{T,U}\ c_{TU}\ (T\,|\,U) \right)
\\
&= [\tilde D_\mu\,|\,\tilde S]_- \left( \sum_{T,U}\ c_{TU}\ (T\,|\,U)
\right) \\
&= [\tilde D_\mu\,|\,\tilde S]_- \left( \sum_{T,U:\ S\geq T}\
c_{TU}\ (T\,|\,U) \right)
\\
&= [\tilde D_\mu\,|\,\tilde S]_- \left( \sum_{U}\ c_{SU}\ (S\,|\,U) \right)
\\
&= c_S\ \left( \sum_{U}\ c_{SU}\ (D_\mu\,|\,U) \right),
\end{align*}
where $c_S$ is a nonzero integer, in contradiction with the linear
independence of standard bitableaux.
\end{proof}



In the language of Subsection~15.1, Theorem~\ref{theorem:
characterization of elements annihilated by strictly upper
polarizations} reads as follows:
\begin{corollary}\label{corollary: characterization of highest weight vectors}
For any element $F$ of $Super_n[\mathcal{L}|\mathcal{P}],$ the
following statements are equivalent.
\begin{enumerate}
\item $F$ is a highest weight vector. \item The expansion of $F$
as a linear combination of standard bitableaux is of the form
$$
F =  \sum_{U\in Stab_\lambda(\mathcal{P})}\ c_{U}\ (D_\lambda\,|\,U),
$$
where $\lambda$ is some partition of $n.$ \item The expansion of
$F$ as a linear combination of standard right symmetrized
bi\-tabl\-eaux is of the form
$$
F =  \sum_{U\in Stab_\lambda(\mathcal{P})}\ d_{U}\
\Big(D_\lambda\,\Big|\,\fbox {$U$}\Big),
$$
where $\lambda$ is some partition of $n.$
\end{enumerate}
\end{corollary}



\subsection{The action of lower polarizations}


The polarizations ${\mathcal D}_{x_i x_j}$ with $m\geq i \geq j
\geq 1$ will be called {\it lower polarizations}; if $i>j,$
${\mathcal D}_{ij}$ will be called a {\it strictly lower
polarization}.



Let $\lambda = (\lambda_1, \ldots, \lambda_p) \vdash n$ be a
partition of the integer $n,$ and let $\tilde \lambda = (\tilde
\lambda_1, \ldots, \tilde \lambda_q)$ be its {\it conjugate}
partition. Let $S = (\omega_1, \ldots, \omega_p)$ be a standard
tableau of shape $\lambda$ on the letter alphabet $\mathcal{L} =
\mathcal{L}_1 = \{x_1, \ldots, x_m \},$ and let $\tilde S =
(\tilde \omega_1, \ldots, \tilde \omega_q)$ be its {\it conjugate}
tableau.

For every pair $(i,j),$ with $i = 1, \ldots, q,$ and  $j = 1,
\ldots, m,$ denote by $ c_{j,i} $ the number of occurrences of
$x_j$ in the word $\tilde \omega_i.$

Let
$$
\mathfrak{P}_S = \mathcal{D}^{c_{2,1}}_{x_2,x_1} \cdots
{\mathcal{D}^{c_{m,1}}_{x_m,x_1}}
{\mathcal{D}^{c_{3,2}}_{x_3,x_2}} \cdots
{\mathcal{D}^{c_{m,2}}_{x_m,x_2}} \cdots
{\mathcal{D}^{c_{q+1,q}}_{x_{q+1,q}x_q}} \cdots
{\mathcal{D}^{c_{m,q}}_{x_m,x_q}}.
$$

\begin{proposition}\label{proposition: action of lower polarizations on bitableaux}
For every standard letter tableau $S$ and every standard place
tableau $T,$ both of shape $\lambda,$ we have
$$
\mathfrak{P}_S \Big( \fbox{$D_\lambda$}\,\Big|\,T) \Big) = \bigg(\prod_{i=1}^q \frac
{\tilde \lambda_i !} {c_{i,i}!}\bigg)\ \Big(\fbox{$S$}\,\Big|\,T\Big).
$$
\end{proposition}




\begin{corollary}\label{corollary: modules generated by highest weight vectors}
Denote by $n^-_m$ be the maximal nilpotent subalgebra of
$sl_m(\BC)$ spanned by all strictly lower triangular matrices.
\begin{enumerate}
\item Let $F$ be a highest weight vector of
$Super_n[\mathcal{L}\vert \mathcal{P}]$ whose expansion as a linear
combination of standard bitableaux is of the form
$$
F =  \sum_{U\in Stab_\lambda(\mathcal{P})}\ c_{U}\ (D_\lambda\mid U),
\quad \lambda \vdash n.
$$
Then, the cyclic submodule $ n^-_m \cdot F $ is an irreducible
$sl_m(\BC)$-module, with basis
$$
\bigg\{\sum_{U\in Stab_\lambda(\mathcal{P})}\ c_{U}\ 
\Big(\fbox{$S$}\,\Big|\,U\Big);\
S\in Stab_{\lambda}(\mathcal{L}) \bigg\}.
$$
Furthermore, $F$ is (up to a scalar factor) the unique highest
weight vector of this irreducible $sl_m(\BC)$-module. \item Let
$F'$ be a highest weight vector of
$Super_n[\mathcal{L}|\mathcal{P}]$ whose expansion  as a linear
combination of standard right symmetrized bitableaux  is of the
form
$$
F' =  \sum_{U\in Stab_\lambda(\mathcal{P})}\ d_{U}\
\Big(D_\lambda\,\Big|\,\fbox{$U$}\Big), \quad \lambda \vdash n.
$$
Then, the cyclic submodule $ n^-_m \cdot F' $ is an irreducible
$sl_m(\BC)$-module, with basis
$$
\bigg\{\sum_{U\in Stab_\lambda(\mathcal{P})}\ d_{U}\
\Big(\fbox{$S$}\,\Big|\,\fbox{$U$}\Big);\ S\in Stab_{\lambda}(\mathcal{L})
\bigg\}.
$$
Furthermore, $F'$ is (up to a scalar factor) the unique highest
weight vector of this irreducible $sl_m(\BC)$-module.
\end{enumerate}
\end{corollary}

\begin{proof}
We prove assertion 1. By the preceding proposition, we infer:
$$
n^-_m \cdot F \supseteq \bigg\langle \sum_{U\in
Stab_\lambda(\mathcal{P})}\ c_{U}\ \Big(\fbox{$S$}\,\Big|\,U\Big);\ S\in
Stab_{\lambda}(\mathcal{L}) \bigg\rangle_{\BK}.
$$
On the other hand, since $F$ equals, up to a scalar factor, we have
$$ \sum_{U\in Stab_\lambda(\mathcal{P})}\ c_{U}\
\Big(\fbox{$D_\lambda$}\,\Big|\,U\Big) \in \bigg\langle \sum_{U\in
Stab_\lambda(\mathcal{P})}\ c_{U}\ \Big(\fbox{$S$}\,\Big|\,U\Big);\ S\in
Stab_{\lambda}(\mathcal{L}) \bigg\rangle_{\BK}
$$
and
$$
b^-_m \cdot F \subseteq \bigg\langle \sum_{U\in
Stab_\lambda(\mathcal{P})}\ c_{U}\ \Big(\fbox{$S$}\,\Big|\,U\Big);\ S\in
Stab_{\lambda}(\mathcal{L}) \bigg\rangle_{\BK}.
$$
Finally, the module
$$
\bigg\langle \sum_{U\in Stab_\lambda(\mathcal{P})}\ c_{U}\
\Big(\fbox{$S$}\,\Big|\,U\Big);\ S\in Stab_{\lambda}(\mathcal{L}) 
\bigg\rangle_{\BK}
$$
is a $gl_m(\BC)$-irreducible module and, hence, an
$sl_m(\BC)$-irreducible module.

Suppose now that $\overline{F}$ is another highest weight vector,
$\overline{F} \in n^-_m \cdot F.$ On the one hand, by 
Corollary~\ref{corollary: characterization of highest weight vectors},
$\overline{F}$ is of the form
$$
\overline{F} =  \sum_{U\in Stab_\mu(\mathcal{P})}\ \overline
c_{U}\ (D_\mu\,|\,U),
$$
for some partition $\mu \vdash n,$ and, on the other hand,
$$
\overline{F} =  \sum_{S\in Stab_\lambda(\mathcal{L})}\ a_S \
\sum_{U\in Stab_\lambda(\mathcal{P})}\ c_{U}\ \Big(\fbox{$S$}\,\Big|\,U\Big).
$$
This implies $\lambda = \mu$ and $a_S = 0$ whenever $S \neq
D_\lambda.$ Therefore, $\overline F$ is a scalar multiple of $F.$
\end{proof}

As a matter of fact, Corollary~\ref{corollary: modules generated
by highest weight vectors} describes {\it all} the irreducible
$sl_m(\BC)$-submodules of $Super_n[\mathcal{L}|\mathcal{P}].$ In
the language of Subsection~15.1, we have:


\begin{proposition}\label{proposition: highest weight vectors in irreducible modules}
Let $W$ be an irreducible $sl_m(\BC)$-submodule of
$Super_n[\mathcal{L}|\mathcal{P}]$. Then, there exists a highest
weight vector $F$ such that
$$
W = n^-_m \cdot F = sl_m(\BC) \cdot F.
$$
\end{proposition}

\begin{proof}
Thanks to the claim at the beginning of this section, we can treat
$W$ as an irreducible $gl_m$-submodule. Let
$$
f = \sum_{\lambda \vdash n} \ \sum_{S,T} c_{S,T} \ \Big(S\,\Big|\,\fbox{$T$}\Big),
\qquad S \in Stab_{\lambda}(\mathcal{L}),\ T \in
Stab_{\lambda}(\mathcal{P})
$$
be a non-zero element of $W.$ Let $S$ be a standard tableau such
that there exists at least one coefficient $c_{S,T}$ different
from zero. Then, the element
$$
Y\Big(D_\lambda, \fbox{$S$}\Big)\ f = \sum_{T} c_{S,T} \
\Big(D_\lambda\,\Big|\,\fbox{$T$}\Big) \neq 0
$$
is a highest weight vector that belongs to $W.$ The cyclic
$gl_m$-submodule (equivalently, the cyclic $sl_m(\BC)$-submodule)
of $Super_n[\mathcal{L}|\mathcal{P}]$ generated by this highest
weight vector is a submodule of $W.$ Since $W$ is supposed to be
irreducible, these modules coincide.
\end{proof}

\begin{claim}
The set of all highest weight vectors of a given weight in
$Super_n[\mathcal{L}|\mathcal{P}]$ is a {\it  Schur irreducible
$pl(\mathcal{P})$-module}, namely, a vector subspace of the form
$$
\langle (D_\lambda\,|\,U),\ \ U \in Stab({\mathcal P}) \rangle_{\BK} =
\Big\langle \Big(D_\lambda\,\Big|\,\fbox{$U$}\Big),\ \ U \in Stab({\mathcal P})
\Big\rangle_{\BK},
$$
with $\lambda \vdash n, \ \lambda \in H(\mathcal{L})\cap
H(\mathcal{P}).$ In particular, this means that each
$sl_m(\BC)$-irreducible submodule has a {\it unique} (up to a
scalar factor) highest weight vector, but it can be expressed in
{\it two ways}, namely, by using the basis $\{(D_\lambda\,|\,U),\ \ U
\in Stab({\mathcal P}) \}$ and by using the basis
$\big\{\big(D_\lambda\,\big|\,\fbox{$U$}\big),\ \ U \in Stab({\mathcal P}) \big\}.$
\end{claim}


\subsection{Highest weight vectors and complete decompositions}


By combining 1) and 3) of Theorem~\ref{theorem:
Clebsch--Gordan--Capelli basis} (Clebsch--Gordan--Capelli bases) and
Corollary~\ref{corollary: characterization of highest weight
vectors}, one immediately infers the following result.

\begin{proposition}\label{proposition: maximal sets of linearly independent highest weight vectors}
The following sets are maximal sets of linearly independent
highest weight vectors in $Super_n[\mathcal{L}|\mathcal{P}]$:
\begin{enumerate}
\item $\{ (D_\lambda\,|\,T);\ \lambda \vdash n, \ T\in
Stab_{\lambda}(\mathcal{P}), \ \lambda \in H(\mathcal{L})\cap
H(\mathcal{P})  \}; $ \item $ \big\{ \big(D_\lambda\,\big|\,\fbox{$T$}\big);\ \lambda
\vdash n, \ T\in Stab_{\lambda}(\mathcal{P}), \ \lambda \in
H(\mathcal{L})\cap H(\mathcal{P}) \big \}. $
\end{enumerate}
\end{proposition}

The preceding facts are to be read as follows:
\begin{enumerate}
\item Given a highest weight vector in
$Super_n[\mathcal{L}|\mathcal{P}]$ of the form
$$
(D_\lambda\,|\,T),\ T\in Stab_{\lambda}(\mathcal{P}), \ \lambda \in
H(\mathcal{L})\cap H(\mathcal{P}),
$$
successive applications of the polarization operators $D_{x_ix_j},
\ i > j,$ generate all the standard left symmetrized bitableaux
$$
\Big( \fbox{$S$}\,\Big|\,T\Big), \ S\in Stab_{\lambda}(\mathcal{L}),\ T\in
Stab_{\lambda}(\mathcal{P}), \ \lambda \in H(\mathcal{L})\cap
H(\mathcal{P});
$$
\item Given a highest weight vector in
$Super_n[\mathcal{L}|\mathcal{P}]$ of the form
$$
\Big(D_\lambda\,\Big|\, \fbox{$T$} \Big),\ T\in Stab_{\lambda}(\mathcal{P}), \
\lambda \in H(\mathcal{L})\cap H(\mathcal{P}),
$$
successive applications of the polarization operators $D_{x_ix_j},
\ i > j,$ generate all the standard doubly symmetrized bitableaux
$$
\Big( \fbox{$S$}\,\Big|\, \fbox{$T$} \Big), \ S\in Stab_{\lambda}(\mathcal{L}),\
T\in Stab_{\lambda}(\mathcal{P}), \ \lambda \in H(\mathcal{L})\cap
H(\mathcal{P});
$$
\end{enumerate}
Therefore:
\begin{enumerate}
\item For every $ T\in Stab_{\lambda}(\mathcal{P}), \ \lambda \in
H(\mathcal{L})\cap H(\mathcal{P}),$ the irreducible
$sl_m(\BC)$-submodule generated by the highest weight vector $(
D_\lambda\,|\,T)$ is the Weyl module $W_{\lambda T}$ parametrized by
the place-tableau $T.$ \item For every $ T\in
Stab_{\lambda}(\mathcal{P}), \ \lambda \in H(\mathcal{L})\cap
H(\mathcal{P}),$ the irreducible $sl_m(\BC)$-submodule generated
by the highest weight vector $\big(D_\lambda\,\big|\, \fbox{$T$}\big)$ is the
Schur module $S_{\lambda T}$ pa\-ra\-me\-trized by the place-tableau $T$
(recall that the set $ \big\{\big( \fbox{$S$}\,\big|\, \fbox{$T$} \big), \ S\in
Stab_{\lambda}(\mathcal{L}),\ T\in Stab_{\lambda}(\mathcal{P}), \
\lambda \in H(\mathcal{L})\cap H(\mathcal{P}) \big\}$ is a basis of
$S_{\lambda T}$).
\end{enumerate}

Finally, we have the following results (compare with 
Theorem~\ref{theorem: decomposition of Super} and 
Proposition~\ref{proposition: maximal sets of linearly independent highest
weight vectors}).

\begin{proposition}\label{proposition: highest weight vectors and complete decompositions}
We have:
\begin{enumerate}
\item The complete decomposition of the semisimple
$sl_m(\BC)$-module $Super_n[{\mathcal L}|{\mathcal P}]$:
$$
Super_n[\mathcal{L}|\mathcal{P}] = \bigoplus_\lambda \quad \bigg(
\bigoplus_{T \in {Stab_{\lambda}(\mathcal{P})}} W_{\lambda T} \bigg)
$$
is the complete decomposition corresponding to the maximal set of
linearly independent highest weight vectors in
$Super_n[\mathcal{L}|\mathcal{P}]$:
$$
\{ (D_\lambda\,|\,T);\ \ T\in Stab_{\lambda}(\mathcal{P}), \ \lambda
\in H(\mathcal{L})\cap H(\mathcal{P})  \};
$$
\item The complete decomposition of the semisimple
$sl_m(\BC)$-module $Super_n[{\mathcal L}|{\mathcal P}]$:
$$
Super_n[\mathcal{L}|\mathcal{P}] = \bigoplus_\lambda \quad \bigg(
\bigoplus_{T \in {Stab_{\lambda}(\mathcal{P})}} S_{\lambda T} \bigg)
$$
is the complete decomposition corresponding to the maximal set of
linearly independent highest weight vectors in
$Super_n[\mathcal{L}|\mathcal{P}]$:
$$
\Big\{\Big(D_\lambda\,\Big|\, \fbox{$T$} \Big); \ T\in Stab_{\lambda}(\mathcal{P}), \
\lambda \in H(\mathcal{L})\cap H(\mathcal{P})\Big  \}.
$$
\end{enumerate}

\end{proposition}

\subsection{ Letterplace algebras and complete sets of pairwise non-isomorphic irreducible
$sl_m(\BC)$-representations}

We know from Subsection~15.1 that every irreducible representation
of $sl_m(\BC)$ is\break uniquely determined by its highest weight
vector, or, equivalently, by its highest weight.

From the ``abstract"  representations theory  of $sl_m(\BC)$, we
get the following result (see, e.g., \cite{FH}, \cite{GW},
\cite{Varad}):

\begin{proposition}
Given an irreducible $sl_m(\BC)$-module, its highest weight is of
the form
$$
a_1\varepsilon_1 + a_2\varepsilon_2 + \cdots + a_m\varepsilon_m,
$$
with $a_i \in \BZ$ and $a_1 \geq a_2 \geq \cdots \geq a_m$.
\end{proposition}

Since a weight in $\textbf{h}^{*}$ is defined up to a sum with a
scalar multiple of the linear functional $\varepsilon_1 +
\varepsilon_2 + \cdots + \varepsilon_m$, the preceding proposition
implies the following assertion.

\begin{corollary}
Given an irreducible $sl_m(\BC)$-module, its highest weight is of
the form
$$
a_1\varepsilon_1 + a_2\varepsilon_2 + \cdots +
a_{m-1}\varepsilon_{m-1},
$$
with $a_i \in \BN$ and $a_1 \geq a_2 \geq \cdots \geq a_{m-1}$.
\end{corollary}

Hence, any irreducible $sl_m(\BC)$-representation can be realized
as  Weyl and Schur  submodules of a suitable letterplace algebra.

More precisely, we have the following result.

\begin{proposition}
Let $\mathcal{L} = \mathcal{L}_1 = \{x_1, \ldots, x_m \}$ be a
negatively signed alphabet of proper letters, and let $\mathcal{P}
= \mathcal{P}_1 = \{y_1, \ldots, y_m \}$ be a negatively signed
alphabet of proper places. Let $\lambda = (\lambda_1 \geq
\lambda_2 \geq \cdots \geq \lambda_p)$ be a partition such that
$\lambda_1 \leq m$, and let $\tilde{\lambda} = (\tilde{\lambda}_1
\geq \tilde{\lambda}_2 \geq \cdots \geq \tilde{\lambda}_q)$ be the
conjugate partition of $\lambda$.

Consider the (irreducible) Weyl submodule  $W_{\lambda T}$ and the
(irreducible) Schur submodule  $S_{\lambda T}$ of
$Super[\mathcal{L}|\mathcal{P}]$,
 $T \in {Stab_{\lambda}(\mathcal{P})}$.

The highest weight of both $W_{\lambda T}$ and $S_{\lambda T}$ is
the element
$$
(\tilde{\lambda}_1 - \tilde{\lambda}_m)\varepsilon_1 +
(\tilde{\lambda}_2 - \tilde{\lambda}_m)\varepsilon_2 + \cdots +
(\tilde{\lambda}_m - \tilde{\lambda}_m)\varepsilon_m \in
\textbf{h}^{*},
$$
where $\tilde{\lambda}_s$ is set to be zero whenever $s > q.$
\end{proposition}



\section{Deruyts' Theory of Covariants\\ (after J.~A.~Green)}



Let $m,n$ be positive integers.  In general, given a positive
integer $r$, we write $\underline r$ to mean the linearly ordered
set $\{1,2, \ldots,r \}$.




Let ${\mathcal A} = \{a_1, \ldots, a_m \}$, ${\mathcal X} = \{x_1,
\ldots, x_n \},$ and  ${\mathcal P} = \underline n = \{1,2,
\ldots,n \}$ be  negatively signed  alphabets.

Consider the (positively signed) letterplace alphabets
\begin{align*}
[{\mathcal A}\,|\,\ \underline n] &= \{(a_i\,|\,\mu); \ i = 1, \ldots, m, \
\mu = 1, \ldots, n \},\\
[\underline n\,|\,\ {\mathcal X}] &= \{(\mu\,|\,x_j);\  \mu, \ j = 1,
\ldots, n \},
\end{align*}
and the tensor product of {\it commutative} letterplace
superalgebras
$$
Super[{\mathcal A}|\ \underline n] \otimes  Super[ \underline n |
\ {\mathcal X}].
$$
In the following we  write $ \BK[{\mathcal A}]$
 for
$Super[{\mathcal A}|\ \underline n],$ and $ \BK[{\mathcal X}] $
for $Super[\underline n|\ {\mathcal X}], $
 respectively,  and
$ \BK[{\mathcal A}] \otimes  \BK[{\mathcal X}] $ for the tensor
product $ Super[{\mathcal A}|\ \underline n] \otimes
Super[\underline n | \ {\mathcal X}]. $


\subsection{Left and right actions of $GL_n(\BK)$ on $\BK[{\mathcal A}]$ and $\BK[{\mathcal X}]$}

Let $S = (s_{\mu \nu})$ be a matrix in $GL_n(\BK).$


We define a left action $GL_n(\BK) \circ \BK[{\mathcal A}]$ by
setting,
 for every matrix $S = (s_{\mu \nu}) \in GL_n(\BK)$:
$$
 S \circ (a_i\,|\,\nu) = \sum_{\mu =1}^n \   (a_i\,|\,\mu) \ s_{\mu \nu}, \quad i = 1, \ldots, m, \quad \nu = 1, \ldots,n,
$$
and extending as an algebra endomorphism.








We define a right action $\BK[{\mathcal X}] \circ GL_n(\BK)  $ by
setting:
$$
 (\mu\,|\,x_j) \circ S = \sum_{\nu =1}^n \ s_{\mu \nu} \ (\nu\,|\,x_j), \quad j, \mu = 1, \ldots,n,
$$
and extending as an algebra endomorphism.

Clearly, the actions of $GL_n(\BK)$ on $\BK[{\mathcal A}]$ and
$\BK[{\mathcal X}]$, regarded as actions on the tensor product
$\BK[{\mathcal A}] \otimes \BK[{\mathcal X}]$, {\it commute} and,
therefore, they may be combined to make $\BK[{\mathcal A}] \otimes
\BK[{\mathcal X}]$ into a  $GL_n(\BK) -  GL_n(\BK)$ - bimodule by
the rule:
$$
S \circ (\alpha \otimes \xi) \circ S' = (S \circ \alpha) \otimes
(\xi \circ S'),
$$
for all $S,S' \in GL_n(\BK)$, $\alpha \in \BK[{\mathcal A}]$, $\xi
\in \BK[{\mathcal X}]$.

\vskip 0.3cm

\subsection{Invariants and covariants}

Given an integer $\omega$, an element $\gamma \in \BK[{\mathcal
A}]$ is called an {\it invariant of weight $\omega$} if
$$
S \circ \gamma = |S|^\omega \ \gamma, \quad \text{for all} \ S \in
GL_n(\BK),
$$
$|S|$ the determinant of $S$.

An element $\varphi \in \BK[{\mathcal A}] \otimes \BK[{\mathcal
X}]$ is called a {\it covariant of weight $\omega$} if
$$
S \circ \varphi \circ S^{-1} = |S|^\omega \ \varphi, \quad \text{for
all} \ S \in GL_n(\BK).
$$


\subsection{Isobaric elements and semiinvariants. Weights}


An element $\gamma \in \BK[{\mathcal A}]$ is said to be {\it
isobaric} of {\it weight} $\pi = (\pi_1, \ldots, \pi_n) \in \BN^n$
if the identity
$$
S \circ \gamma  = d_1^{\pi_1} \cdots d_n^{\pi_n} \cdot \gamma
$$
holds for all diagonal matrices $S = Diag(d_1, \ldots, d_n)$,
$d_1, \ldots, d_n \in \BK^*$.


\vskip 0.3cm




The {\it Borel} subgroup $B_n$ of $GL_n(\BK)$ is the subgroup
which consists of all ``upper triangular" (non-singular) $n \times
n$ matrices.




An element $\gamma \in \BK[{\mathcal A}]$ is called a {\it
semiinvariant} of weight $\pi$ if
$$
S \circ \gamma  = d_1^{\pi_1} \cdots d_n^{\pi_n} \cdot \gamma,
$$
for  all  $S \in B_n$, $(d_1, \ldots, d_n) \in ({\BK^*})^n$ the
$n$-tuple of diagonal entries of $S$.

\begin{remark}\label{remark: isobaric and semiinvariants}
 Clearly, semiinvariants are isobaric elements. In particular, a
semiinvariant of weight $\pi$ is a homogeneous polynomial of
degree $d = \pi_1 + \pi_2 + \cdots + \pi_n$ of $\BK[{\mathcal
A}].$
\end{remark}


Note that the remarks in Subsection~15.4 apply to the study of
semiinvariants, since they are homogenous elements of
$\BK[{\mathcal A}];$ the next result is just a reformulation of
Theorem~\ref{theorem: characterization of elements annihilated by
strictly upper polarizations} and Corollary~\ref{corollary:
characterization of highest weight vectors}.

We recall that, given a shape $ \lambda = (\lambda_1, \ldots,
\lambda_p) \vdash d, \ \lambda_1 \leq n, $ we denote by
$D_\lambda$ the Deruyts tableau of shape $\lambda$ filled, in
order, with the first $\lambda_1$ symbols of the alphabet.


\begin{theorem}\label{theorem: characterization of semiinvariants}
Let $\gamma$ be a homogeneous element of degree $d$  of
$\BK[{\mathcal A}].$

The following statements are equivalent.
\begin{enumerate}
\item $\gamma$ is a semiinvariant of weight $\pi = (\pi_1, \pi_2,
\ldots,   \pi_n) $. \item The expansion of $\gamma$ as a linear
combination of standard bitableaux is of the form
$$
\gamma =  \sum_{\sh(V) = \lambda}\ c_{V}\ (V\,|\,D_\lambda), \qquad
\lambda \vdash d, \quad \lambda_1 \leq m,
$$
with $\pi = \tilde \lambda.$ \item The expansion of $\gamma$ as a
linear combination of standard left symmetrized bi\-tabl\-eaux is of
the form
$$
\gamma =  \sum_{\sh(V) = \lambda}\ d_{V}\ \Big(\fbox {$V$}\,\Big|\,D_\lambda\Big),
\qquad \lambda \vdash d, \quad \lambda_1 \leq m,
$$
with $\pi = \tilde \lambda.$
\end{enumerate}

\end{theorem}

\begin{claim}
The preceding result implies that the weight $\pi$ of a
semiinvariant is an $n$-tuple of natural integers $\pi = (\pi_1,
\ldots, \pi_n)$, with $\pi_1 \geq \pi_2 \geq \cdots \geq \pi_n$.
\end{claim}



\subsection{The map $\sigma : \BK[{\mathcal A}]  \rightarrow \BK[{\mathcal A}] \otimes \BK[{\mathcal X}]$}

Given $a_i \in {\mathcal A}$ and $x_j \in {\mathcal X}$, we set:
$$
\langle a_i\,|\,x_j \rangle = \sum_{\mu=1}^n (a_i\,|\,\mu) \otimes
(\mu\,|\,x_j), \quad i = 1, \ldots,m \quad j = 1, \ldots, n.
$$
We explicitly note that the elements $\langle a_i\,|\,x_j \rangle$ are
covariants of weight zero in $\BK[{\mathcal A}] \otimes
\BK[{\mathcal X}].$

Define a map
$$
\sigma : \BK[{\mathcal A}]  \rightarrow \BK[{\mathcal A}] \otimes
\BK[{\mathcal X}]
$$
by setting
$$
\sigma : (a_i\,|\,j) \rightarrow \langle a_i\,|\,x_j \rangle, \quad i = 1,
\ldots,m, \quad j = 1, \ldots, n
$$
and extending as an algebra morphism.


\begin{remark}\label{remark: polynomial function}
Since $\BK$ is an infinite field, we regard an element $\varphi
\in \BK[{\mathcal A}]\otimes \BK[{\mathcal X}]$ as a polynomial
function in the $n^2$ variables $(\mu\,|\,x_j)$ from the space
$M_n(\BK)$ to $\BK[{\mathcal A}]$ (evaluate any ``variable"
$(\mu\,|\,x_j)$ on the entry $h_{\mu j}$ of a matrix $H \in M_n(\BK),
\ \mu, \ j = 1, \ldots, n$).

The following facts follow from the definitions.
\begin{enumerate}
\item Let $\gamma \in \BK[{\mathcal A}]$. Then $(\sigma \gamma)
(S) = S \circ \gamma$, for all $S \in GL_n(\BK)$. \item Let
$\varphi \in \BK[{\mathcal A}] \otimes \BK[{\mathcal X}].$ Then
$$
(\varphi \circ S) (H) =  \varphi (SH) \in \BK[{\mathcal A}],
$$
for  all  $S \in GL_n(\BK), \ H \in M_n(\BK).$
\end{enumerate}
\end{remark}

\begin{theorem}\label{theorem: the map sigma}
The map $\sigma$ is a $\BK$-algebra monomorphism.

Furthermore, $Im [\sigma]$ is the set of all $\varphi \in
\BK[{\mathcal A}] \otimes \BK[{\mathcal X}]$ which satisfy
$$
S \circ \varphi = \varphi \circ S, \quad for \ all \ S \in
GL_n(\BK).
$$
In other words, $Im [\sigma]$ is the set of all covariants of
weight zero in $\BK[{\mathcal A}] \otimes \BK[{\mathcal X}].$
\end{theorem}

\begin{proof}


Notice that, since $\sigma$ is an algebra morphism, it maps any
element $\gamma \in \BK[{\mathcal A}]$ to a covariant of weight
zero.

Since
$$
\sigma (a_i\,|\,j) (S) = \langle a_i\,|\, x_j \rangle (S) = S \circ
(a_i\,|\,j), \qquad \text{for all } S \in GL_n(\BK),
$$
it follows that
$$
(\sigma \gamma) (S) = S  \circ \gamma, \qquad \text{for all } \gamma \in
\BK[{\mathcal A}],\ \text{for all } S \in GL_n(\BK).
$$
In particular, the ``evaluation of the covariant $(\sigma \gamma)
(I)$ at the identity matrix $I \in GL_n(\BK)$" equals $\gamma$, in
symbols:
$$
(\sigma \gamma) (I) = \gamma, \qquad \text{for all } \gamma \in
\BK[{\mathcal A}],
$$
Hence, the map $\sigma$ is injective.

Let $\varphi$ be a covariant of weight zero. By the previous
remark, we know that
$$
(S \circ \varphi)(H) = (\varphi \circ S) (H) = \varphi (SH),
$$
for  all  $S \in GL_n(\BK), \ H \in M_n(\BK).$

By setting $H = I$, we have
$$
\varphi(S)= (S \circ \varphi) (I) = S \circ \varphi(I), \qquad
\text{for all } S\in GL_n(\BK),
$$
with $ \varphi(I) \in \BK[{\mathcal A}].$ Since
$$
\varphi(S) = S \circ \varphi(I) = [\sigma (\varphi(I))](S), \qquad
\text{for all } S\in GL_n(\BK),
$$
and $GL_n(\BK)$ is a Zariski open set in $M_n(\BK),$ this implies
the identity
$$
\varphi = \sigma (\varphi(I)) \quad \text{in}\quad \BK[{\mathcal
A}]\otimes \BK[{\mathcal X}],
$$
and, therefore, the map $\sigma$ is surjective on the subalgebra
of covariants of weight zero.
\end{proof}

If $\varphi$ is a covariant of weight zero, then,  by 
Theorem~\ref{theorem: the map sigma}, there exists a unique element
$\gamma \in \BK[{\mathcal A}]$ such that $\sigma(\gamma) =
\varphi$. The element $\gamma$ is called the {\it source} of the
covariant $\varphi,$ and $\gamma = \varphi(I).$



\subsection{Left and right spans of covariants of weight zero}

Consider an element $\varphi \in \BK[{\mathcal A}] \otimes
\BK[{\mathcal X}],$ and write
$$
\varphi = \alpha_1 \otimes \beta_1 + \cdots + \alpha_r \otimes
\beta_r, \quad \alpha_s \in \BK[{\mathcal A}], \quad \beta_s \in
\BK[{\mathcal X}];
$$
let $r(\varphi)$ be the minimal length of all such expansions for
$\varphi$. If $r = r(\varphi)$, we say that the above expansion is
{\it minimal}. As a matter of fact an expansion is minimal if and
only if $\{\alpha_1, \ldots, \alpha_r \}$ and $\{\beta_1, \ldots,
\beta_r \}$ are linearly independent sets. The vector spaces
$$L(\varphi) = \langle \alpha_1, \ldots, \alpha_r \rangle_{\BK}$$
and
$$R(\varphi) = \langle \beta_1, \ldots, \beta_r \rangle_{\BK}$$
are called the {\it left span} and the {\it right span} of
$\varphi$, respectively.

The following result follows directly from the definitions.

\begin{proposition}\label{proposition: spans of covariants and modules}
Let $\varphi \in \BK[{\mathcal A}] \otimes \BK[{\mathcal X}]$ be a
covariant of weight zero. Then $L(\varphi)$ and $R(\varphi)$ are
left and right $\BK [GL_n(\BK)]$- submodules of $\BK[{\mathcal
A}]$ and $\BK[{\mathcal X}]$, respectively.
\end{proposition}



The following result easily follows from a standard argument (see,
e.g., Green \cite[p.~261]{Green1}).

\begin{proposition}\label{proposition: spans of covariants and cyclic modules}
If $\varphi = \sigma(\gamma)$, $\gamma \in \BK[{\mathcal A}]$,
then
$$
L(\varphi) = \BK [GL_n(\BK)] \circ \gamma,
$$
the cyclic   $\BK [GL_n(\BK)]$- submodule of $\BK[{\mathcal A}]$
generated by $\gamma$.
\end{proposition}


\subsection{Primary covariants and irreducible $GL_n(\BK)$-re\-pre\-sen\-ta\-tions}


A non-zero function $\varphi \in \BK[{\mathcal A}] \otimes
\BK[{\mathcal X}]$ is called a {\it primary covariant} if
$\varphi$ is a covariant of weight zero and $\varphi =
\sigma(\gamma)$, where the source $\gamma$ is a semiinvariant.

\begin{theorem}\label{theorem: primary covariants and irreducible modules}
If $\varphi \in \BK[{\mathcal A}] \otimes \BK[{\mathcal X}]$ is a
primary covariant, then $L(\varphi)$ is an irreducible left $\BK
[GL_n(\BK)]$- submodule of $\BK[{\mathcal A}]$.
\end{theorem}

\begin{proof}
By the preceding proposition,
$$
L(\varphi) = \BK [GL_n(\BK)] \circ \gamma,
$$
and the source $\gamma$ is a semiinvariant of a given weight
$\pi$. Therefore, by Theorem~\ref{theorem: characterization of
semiinvariants}, $\gamma$ is of the form
$$
\gamma =  \sum_{\sh(V) = \lambda}\ c_{V}\ (V\,|\,D_\lambda),
$$
where $\lambda$ is the conjugate shape of $\pi$.

By Corollary~\ref{corollary: modules generated by highest weight
vectors}, the cyclic module $\BK [GL_n(\BK)] \circ \gamma$ is the
irreducible module spanned, as a vector space, by the basis
$$
\bigg\{  \sum_{\sh(V) = \lambda}\ c_{V}\ \Big(V\,\Big|\,\fbox{$T$}\Big); T \ \text{standard},
\ \sh(T) = \lambda \bigg\}.
$$
\end{proof}

\vskip 0.3cm

Given a weight $\pi = (\pi_1 \geq \pi_2 \geq \cdots \geq \pi_n)$,
$\pi_h \in \BN$, consider the semiinvariant
$$
a(\pi) = (D_\lambda\,|\,D_\lambda) \in \BK[{\mathcal A}],
$$
where the left-most $D_\lambda$ denotes the Deruyts tableau of
shape $\lambda$ filled, in  order, with the first letters of
${\mathcal A},$ the right-most $D_\lambda$ denotes the Deruyts
tableau of shape $\lambda$ filled, in  order, with the first
places of $\underline n,$ and $\lambda$ is the conjugate shape of
$\pi$.

\begin{example}
Let $\pi = (3,3,1)$. Then
$$
a(\pi) = (D_{(3,2,2)}\,|\,D_{(3,2,2)}) = \left(
\begin{array}{l|l}
a_1a_2a_3 & 123 \\
a_1a_2 & 12 \\
a_1a_2 & 12 \\
\end{array}
\right).
$$
\end{example}


\begin{proposition}\label{proposition: types of irreducible modules}
We have:
\begin{itemize}
\item Given any semiinvariant $\gamma = \sum_{\sh(V) = \lambda}\
c_{V}\ (V\,|\,D_\lambda) \in \BK[{\mathcal A}]$ of weight $\pi =
\tilde \lambda$, we have the following identity:
$$
\gamma = \left( \sum_{\sh(V) = \lambda}\ c_{V}\
Y\Big(V\,\Big|\,\fbox{$D_\lambda$}\Big) \right) (a(\pi)),
$$
where the ${\mathcal A}$-letter polarization operator
$\sum_{\sh(V) = \lambda}\ c_{V}\ Y\big(V\,\big|\,\fbox{$D_\lambda$}\big)$ is a
$\BK [GL_n(\BK)]$- equivariant operator. 
\item Every non-zero
semiinvariant $\gamma \in \BK[{\mathcal A}]$ of weight $\pi$
generates an irreducible left $\BK [GL_n(\BK)]$- submodule $\BK
[GL_n(\BK)] \circ \gamma$ which is isomorphic to $\BK [GL_n(\BK)]
\circ a(\pi)$.
\end{itemize}

\end{proposition}

The proof follows immediately from Theorem~\ref{theorem:
characterization of semiinvariants} and Theorem~\ref{theorem:
primary covariants and irreducible modules}.









\subsection{The Deruyts--Capelli expansion}


\begin{theorem}\label{theorem: decomposability of left spans}

Let $\gamma$ be an element of $\BK[{\mathcal A}]$, homogeneous of
degree $d \in {\BZ}^+$. Consider the canonical expansion
$$
\gamma = \sum_{\scriptsize \begin{array}{c} \lambda \vdash d \\
\lambda_1 \leq  min (m,n) \end{array}} \sum_{\sh(T)=\lambda} \
 \sum_{\sh(S)=\lambda} \ c_{ST}^\gamma \ \Big(S\,\Big|\,\fbox{$T$}\Big)
$$
with respect to  the Clebsch--Gordan--Capelli basis of standard
right symmetrized bi\-tabl\-eaux, and set
\begin{align*}
\gamma_{ \lambda, T} =  \sum_{\sh(S)=\lambda} \ c_{ST}^\gamma \
\Big(S\,\Big|\,\fbox{$T$}\Big),
\\
\gamma_{ \lambda, T}^{D_\lambda} = \sum_{\sh(S)=\lambda} \
c_{ST}^\gamma \ (S\,|\,D_\lambda),
\end{align*}
for every $\lambda \vdash d$ and $T \in Stab_\lambda(\mathcal P)$.



Then:
\begin{enumerate}
\item $\gamma_{ \lambda, T}$ belongs to the cyclic
$\BK[GL_n(\BK)]$-module generated by $\gamma \in \BK[{\mathcal
A}]$; \item $\gamma_{ \lambda, T}$ belongs to the irreducible
cyclic $\BK[GL_n(\BK)]$-module generated by the semiinvariant
$\gamma_{ \lambda, T}^{D_\lambda}$; \item the semiinvariant
$\gamma_{ \lambda, T}^{D_\lambda}$
 belongs to the cyclic $\BK[GL_n(\BK)]$-module generated
by $\gamma_{ \lambda, T}$; \item the cyclic modules
$\BK[GL_n(\BK)] \circ \gamma_{ \lambda, T}$ and $\BK[GL_n(\BK)]
\circ \gamma_{ \lambda, T}^{D_\lambda}$ are the same.
\end{enumerate}

\end{theorem}

\begin{proof}
By Proposition~\ref{proposition: letter and place orthonormal
generators}, we have:
\begin{enumerate}
\item $\gamma_{ \lambda, T}$ is the image of $\gamma$ under the
operator $\big(T, \fbox{$T$}\big)Y $ induced by the (place) action of
$\BK[GL_n(\BK)]$. \item $\gamma_{ \lambda, T}$ is the image of
$\gamma_{ \lambda, T}^{D_\lambda}$ under  the operator
$\big(D_\lambda, \fbox{$T$}\big)Y $ induced by the (place) action of
$\BK[GL_n(\BK)]$, up to a non-zero scalar factor. \item $\gamma_{
\lambda, T}^{D_\lambda}$ is the image of $\gamma_{ \lambda, T}$
under  the operator $\big(T, \fbox{$D_\lambda$}\big)Y $ induced by the
(place) action of $\BK[GL_n(\BK)]$, up to a non-zero scalar
factor.
\end{enumerate}
\end{proof}

The next reformulation of Theorem~\ref{theorem: decomposability of
left spans} describes the {\it Deruyts--Capelli expansion} of a
polynomial $\gamma \in \BK[{\mathcal A}].$ As a matter of fact,
the first statement is properly the Deruyts Expansion (cfr. Green,
\cite[Theorem~12.1]{Green1}), and the second statement is the core
of the crucial Capelli's refinement of the result of Deruyts
(Capelli's polar expansion formula \cite{CAP}).




\begin{corollary}[\sc The Deruyts--Capelli Expansion]\label{corollary: Deruyts expansion}
Let $\gamma$ be an element of
$\BK[{\mathcal A}]$, $\gamma$ homogeneous of degree $d$. Then:
\begin{enumerate}
\item for every $\lambda \vdash d$ and $T \in
Stab_\lambda(\mathcal P)$, there exist semiinvariants
$$
\gamma_{ \lambda, T}^{D_\lambda},
$$
and elements
$$
\Lambda_{ \lambda, T} \in \BK[GL_n(\BK)],
$$
such that:
$$
\gamma = \sum_{\lambda \vdash d} \sum_{\sh(T)=\lambda} \ \Lambda_{
\lambda, T} \circ \gamma_{ \lambda, T}^{D_\lambda};
$$
\item the semiinvariants
$$
\gamma_{ \lambda, T}^{D_\lambda}
$$
belong to the cyclic $\BK[GL_n(\BK)]$-module generated by $\gamma
\in \BK[{\mathcal A}].$
\end{enumerate}
\end{corollary}

A more explicit formulation and a more direct proof of the
preceding result can be found in \cite{BRT4}.

\subsection{On  the  complete reducibility of  the span of covariants of weight zero.
A ``pre-Schur" description of polynomial $GL_n(\BK)$- irreducible
representations}

In this subsection, we  will show that any left span $L(\varphi)$
can be decomposed into a direct sum of irreducible
representations.

Since $L(\varphi) = \BK [GL_n(\BK)] \circ \gamma$, $\sigma (
\gamma) = \varphi$, the problem can be solved in the algebra
$\BK[{\mathcal A}]$.

Let $\varphi \in \BK[{\mathcal A}] \otimes \BK[{\mathcal X}]$ be a
covariant of weight zero, and let $\gamma \in \BK[{\mathcal A}]$
be its source. Without loss of generality, we assume that $\gamma$
is a ``homogeneous polynomial" of degree $d$.


Let
\begin{align*}
\gamma &= \sum_{\lambda \vdash d} \sum_{\sh(T)=\lambda} \  \gamma_{
\lambda, T}, \quad \gamma_{ \lambda, T} \neq 0,\\
\gamma_{ \lambda, T} &=  \sum_{\sh(S)=\lambda} \ c_{ST}^\gamma
\cdot \Big(S\,\Big|\,\fbox{$T$}\Big),
\end{align*}
be the canonical expression of $\gamma$ as a linear combination of
standard right symmetrized bitableaux.






\begin{corollary}[\sc Decomposition theorem]\label{corollary: decomposition of left spans} 
Let $\varphi \in \BK[\mathcal A] \otimes \BK[\mathcal X]$ be a
covariant of weight zero, and let $\gamma \in \BK[\mathcal A]$ be
its source, that is $\varphi = \sigma (\gamma)$. Then:
\begin{align*} L(\varphi) &= \BK[GL_n(\BK)] \circ \gamma \\
&=
\bigoplus_{\lambda \vdash d} \ \bigg( \sum_{\sh(T)=\lambda} \
\BK[GL_n(\BK)] \circ \gamma_{\lambda T}^{D_\lambda} \bigg) \\
&=
\bigoplus_{\lambda \vdash d} \ \bigg( \sum_{\sh(T)=\lambda} \ L(\sigma(
\gamma_{\lambda T}))\bigg),
\end{align*}
where $L(\sigma(  \gamma_{\lambda T})) = \BK[GL_n(\BK)] \circ
\gamma_{\lambda T}^{D_\lambda}.$

\end{corollary}

\subsection{ Weyl's First Fundamental Theorem}

Let $V$ be a vector space of finite dimension $m$. Since
$char(\BK) = 0$, we may identify the algebra $\BK[{\mathcal A}]$
with the algebra $\BK[V^{\oplus n}]$ of polynomial functions on
the vector space $V^{\oplus n}$, by reading the variable
$(a_i\,|\,\mu)$ as the {\it $i$-th coordinate function} on the
$\mu$-th copy of $V$ in the direct sum $V^{\oplus n}$, for every
$i = 1,2, \ldots , m$, $\mu = 1,2, \ldots, n.$

In plain words, we identify the algebra $\BK[{\mathcal A}]$ with
the algebra $\BK[V^{\oplus n}] = Sym[(V^*)^{\oplus n}].$


We recall the {\it contravariant} action of the general linear
group $GL_m(\BK)$  on
$$\BK[{\mathcal A}] = \BK[V^{\oplus n}] = Sym[(V^*)^{\oplus n}]
$$
is defined in the following way:
$$
(\textbf{g} \circ \gamma) (v_1, \ldots, v_n) =
\gamma(\textbf{g}^{-1}(v_1), \ldots, \textbf{g}^{-1}(v_n)),
$$
for all $\textbf{g} \in GL_m(\BK)$, $\gamma \in \BK[{\mathcal
A}]$,  $(v_1, \ldots, v_n) \in V^{\oplus n}.$

In matrix notation, the {\it diagonal} action of a matrix $T =
(t_{hk}) \in GL_m(\BK)$ on $\BK[{\mathcal A}]$ is given by
$$
 T \circ (a_i\,|\,\mu) = \sum_{k =1 }^m \ \bar{t}_{i k} \ (a_k\,|\,\mu)  , \quad i = 1, \ldots, m, \quad \mu = 1, \ldots, n,
$$
where
$$
( \bar{t}_{hk } ) = T^{-1}.
$$

The algebra $\BK[{\mathcal A}]$ is  both a left $GL_n(\BK)$-module
(see Subsection~16.1) and a left $GL_m(\BK)$-module, and the two
actions clearly commute.



Furthermore, the  action of the general linear group $GL_m(\BK)$
on every homogeneous component of $\BK[{\mathcal A}]$ is
implemented by the algebra of ${\mathcal A}$-letter polarizations,
and the  action of the general linear group $GL_n(\BK)$ is
implemented by the algebra of ${\mathcal P}$-place polarizations,
${\mathcal P} = \underline n = \{1,2, \ldots,n \}.$


\vskip 0.2cm




Let  $m \leq n.$

Informally speaking, the Deruyts--Capelli expansion formula says
that any homogeneous polynomial function $\gamma$ in $n$ vector
variables in dimension $m$ may be expressed as a linear
combination of polarized
 $GL_n(\BK)$-semiinvariants and, by Theorem~\ref{theorem: characterization of
semiinvariants}, these $GL_n(\BK)$-semiinvariants involve only the
first $m^2$ letterplace variables
$$
(a_1\,|\,\mu), \ldots, (a_m\,|\,\mu), \quad \mu = 1, \ldots, m;
$$
furthermore,  these $GL_n(\BK)$-semiinvariants may be obtained, in
turn, by applying place polarization operators to the original
polynomial function $\gamma.$

Since the place polarization process is a $GL_m(\BK)$-invariantive
process (i.e., the actions of $GL_m(\BK)$ and $GL_n(\BK)$
commute), the study of $GL_m(\BK)$-invariant  polynomial functions
in $n$ vector variables in dimension $m$ is reduced to the study
of invariant homogeneous polynomial functions  in $m-1$ vector
variables.

To be precise, we recall the following definitions:
\begin{itemize}

\item Let $G$ be a subgroup of the general linear group
$GL_m(\BK)$.

A polynomial function $\gamma \in \BK[{\mathcal A}] =
\BK[V^{\oplus n}]$ is said to be a (formal) {\it relative
$G$-invariant} if and only if the following condition holds:
$$
(\textbf{g} \circ \gamma) = \lambda(\textbf{g}) \gamma,
$$
for all $\textbf{g} \in G,$ $\lambda(\textbf{g}) \in \BK.$ \item
The {\it bracket}
$$
[j_1, \ldots, j_m], \quad j_h \in {\mathcal P} = \underline n =
\{1,2, \ldots,n \}
$$
is defined as follows:
$$
[j_1, \ldots, j_m] = \det [ (a_i\,|\,j_h ) ]_{i=1, \ldots, m, \ h=1,
\ldots, m,} = (-1)^{\binom m  2} (a_1a_2 \cdots a_m\,|\,j_1j_2 \cdots
j_m).
$$
\end{itemize}

Clearly, any bracket $[j_1, \ldots, j_m]$ is a relative
$G$-invariant, $G$ a subgroup of the general linear group
$GL_m(\BK).$

Let $\gamma \in \BK[{\mathcal A}] = \BK[V^{\oplus n}]$ be a
polynomial function, $\gamma$ homogeneous of degree $d.$  Thanks
to the Deruyts--Capelli expansion formula 
(Corollary~\ref{corollary: Deruyts expansion}), $\gamma$ is expanded in the
following form:
$$
\gamma = \sum_{\scriptsize \begin{array}{l} \lambda \vdash d \\
\lambda_1 \leq m \end{array}} \  \sum_{\sh(T)=\lambda} \ \Lambda_{
\lambda, T} \circ \gamma_{ \lambda, T}^{D_\lambda}, \qquad
\Lambda_{ \lambda, T} \in \BK[GL_n(\BK)], \quad T \in
Stab({\mathcal P}),
$$
where any
$$
\gamma_{ \lambda, T}^{D_\lambda} = \sum_{\sh(S)=\lambda} \
c_{ST}^\gamma \ (S\,|\,D_\lambda)
$$
belongs to the cyclic $\BK[GL_n(\BK)]$-module generated by $\gamma
\in \BK[{\mathcal A}].$

Since the actions of $GL_m(\BK)$ and $GL_n(\BK)$ commute, if
$\gamma$ is a relative $G$-invariant ($G$ a subgroup of
$GL_m(\BK)$), the $GL_n(\BK)$-semiinvariants $\gamma_{ \lambda,
T}^{D_\lambda}$ are relative $G$-invariants.



Note that any $\gamma_{ \lambda, T}^{D_\lambda}$ can be written in
the form
$$
\gamma_{ \lambda, T}^{D_\lambda} = [1, 2, \ldots ,m]^{q_{\lambda}}
\varphi_T,
$$
where
$$
q_{\lambda} = \# \{i;\ \lambda_i = m\},
$$
and
$$
\varphi_T = \pm \sum_S \ c^\gamma_{ST} \ (S^{*}\,|\,D_{\lambda^{*}}),
$$
where $\lambda^{*} = (\lambda_{q_{\lambda}+1},
\lambda_{q_{\lambda}+2}, \ldots )$ and the tableaux $S^{*}$  of
shape $\lambda^{*}$ are obtained from the tableaux $S$ of shape
$\lambda$ by deleting the first $q_{\lambda}$ top rows.

Since $\lambda^{*}_1 < m$, any $\varphi_T$ is a polynomial in the
$m \times {m-1}$ letterplace variables
$$
\begin{array} {cccc}
(a_1\,|\,1), & (a_2\,|\,1), & \ldots, & (a_m\,|\,1)    \\
(a_1\,|\,2), & (a_2\,|\,2), & \ldots, & (a_m\,|\,2)  \\
 \ \vdots & \vdots  &   & \vdots  \  \\
(a_1\,|\,m-1), & (a_2\,|\,m-1), & \ldots, & (a_m\,|\,m-1)  \\
\end{array}
$$
or, equivalently, $\varphi_T$ is a polynomial function
$$
 \varphi_T(v_1, v_2, \ldots, v_{m-1}) \in \BK[V^{\oplus (m-1)}]
$$ in the  vector variables $v_1, v_2, \ldots, v_{m-1}$.

 Hence, we have the following result.





\begin{corollary}[\sc Weyl's First Fundamental Theorem]
Let $G$ be a subgroup of $GL_m(\BK),$ and let $\gamma$ be a
$d$-homogeneous relative $G$-invariant polynomial function in $n$
vector variables, $n\geq m.$ Then,
$$
\gamma = \sum_{\scriptsize \begin{array}{l} \lambda \vdash d \\
\lambda_1 \leq m \end{array}} \ \sum_{T\in Stab_{\lambda}(P)}
\Lambda_{ \lambda, T} \circ ([1, 2, \ldots m]^{q_{\lambda}}
\varphi_T),
$$
where
$$
\Lambda_{ \lambda, T} \in \BK[GL_n(\BK)],
$$
and the $\varphi_T$'s are both relative $G$-invariant polynomial
functions in the  vector variables $v_1, v_2, \ldots, v_{m-1}$ and
$GL_n(\BK)$-semiinvariants.
\end{corollary}

For example, since the $GL_m(\BK)$-invariant polynomial functions
which involve at most $m-1$ vector variables are just the constant
functions (see, e.g., Theorem~\ref{theorem: action of orthorormal
generators on symmetrized bitableaux} and Remark~\ref{remark:
action of Young--Capelli}) and polarizations of  brackets yield
linear combinations of brackets, the following well-known result
immediately follows.

\begin{corollary}
[\sc The first fundamental theorem for vector $GL_m(\BK)$-in\-va\-ri\-ants]
Let $\gamma$ be a relative $GL_m(\BK)$-invariant  polynomial
function on $n$ vector variables, $n\geq m.$ Then $\gamma$ can be
written as a homogeneous polynomial in the brackets
$$
[j_1, \ldots, j_m], \quad j_h \in {\mathcal P} = \underline n =
\{1,2, \ldots,n \}.
$$
\end{corollary}

\subsection{ The Capelli identities}

For several decades, the Capelli identities were regarded as the
main technical tool to prove the Deruyts--Capelli Expansion formula
and Weyl's theorem; however, as we recognized in Subsection~16.7
and 16.9, these results can be easily derived from general
arguments.

The standard proofs of the Capelli identities are rather
complicated (see., e.g., \cite{FH}, \cite{Weyl}; in this
subsection, by way of application of the method of virtual
variables, we provide an elementary and mechanical proof of them.




Let ${\mathcal A} = \{a_1, a_2, \ldots, a_m\}$ and ${\mathcal P} =
{\underline n} = \{1, 2, \ldots, n\}$ be negatively signed
alphabets and let $ \BK[{\mathcal A}] = Super[{\mathcal A}|\
\underline n]$ be the {\it commutative} letterplace algebra
generated by the positively signed letterplace alphabet
$[{\mathcal A}\,|\,\ {\underline n}]$.



Let $H_m$ be the {\it Capelli operator} in $m$ variables:
$$
H_m = \det\left[
\begin{array}{llll}
{\mathcal D}_{a_1 a_1} + (m-1)I & {\mathcal D}_{a_1 a_2} & \ldots
& {\mathcal D}_{a_1 a_m}
\\{\mathcal D}_{a_2 a_1} & {\mathcal D}_{a_2 a_2}+(m-2)I & \ldots & {\mathcal D}_{a_2 a_m}
\\ \vdots & \vdots &  \ddots & \vdots
\\ {\mathcal D}_{a_m a_1} & {\mathcal D}_{a_m a_2} & \ldots & {\mathcal D}_{a_m a_m}
\end{array}\right],
$$
where the expansion of the determinant is by column from left to
right.

Let $\Omega_n$ be the  {\it Cayley operator} in dimension $n$:
$$\Omega_n = \det\left[ \begin{array} {llll}

\frac {\partial} {\partial (a_1\,|\,1)} & \frac {\partial} {\partial
(a_1\,|\,2)} & \ldots & \frac {\partial} {\partial (a_1\,|\,n)}

\\ \frac {\partial} {\partial (a_2\,|\,1)} & \frac {\partial} {\partial (a_2\,|\,2)} & \ldots
& \frac {\partial} {\partial (a_2\,|\,n)}

\\ \vdots & \vdots & \vdots & \vdots

\\ \frac {\partial} {\partial (a_n\,|\,1)} & \frac {\partial} {\partial (a_n\,|\,2)} & \ldots
& \frac {\partial} {\partial (a_n\,|\,n)} \end{array} \right],
$$
where the symbol $\frac {\partial} {\partial (a_i\,|\,\mu)}$ denotes
the (formal) partial derivative with respect to the variable
$(a_i\,|\,\mu)$.


The following result is due to Capelli (see, e.g., \cite{CAP},
\cite{Weyl}).

\begin{theorem}  Let $\gamma \in \BK[{\mathcal A}].$
We have the following identities:
$$
H_m(\gamma) = \begin{cases} 0 & \text{if }m > n,
\\  [a_1, a_2, \ldots , a_n] \ \Omega_n(\gamma) & \text{if }m=n, \end{cases}
$$
where the bracket $[a_1, a_2, \ldots , a_n]$ denotes the
determinant
$$ \det [ (a_i\,|\,\mu) ]_{i,\ \mu =1,2, \ldots ,m = n}.$$

\end{theorem}


\begin{proof} We regard the algebra $ \BK[{\mathcal A}] =
Super[{\mathcal A}|\ \underline n]$ as a subalgebra of the
letterplace superalgebra $Super[{\mathcal A} \cup \{ \alpha \}|\
\underline n]$, $\alpha$ a (virtual) positive letter.

From Subsection~6.6, we recall that the action on $ \BK[{\mathcal
A}]$ of the Capelli operator $H_m$ is the same as the action of
the operator
$$
{\mathcal D}_{a_1,\alpha} {\mathcal D}_{a_2,\alpha} \ldots
{\mathcal D}_{a_m,\alpha} {\mathcal D}_{\alpha,a_m} {\mathcal
D}_{\alpha,a_{m-1}} \ldots {\mathcal D}_{\alpha,a_1}.
$$






Consider a momomial $\textbf{m}  = \prod_{i,\mu} (a_i\,|\,\mu)^{d_{i
\mu}}$, with $d_{i \mu} \in \BN$, in $ \BK[{\mathcal A}].$


Clearly, all the monomials in the  polynomial
\begin{equation}
{\mathcal D}_{\alpha,a_m} {\mathcal D}_{\alpha,a_{m-1}} \ldots
{\mathcal D}_{\alpha,a_1} (\textbf{m}) \in Super[{\mathcal A} \cup
\{ \alpha \}|\ \underline n]
\tag*{$(*)$}
\end{equation}
contain exactly $m$ occurrences of the positive letter $\alpha$.

Note that the letterplace variables  $(\alpha\,|\,\mu)$ are of
$\BZ_2$-degree 1.

In the case $m > n$, all the monomials in the  polynomial $(*)$
 must contain at least a
square of such a letterplace variable $(\alpha\,|\,\mu)$; thus,
$$
 {\mathcal D}_{\alpha,a_m} {\mathcal D}_{\alpha,a_{m-1}} \ldots
{\mathcal D}_{\alpha,a_1} (\textbf{m}) = 0,
$$
and the first assertion is proved, by linearity.

Let us now consider the case $m = n$. Set $(a_i\,|\,\mu)^{d_{i \mu} -1} =
0$ whenever $d_{i \mu} = 0$, and recall that $(\alpha\,|\,\mu)^2 = 0$
for $\mu = 1, 2, \ldots,n.$ We have the identity
\begin{multline*}
{\mathcal D}_{\alpha, a_n} {\mathcal D}_{\alpha,a_{n-1}} \ldots
{\mathcal D}_{\alpha,a_1} (\textbf{m})\\ = 
\sum_{\sigma} \bigg( \prod_{i,\mu} d_{i,\sigma(i)}
(a_i\,|\,\mu)^{d_{i,\mu}- \delta_{i, \sigma(i)}} (\alpha\,|\,\sigma(n))
(\alpha\,|\,\sigma(n-1)) \ldots (\alpha\,|\,\sigma(1)\bigg),
\end{multline*}
where the summation is over all the permutations $\sigma$ of the
set $\{1, 2, \ldots, n \}$ and $\delta$ is the Kronecker symbol.

The last expression can be rewritten in the form
\begin{multline*}
\sum_{\sigma} \Bigg[\frac {\partial} {\partial (a_1\,|\,\sigma(1))}a
\bigg(\prod_\mu (a_1\,|\,\mu)^{d_{1 \mu}}\bigg) \cdots \frac {\partial}
{\partial (a_n\,|\,\sigma(n))} \bigg(\prod_\mu (a_n\,|\,\mu)^{d_{n \mu}}\bigg)\Bigg] \\
(\alpha\,|\,\sigma(n))
 \ldots (\alpha\,|\,\sigma(1)).
\end{multline*}
Since the variables
$(\alpha\,|\,\sigma(\mu))$ anticommute, we get the 
identities
\begin{align*}
&{\mathcal D}_{\alpha, a_n} {\mathcal D}_{\alpha,a_{n-1}} \ldots
{\mathcal D}_{\alpha,a_1} (\textbf{m})\\
&\kern2cm = 
(\alpha\,|\,n) (\alpha\,|\,n-1) \ldots (\alpha\,|\,1)\ \bigg( \sum_{\sigma}
(-1)^{|\sigma|} \frac {\partial} {\partial (a_1\,|\,\sigma(1))} \cdots
\frac {\partial} {\partial (a_n\,|\,\sigma(n))}\bigg) \ (\textbf{m})\\
&\kern2cm =
(\alpha\,|\,n) (\alpha\,|\,n-1) \ldots (\alpha\,|\,1) \ \Omega_n (\textbf{m}).
\end{align*}
Since
$${\mathcal D}_{a_1,\alpha} {\mathcal D}_{a_2,\alpha} \cdots {\mathcal D}_{a_n, \alpha} (\alpha\,|\,n)
(\alpha\,|\,n-1) \ldots (\alpha\,|\,1) = [a_1,a_2, \ldots ,a_n] ,$$
the second assertion is proved, by linearity.
\end{proof}


\section{$\BZ_2$-Graded Tensor Representations: the Berele--Regev Theory}


From now on let ${\mathcal L}= \{1,2, \ldots, n\}= {\mathcal L}_0$
be an alphabet of $n$ positive letters, and ${\mathcal P}=
{\mathcal P}_0\cup {\mathcal P}_1$ a finite alphabet of places,
with $|{\mathcal P}_0| = r,$ $|{\mathcal P}_1| = s.$

A monomial of $Super_n[{\mathcal L}|{\mathcal P}]$ is said to be
letter-{\it multilinear} whenever it contains each letter of
$\mathcal L$ exactly once; a Young tableau $T$ over ${\mathcal L}$
is said to be {\it multilinear} whenever each letter of ${\mathcal
L}$ appears exactly once in $T$; obviously, if $T$ is multilinear
over ${\mathcal L}$, then $\sh(T) \vdash n.$

We consider the following structures:
\begin{itemize}
\item $Super_n[\underline {\mathcal L}|{\mathcal P}],$ the
subspace of $Super_n[{\mathcal L}|{\mathcal P}]$ freely generated
by the letter-mul\-ti\-linear monomials or, equivalently, by the
standard symmetrized bitableaux $\big(S\,\big|\,\fbox{$T$}\big),$ where $S$
is a multilinear tableau on $\mathcal{L};$ \item $\underline{\mathcal
B}_n,$ the operator algebra linearly generated by the orthonormal
letter generators $Y_n\big(S',\fbox{$S$}\big),$ with $S$ and $S'$
multilinear tableaux over $\mathcal{L},$ restricted to the
subspace $Super_n[\underline{\mathcal L}|{\mathcal P}].$ \item
$_n{\mathcal B}',$ the operator algebra generated by the proper
place polarizations, or, equivalently, by the place orthonormal
generators $\big(T,\fbox{$T'$}\big) _nY,$ $ T, T' \in
Stab_\lambda(\mathcal{P}),$ restricted to the subspace
$Super_n[\underline{\mathcal L}|{\mathcal P}].$
\end{itemize}

Note that $Super_n[\underline{\mathcal L}|{\mathcal P}]$ is
invariant under the action of $\underline{\mathcal B}_n,$
$_n{\mathcal B}',$ so we have the bimodule
$$
\underline {\mathcal B}_n \cdot Super_n[\underline {\mathcal
L}|{\mathcal P}] \cdot _n{\mathcal B}'.
$$

By specialization of the general theory, we have that: the space
$Super_n[\underline {\mathcal L}|{\mathcal P}]$ is a semisimple
$\underline{\mathcal B}_n$-module, and it is a semisimple
$_n{\mathcal B}'$-module; the operator algebras $\underline
{\mathcal B}_n$ and $_n{\mathcal B}'$ are semisimple, and  one is
the centralizer of the other in the endomorphism algebra of
$Super_n[\underline {\mathcal L}|{\mathcal P}].$ The basic
elements and operators, that is, the orthonormal letter generators
$Y_n\big(S',\fbox{$S$}\big),$ the symmetrized bitableaux 
$\big(S\,\big|\,\fbox{$T$}\big),$
and the place orthonormal generators $\big(T,\fbox{$T'$}\big) _nY,$ are
parametrized by tableaux such that
$$
S',S\in Stab_\lambda(L\mathcal{}),\ \ S',S\ \text{multilinear}, \qquad
T,T'\in Stab_\lambda(\mathcal{P}),
$$
where the partitions $\lambda \vdash n$ satisfy the hook condition
$$
\lambda_{r+1}<s+1.
$$


\subsection{The letter multilinear subspace as a $\BK[{\bf S}_n]$-module}


Note that a group action of the symmetric group ${\bf S}_n$ on
$Super_n[\underline {\mathcal L}|{\mathcal P}]$ is consistently
defined by setting
$$
\sigma \cdot (i_1\,|\,y_{j_1}) (i_2\,|\,y_{j_2}) \ldots (i_n\,|\,y_{j_n}) =
(\sigma(i_1)\,|\,y_{j_1}) (\sigma(i_2)\,|\,y_{j_2}) \ldots
(\sigma(i_n)\,|\,y_{j_n}).
$$
This action defines a representation
$$
\rho: \BK[{\bf S}_n]\rightarrow End_{\BK}[Super_n[\underline
{\mathcal L}|{\mathcal P}]].
$$
Notice that the representation of any permutation $\sigma$ can be
regarded as a multilinear letter Capelli column:
$$
\left[
\begin{array}{c}
\sigma(1)\\
\vdots\\
\sigma(n)
\end{array}
\right| \left.
\begin{array}{c}
1\\
\vdots\\
n
\end{array}
\right] = \mathcal{D}_{\sigma(1) \alpha_1} \cdots
\mathcal{D}_{\sigma(n) \alpha_n} \mathcal{D}_{\alpha_n n}  \cdots
\mathcal{D}_{\alpha_1 1} = \rho(\sigma).
$$
Indeed, we have the following result.
\begin{proposition}\label{proposition: operator algebra induced by action of symmetric group on letter multilinear subspace}
The operator algebra induced by the action of the symmetric group
coincides with the operator algebra generated by the multilinear
orthonormal letter generators:
$$
\underline {\mathcal B}_n = \rho(\BK[S_n]).
$$
\end{proposition}

\begin{proof}\cite{BRT3}
To begin with, we show that each operator associated to a
permutation belongs to the algebra $\underline {\mathcal B}_n.$
Indeed, for any permutation $\sigma\in S_n,$ and for any
multilinear standard symmetrized bitableau 
$\big(T\,\big|\,\fbox{$U$}\big),$ we
have
\begin{align*} \sigma \cdot \Big(T\,\Big|\,\fbox{$U$}\Big) & = 
\Big(\sigma \cdot T\,\Big|\,\fbox{$U$}\Big)
\\ & = \sum_{S'} c^{\sigma}_{TS'} \Big(S'\,\Big|\,\fbox{$U$}\Big)
\\ & = \bigg(\sum_{S', S} c^{\sigma}_{SS'}\ Y_n(S',S)\bigg)
\Big(T\,\Big|\,\fbox{$U$}\Big),
\end{align*}
where $S'$ and $S$ range over all the multilinear standard letter
tableaux.

On the other hand, any multilinear Young--Capelli symmetrizer
belongs to the subalgebra $\rho(\BK[S_n]).$ Indeed, by 
Proposition~\ref{proposition: factorization of Young--Capelli symmetrizers}
(Factorization Theorem), the action of $\gamma_n(S',S)$ on
$Super_n[\underline{\mathcal{L}}|\mathcal{P}]$ is the same, up to
a scalar factor, as the action of an operator of the form
$$
{\mathcal D}_{S'C} {\mathcal D}_{CT} {\mathcal D}_{TD} {\mathcal
D}_{DS},
$$
where $T$ denotes any multilinear standard tableau of the same
shape as $S'$ and $S$, filled with positive virtual letters
$\alpha_1, \ldots, \alpha_n$ not appearing in the tableaux $C$ and
$D.$ Furthermore, by the remark at the end of Subsection~10.4 and
the third example in Subsection~6.6, the action of this operator
is the same, up to a scalar factor, as the action of an operator
of the form
\begin{multline*}
\sum_{\sigma} {\mathcal D}_{\sigma(1)\alpha_1} {\mathcal
D}_{\sigma(2)\alpha_2} \ldots {\mathcal D}_{\sigma(n)\alpha_n}
\sum_{\tau} (-1)^{\tau} {\mathcal D}_{\alpha_1\tau(1)} {\mathcal
D}_{\alpha_2\tau(2)} \ldots {\mathcal D}_{\alpha_n\tau(n)} 
\\=
\sum_{\sigma, \tau} \pm {\mathcal D}_{\sigma(1)\alpha_1} {\mathcal
D}_{\sigma(2)\alpha_2} \ldots {\mathcal D}_{\sigma(n)\alpha_n}
{\mathcal D}_{\alpha_1\tau(1)} {\mathcal D}_{\alpha_2\tau(2)}
\ldots {\mathcal D}_{\alpha_n\tau(n)}.
\end{multline*}
where $\sigma, \tau$ range over suitable sets of permutations.
Finally, notice that each summand acts on $Super_n[\underline
{\mathcal L}|{\mathcal P}]$ in the same way as a permutation.
\end{proof}

\begin{remark}\label{remark: Young symmetrizers}
\begin{enumerate}
\item The action of $\gamma_n\big(S',\fbox{$S$}\big)$ is, up to sign, the
same as the action of
$$
\pi_{S'S}  \sum_{\scriptsize\begin{array}{c} \tau\in R(S)\\ \xi\in
C(S)\end{array}} (-1)^{|\xi|} \tau \xi ,
$$
where $R(S)$ and $C(S)$ are the row-stabilizer and the
column-stabilizer of the tableau $S,$ respectively, and $\pi_{S'S}
\in {\bf S}_n$ is the permutation such that $\pi_{S'S}(S) = S'.$

The element
$$
e^{*}_{S'S} = \pi_{S'S}  \sum_{\scriptsize\begin{array}{c} \tau\in
R(S)\\ \xi\in C(S)\end{array}} (-1)^{|\xi|} \tau \xi \in \BK[{\bf
S}_n]
$$
is called a generalized (dual) Young symmetrizer.
Clearly, $e^{*}_{S'S} = \pi_{S'S}e^{*}_{SS}$, for any pair of multilinear tableaux $(S',S).$

\item Let $S' \in Tab({\mathcal L})$ be any multilinear tableau.
Then,we have the following result:
$$
\Big(S'\,\Big|\,\fbox{$T$}\Big) = \pm \gamma_n\Big(S',\fbox{$S$}\Big) \langle S\,|\,T \rangle,
$$
where $\langle S\,|\,T \rangle$ is the tableau monomial in
$Super_n[\underline {\mathcal L}|{\mathcal P}]$ associated to the
pair $(S,T)$ and $S \in Tab({\mathcal L})$ is {\it any}
multilinear tableau such that $sh(S) = sh(T)$. As a matter of
fact, (in short notation) we have
$$\Big(S'\,\Big|\,\fbox{$T$}\Big) = S'C \ CD \ \underline{DT} = \pm  S'C \ CD \ DS \ \underline{ST} =
\pm \gamma_n\Big(S',\fbox{$S$}\Big) \langle S\,|\,T \rangle.$$
\end{enumerate}
\end{remark}


\subsection{The $\BZ_2$-graded tensor representation theory}


Let
$$
W = \langle {\mathcal P} \rangle_{\BK} = \langle {\mathcal P_0}
\rangle_{\BK} \oplus \langle {\mathcal P_1} \rangle_{\BK} = W_0
\oplus W_1,
$$
then
$$
Super_n[\underline{\mathcal L}|{\mathcal P}] \cong T^n[W_0\oplus
W_1]
$$
via the linear isomorphism
$$
{\cal F} : (1\,|\,y_{i_1}) \ldots (n\,|\,y_{i_n}) \mapsto y_{i_1}\otimes
\ldots \otimes y_{i_n}.
$$
Therefore, any construction and any result pertaining to the
multilinear letter subspace $ Super_n[\underline {\mathcal
L}|{\mathcal P}] $ may be carried over to the space $
T^n[W_0\oplus W_1] $ of the $n$-tensors over the ${\BZ}_2$-graded
vector space $W= W_0\oplus W_1.$

The natural action of the symmetric group ${\bf S}_n$ on
$Super_n[\underline {\mathcal L}|{\mathcal P}],$ gives rise to a
representation
$$
\BK[S_n]\rightarrow End_{\BK}[T^n[W_0\oplus W_1]],
$$
which is called a {\it Berele--Regev ${\BZ}_2$-graded representation} of
${\bf S}_n.$

\begin{example} \label{BR action}
Let $y_1, y_2, y_3 \in  {\mathcal P}$,  $|y_1| = 0, |y_2| = |y_3| = 1$.
Let $n = 3.$
We have:

\begin{align*}
(12) \cdot y_1 \otimes y_2 \otimes y_3
&= {\cal F} \left( (12) \cdot (1\,|\,y_1)(2\,|\,y_2)(3\,|\,y_3) \right)
\\
&=
{\cal F} \left( (2\,|\,y_1)(1\,|\,y_2)(3\,|\,y_3) \right)
\\
&=
{\cal F} \left( (1\,|\,y_2)(2\,|\,y_1)(3\,|\,y_3) \right)
= y_2 \otimes y_1 \otimes y_3.
\\
\\
(23) \cdot y_1 \otimes y_2 \otimes y_3
&= {\cal F} \left( (23) \cdot (1\,|\,y_1)(2\,|\,y_2)(3\,|\,y_3) \right)
\\
&=
{\cal F} \left( (1\,|\,y_1)(3\,|\,y_2)(2\,|\,y_3) \right)
\\
&=
{\cal F} \left( -(1\,|\,y_1)(2\,|\,y_3)(3\,|\,y_2) \right)
= - y_1 \otimes y_3 \otimes y_2.
\end{align*}
\end{example}

On the other hand, the action of $pl({\cal P})$ on
$Super_n[\underline {\mathcal L}|{\mathcal P}]$ gives rise to a
representation
$$
End_{\BK}[T^n[W_0\oplus W_1]] \leftarrow pl(W)
$$
of the general linear Lie superalgebra $pl(W).$ Therefore, by specializing the
results of Section~12  to the bimodule
$$
\BK[S_n] \cdot T^n[W_0\oplus W_1] \cdot pl(W_0\oplus W_1),
$$
we get  the following results.

\begin{theorem}\label{theorem: Berele--Regev representations}
$\ $
\begin{enumerate}
\item The subalgebras of $End_{\BK}[T^n[W_0\oplus W_1]$ induced by
the actions of
$$
\BK[{\bf S}_n]  \quad and \quad pl(W_0\oplus W_1)
$$
are the
centralizers of each other.
\item A complete decomposition of
$T^n[W_0\oplus W_1]$ with respect to the action of the symmetric
group $S_n$ is given by:
\begin{multline*}
T^n[W_0\oplus W_1] = {\cal F}\left[ Super_n[\underline {\mathcal
L}|{\mathcal P}] \right]
\\=
\bigoplus_{\stackrel{\scriptstyle{\lambda \in H(\mathcal{P})}}
{\lambda \vdash n}} \quad \bigoplus_{\stackrel{\scriptstyle{T \in
Stab(\mathcal{P}})} {\sh(T)=\lambda}} {\cal F}\left[ \Big\langle
\Big(S\,\Big|\,\fbox{$T$}\Big),\ \ S\in Stab(\mathcal{L}),\ S\ \text{multilinear} \Big\rangle
\right];
\end{multline*}
\item A complete decomposition of $T^n[W_0\oplus W_1]$ with
respect to the action of the general linear Lie superalgebra
$pl(W_0\oplus W_1)$ is given by:
\begin{multline*}
T^n[W_0\oplus W_1]  = {\cal F}\left[ Super_n[\underline {\mathcal
L}|{\mathcal P}] \right]
\\=
\bigoplus_{\stackrel{\scriptstyle{\lambda \in H(\mathcal
{P})}}{\lambda \vdash n}} \quad
\bigoplus_{\stackrel{\scriptstyle{S \in Stab(\mathcal{L}),\ S\
\text{multilinear}}} {\sh(S)=\lambda}} {\cal F}\left[ \Big\langle
\Big(S\,\Big|\,\fbox{$T$}\Big),\ \ T\in Stab({\mathcal P}) 
\Big\rangle \right].
\end{multline*}
\end{enumerate}
\end{theorem}


The  irreducible $\BK[{\bf S}_n]$-submodules and $pl(W_0\oplus W_1)$-submodules
which appear in the preceding complete
decomposition theorem admit a direct description as subspaces of the tensor space $T^n[W_0\oplus W_1]$.

First of all, we notice that any decomposable tensor
$y_{i_1}\otimes \ldots \otimes y_{i_n} \in T^n[W_0\oplus W_1]$ is
(up to a sign) the image under ${\cal F}$ of a bitableau monomial
$\langle S\,|\,T \rangle$, where $S \in Tab({\mathcal L})$ is a
multilinear tableau, $T \in Tab({\mathcal P})$, $sh(S) = sh(T)
\vdash n$. Clearly, this representation is not unique.

For example, let $y_1, y_2, y_3 \in  {\mathcal P}$,  $|y_1| = 0, |y_2| = |y_3| = 1$; then
\begin{align*}
y_1 \otimes y_2 \otimes y_3 &= {\cal F}\left( \left\langle
\begin{array}{l|l}
123 & y_1 y_2 y_3
\end{array}
\right\rangle \right) = {\cal F}
\left((1\,|\,y_1)(2\,|\,y_2)(3\,|\,y_3)\right)
\\
&=
 {\cal F}
\left( - \left\langle
\begin{array}{l|l}
13 & y_1 y_3 \\
2 & y_2
\end{array}
\right\rangle \right) = {\cal F} \left( -
(1\,|\,y_1)(3\,|\,y_3)(2\,|\,y_2)\right).
\end{align*}

Thanks to Remark~\ref{remark: Young symmetrizers}, we
get the following results.

\begin{remark} \label{remark: $S_n$-tensor interpretation}
The irreducible  $\BK[{\bf S}_n]$-submodule
$$
{\cal F}\left[ \Big\langle \Big(S\,\Big|\,\fbox{$T$}\Big),\ \ S\in
Stab_\lambda(\mathcal{L}),\ S\ \text{multilinear} \Big\rangle \right]
$$
equals
$$
\left\{ \alpha \cdot {\cal F}\left( \langle U\,|\,T \rangle \right); \
\alpha \in \BK[{\bf S}_n]e^{*}_{UU} \right\} \subseteq
T^n[W_0\oplus W_1],
$$
for every multilinear tableau $U \in Stab_\lambda(\mathcal{L})$
and, thus, it is $\BK[{\bf S}_n]$-isomorphic to the (minimal) left
ideal $\BK[{\bf S}_n]e^{*}_{UU}$ of $\BK[{\bf S}_n].$
\end{remark}

\begin{example}
Let $y_1, y_2, y_3 \in {\mathcal P}$, and let $|y_1| = 0, |y_2| =
|y_3| = 1$. Consider the tableaux $U, U' \in Tab(\mathcal{L})$ and
$T \in Stab(\mathcal{P})$,
$$
U = \begin{array}{l}
12 \\
34
\end{array}
\quad U' = \begin{array}{l}
13 \\
24
\end{array}
\quad T = \begin{array}{l}
y_1y_3 \\
y_2y_3
\end{array}.
$$
Note that
$$
{\cal F} \left( \left\langle
\begin{array}{l|l}
12 & y_1y_3 \\
34 & y_2y_3
\end{array}
\right\rangle \right) = y_1 \otimes y_3 \otimes y_2 \otimes y_3,
\quad
{\cal F} \left( \left\langle
\begin{array}{l|l}
13 & y_1y_3 \\
24 & y_2y_3
\end{array}
\right\rangle \right) = - y_1 \otimes y_2 \otimes y_3 \otimes y_3.
$$
The irreducible  $\BK[{\bf S}_n]$-submodule
$$
{\cal F}\left[ \Big\langle \Big(S\,\Big|\,\fbox{$T$}\Big),\ \ S\in
Stab_{(2,2)}(\mathcal{L}),\ S\ \text{multilinear} \Big \rangle \right]
\subseteq T^4[W_0\oplus W_1]
$$
equals
$$
\left\{ \alpha \cdot (y_1 \otimes y_3 \otimes y_2 \otimes y_3); \
\alpha \in \BK[{\bf S}_n]e^{*}_{UU} \right\} \! =\! 
\left\{ \alpha \cdot (- y_1 \otimes y_2 \otimes y_3 \otimes y_3); \
\alpha \in \BK[{\bf S}_n]e^{*}_{U'U'} \right\}.
$$
\end{example}

\begin{remark} \label{remark: $pl(V)$-tensor interpretation}
The irreducible $pl(W_0\oplus W_1)$-submodule
$$
{\cal F}\left[ \Big\langle \Big(S\,\Big|\,\fbox{$T$}\Big),\ \ T\in
Stab_\lambda({\mathcal P}) \Big\rangle \right]
$$
equals
$$
e^{*}_{SS} \left[ T^n[W_0\oplus W_1] \right],
$$
the image of the tensor space $T^n[W_0\oplus W_1]$ under the
action of the (dual) Young symmetrizer $e^{*}_{SS}.$
\end{remark}

\begin{remark} \label{the classical theory}
If  ${\mathcal P}$ is trivially $\BZ_2$-graded, it follows from Subsection~4.5 that the preceding results
yield the classical Schur--Weyl
tensor representation theory of the general linear group $GL(W)$ and of the symmetric group
${\bf S}_n$ (\cite{Schur1}, \cite{Schur2}, \cite{Weyl}).
\end{remark}

\section{The Symmetric Group}

From now on let ${\mathcal L}= {\mathcal L}_1 = {\mathcal P}=
{\mathcal P}_1 = \{1,2, \ldots, n\},$ an alphabet of $n$ negative
symbols, and consider the letterplace algebra (in {\it
commutative} letterplace variables)
$Super_n[\mathcal{L}|\mathcal{P}].$

In the sequel, given a partition $\lambda \vdash n,$ we will write
$S_{\lambda 1}, S_{\lambda 2}, \ldots, S_{\lambda {\underline
f}_\lambda}$ to mean the list of all multilinear standard tableaux
of shape $\lambda$ over $\mathcal{L} = \mathcal{P} = \{1,2,
\ldots, n\}$ sorted with respect to the linear order defined in
Subsection~5.5. When the shape is clear from the context, we write
simply $S_{i}$ instead of $S_{\lambda i}.$

We recall that the ``doubly indexed" {\it Young symmetrizers} (of
shape $\lambda \vdash n$) are the elements of the group algebra
$\BK[\bf S_n]$ defined as follows:
$$
e_{ij} = \pi_{ij} \bigg( \sum_{\scriptsize \begin{array}{c} \tau\in
R(S_j)\\ \xi\in C(S_j)\end{array}} (-1)^{|\tau|} \tau \xi\bigg) ,
$$
where $R(S_j)$ and $C(S_j)$ are the row-stabilizer and the
column-stabilizer of $S_j,$ respectively, and $\pi_{ij}\in {\bf
S}_n$ is the (unique) permutation such that
$$
\pi_{ij} (S_j) = S_i.
$$

In the classical notation, $e_{ij} = \pi_{ij} e_j ,$ where $e_j$
denotes the {\it Young symmetrizer} associated to the tableau
$S_j$ (see, e.g., \cite {JK}).

We consider the module
$$
\underline{\underline {\mathcal{B}}}_n \cdot Super_n[\underline
{\mathcal L}| \underline {\mathcal P}],
$$
 where
\begin{itemize}
\item $Super_n[\underline {\mathcal L}| \underline {\mathcal P}]$
is the subspace of $Super_n[{\mathcal L}|{\mathcal P}]$ spanned by
all the doubly multilinear monomials, that is, monomials of the
form
$$
(\tau(1)\,|\, 1) (\tau(2)\,|\, 2) \cdots (\tau(n)\,|\, n), \quad \tau \in {\bf
S}_n,
$$
or, equivalently, by the doubly multilinear symmetrized bitableaux
$\big(S\,\big|\,\fbox{$T$}\big),$ $S$ and $T$ multilinear tableaux on
$\mathcal{L}= \mathcal{P}.$ Notice that the set of right
symmetrized bitableux
$$
\Big(S_{\lambda i},\fbox{$S_{\lambda j}$}\Big), \qquad i,j= 1, 2, \ldots,
{\underline f}_\lambda, \quad \lambda \vdash n,
$$
is a ${\mathbb K}$-linear basis of $Super_n[\underline {\mathcal
L}| \underline {\mathcal P}].$ \item $\underline{\underline
{\mathcal{B}}}_n$ is the  algebra of operators on
$Super_n[\underline {\mathcal L}| \underline {\mathcal P}]$
linearly generated by the restrictions of the orthonormal letter
generators $Y_n\big(S',\fbox{$S$}\big),$ $S$ and $S'$ multilinear tableaux
on $\mathcal{L}$.
%(Note that
%$Super_n[\underline{\mathcal L}|\underline{\mathcal P}] \subseteq Super_n[{\mathcal L}|{\mathcal P}]$
%is an invariant subspace under the action of the operators 
$Y_n\big(S',\fbox{$S$}\big),$
%$S$ and $S'$ multilinear tableaux on $\mathcal{L}$).
We hardly need to recall that the set of restricted operators
$$
Y_n\Big(S_{\lambda i},\fbox{$S_{\lambda j}$}\Big), \qquad i,j = 1, \ldots,
{\underline f}_\lambda \quad \lambda \vdash n
$$
is a $\BK$-linear basis of $\underline{\underline
{\mathcal{B}}}_n$. Therefore, this algebra is a semisimple
subalgebra of $End_{\BK}(Super_n[\underline {\mathcal
L}|\underline {\mathcal P}]),$ and admits a complete decomposition
$$
\underline{\underline {\mathcal{B}}}_n= \bigoplus_{\lambda \vdash
n}\ \bigoplus_{j= 1, 2, \ldots, {\underline f}_\lambda} \Big\langle
Y_n\Big(S_{\lambda i},\fbox{$S_{\lambda j}$}\Big),\ i= 1, 2, \ldots,
{\underline f}_\lambda \Big\rangle_{\mathbb K} = \bigoplus_{\lambda
\vdash n}\ \underline{\underline {\mathcal{B}}}_\lambda,
$$
where each simple component  $\underline{\underline
{\mathcal{B}}}_\lambda$ is isomorphic to $M_{{\underline
f}_\lambda},$ the full ${\mathbb K}$-algebra of square matrices of
order ${\underline f}_\lambda.$
\end{itemize}



\subsection{The doubly multilinear subspace as a $\BK[\bf S_n]$-module}


The action of the symmetric group ${\bf S}_n$ on
$Super_n[\underline {\mathcal L}| \underline {\mathcal P}]$
defined by setting
$$
\sigma \cdot ((\tau(1)\,|\, 1) ((\tau(2)\,|\, 2) \cdots (\tau(n)\,|\, n)) =
(\sigma \tau(1)\,|\, 1) (\sigma \tau(2)\,|\, 2) \cdots (\sigma \tau(n)\,|\,n)
$$
induces a faithful representation
$$
\underline{\rho} :\ {\mathbb K}[{\bf S}_n] \rightarrow
End_{\mathbb K}[Super_n[\underline {\mathcal L}| \underline
{\mathcal P}]].
$$
By specializing the argument of Proposition~\ref{proposition:
operator algebra induced by action of symmetric group on letter
multilinear subspace}, we get:

\begin{proposition}\label{proposition: operator algebra induced by action of symmetric group on doubly multilinear subspace}
The operator algebra induced by the action of the symmetric group
coincides with the operator algebra generated by the multilinear
orthonormal letter generators:
$$
\underline{\rho} [{\mathbb K}[{\bf S}_n]] = \underline{\underline
{\mathcal{B}}}_n,
$$
Furthermore, $\underline{\rho}$ induces a ${\mathbb K}$-algebra
isomorphism ${\mathbb K}[{\bf S}_n] \cong \underline{\underline
{\mathcal{B}}}_n.$
\end{proposition}

We remark that the Young--Capelli symmetrizers are, up to a sign,
the representations of the classical two-pa\-ra\-me\-ter Young
symmetrizers:
$$
\underline{\rho} (e_{ij}) = (-1)^{(^n_2)} \gamma_n\Big(S_i,
\fbox{$S_j$}\Big).
$$






\subsection{Complete decompositions of the group algebra $\BK[S_n]$ as a left regular module.}


The map
$$
F : Super_n[\underline {\mathcal L}| \underline {\mathcal P}]
\rightarrow {\mathbb K}[{\bf S}_n], \qquad (\tau(1)\,|\, 1) (\tau(2)\,|\,
2) \cdots (\tau(n)\,|\, n) \mapsto \tau
$$
is an isomorphism from the module
$$
{\mathbb K}[{\bf S}_n] \cdot Super_n[\underline {\mathcal L}|
\underline {\mathcal P}]
$$
to the module
$$
{\mathbb K}[{\bf S}_n] \cdot {\mathbb K}[{\bf S}_n].
$$
The symmetrized bitableaux are sent, by this isomorphism to the
classical two-pa\-ra\-me\-ter Young symmetrizers:
$$
F\left( \Big(S_i\,\Big|\, \fbox{$S_j$}\Big)\right) = (-1)^{(^n_2)}e_{ij}.
$$
By combining Theorem~\ref{theorem: decomposition of Super} with
Proposition~\ref{proposition: operator algebra induced by action
of symmetric group on doubly multilinear subspace}, we get
immediately the following theorem.

\begin{theorem}[\sc Complete decomposition of {${\mathbb K}[{\bf S}_n]$} as a left
regular module]
\label{theorem: complete decomposition of symmetric group algebra}
\begin{align*}
{\mathbb K}[{\bf S}_n] &= F [Super_n[\underline {\mathcal
L}|\underline {\mathcal P}]]
\\
&= F \Bigg[ \bigoplus_{\lambda \vdash n}\ \bigoplus_{j= 1, 2,
\ldots, {\underline f}_\lambda} \Big\langle \Big(S_{\lambda
i}\,\Big|\,\fbox{$S_{\lambda j}$}\Big),\ i= 1, 2, \ldots, {\underline
f}_\lambda \Big\rangle_{\mathbb K} \Bigg]
\\
&= \bigoplus_{\lambda \vdash n}\ \bigoplus_{j= 1, 2, \ldots,
{\underline f}_\lambda} F \left[ \Big\langle \Big(S_{\lambda
i}\,\Big|\,\fbox{$S_{\lambda j}$}\Big),\ i= 1, 2, \ldots, {\underline
f}_\lambda \Big\rangle_{\mathbb K} \right]
\\
&= \bigoplus_{\lambda \vdash n}\ \bigoplus_{j= 1, 2, \ldots,
{\underline f}_\lambda} \langle \pi_{ij} e_j;\ i= 1, 2, \ldots,
{\underline f}_\lambda \rangle_{\mathbb K}
\\
&= \bigoplus_{\lambda \vdash n}\ \bigoplus_{j= 1, 2, \ldots,
{\underline f}_\lambda} {\mathbb K}[{\bf S}_n] e_j,
\end{align*}
where the outer sum is the isotypic decomposition, and
$$
\langle \pi_{ij} e_j;\ i= 1, 2, \ldots, {\underline f}_\lambda
\rangle_{\mathbb K} = {\mathbb K}[{\bf S}_n] e_j, \qquad j= 1, 2,
\ldots, f_\lambda^M,
$$
are minimal left ideals of ${\mathbb K}[{\bf S}_n].$
\end{theorem}

The irreducible ${\mathbb K}[{\bf S}_n]$-module
$$
{\mathcal S}_{\lambda j} = \Big\langle \Big(S_{\lambda
i}\,\Big|\,\fbox{$S_{\lambda j}$}\Big),\ i= 1, 2, \ldots, {\underline
f}_\lambda \Big\rangle_{\mathbb K} = F^{-1} [{\mathbb K}[{\bf S}_n]
\cdot e_j]
$$
is the {\it Specht module} (of the first kind) associated to the
multilinear standard tableau $S_{\lambda j}$ of shape $\lambda$
(see, e.g., \cite{Clausen1}, \cite{Sag}).

\subsection{On the coefficients $h_\lambda$ (Lemma~\ref{lemma: nondegeneracy}
 and Theorem~\ref{theorem: triangularity of Young--Capelli symmetrizers
and symmetrized bitableaux}).}
\begin{proposition}
The coefficient $h_\lambda$ which appears in Lemma~\ref{lemma:
nondegeneracy} and Theorem~\ref{theorem: triangularity of
Young--Capelli symmetrizers and symmetrized bitableaux} equals the
product of hook lengths of the shape $\lambda$.
\end{proposition}

\begin{proof}
We already know that the coefficient $h_\lambda$ depends only on
the shape $\lambda$.

Let $S_i$ be the $i$-th multilinear standard tableau of shape
$\lambda$.

The Triangularity Theorem (Theorem~\ref{theorem: triangularity of
Young--Capelli symmetrizers and symmetrized bitableaux} )
specializes to the following identity:
$$
\gamma_n\Big(S_i,\fbox{$S_i$}\Big) \Big(S_i\,\Big|\,\fbox{$S_i$}\Big) = (-1)^{(^n_2)}
\theta^{-+}_{S_i S_i} h_\lambda \Big(S_i\,\Big|\,\fbox{$S_i$}\Big),
$$
with $\theta^{-+}_{S_i S_i} = 1.$

Then, the following identities hold:
\begin{align*} 
F\left( \underline{\rho}(e_{ii}) \Big(S_i\,\Big|\,\fbox{$S_i$}\Big) \right) &=
F\left( (-1)^{(^n_2)} \gamma_n\Big(S_i,\fbox{$S_i$}\Big)
\Big(S_i\,\Big|\,\fbox{$S_i$}\Big) \right) \\
&= F\left( h_\lambda \Big(S_i\,\Big|\,\fbox{$S_i$}\Big)
\right) \\
&= (-1)^{(^n_2)} h_\lambda e_{ii}.
\end{align*}
On the other hand, we have:
$$ F\left( \underline{\rho}(e_{ii}) \Big(S_i\,\Big|\,\fbox{$S_i$}\Big) \right) =
e_{ii}\cdot F\left( \Big(S_i\,\Big|\,\fbox{$S_i$}\Big) \right) = (-1)^{(^n_2)}
e_{ii}^2.$$
Hence $e_{ii}^2 =  h_\lambda e_{ii}.$

Since it is well-known that  $e_{ii}^2 =   \prod \left(\text{hook 
lenghts of} \ \lambda \right) e_{ii},$ (see, e.g., \cite{FH},
\cite{Sag}, \cite{JK}, \cite{Rut}), the assertion follows.

\end{proof}

\subsection{The Young natural form of irreducible representations}


In the following, we specialize the constructions and the results
of Section~13.

Given any partition $\lambda \vdash n$ and any multilinear
standard place tableau $S_{\lambda j}$ of shape $\lambda,$ the
module structure ${\mathbb K}[{\bf S}_n] \cdot {\mathcal
S}_{\lambda j}$ induces a surjective algebra morphism
$$
\underline{\nu}_{\lambda j} :\ {\mathbb K}[{\bf S}_n] \rightarrow
End_{\mathbb K}[{\mathcal S}_{\lambda j}].
$$
By choosing the basis of the $\big(S_{\lambda i}\,\big|\,\fbox{$S_{\lambda
j}$}\big)$'s in ${\mathcal S}_{\lambda j},$ the morphism
$\underline{\nu}_{\lambda,j}$ induces an irreducible matrix
representation
$$
\overline{\underline{\nu}}_{\lambda j} :\ {\mathbb K}[{\bf S}_n]
\rightarrow M_{{\underline f}_\lambda},
$$
where, for every $\sigma \in {\bf S}_n,$
$$
\overline{\underline{\nu}}_{\lambda j} (\sigma) = [ c_{hk}^\lambda
(\underline{\nu}_{\lambda j} (\sigma)) ].
%\qquad
%\nu_{\lambda j}(\sigma) \Big(S_k\,\Big|\,\fbox{$S_j$}\Big) =
%\sum_{h,k} c_{hk}^\lambda (\nu_{\lambda j} (\sigma)) \Big(S_h\,\Big|\,\fbox{$S_j$}\Big).
$$


For every $\lambda \vdash n,$ the module structure ${\mathbb
K}[{\bf S}_n] \cdot
Super_n^M[\underline{\mathcal{L}}|\underline{\mathcal{P}}]$
induces a surjective algebra morphism
$$
\underline{\rho}_\lambda :\ {\mathbb K}[{\bf S}_n] \rightarrow
\underline{\underline{\mathcal{B}}}_\lambda;
$$
by choosing the basis of the 
$Y\big(S_{\lambda i}\,\big|\,\fbox{$S_{\lambda
j}$}\big)$ in $\underline{\underline{\mathcal{B}}}_n,$ the morphism
$\underline{\rho}_\lambda$ induces an irreducible matrix
representation
$$
\overline{\underline{\rho}}_\lambda :\ {\mathbb K}[{\bf S}_n]
\rightarrow M_{{\underline f}_\lambda},
$$
where, for every $\sigma \in {\bf S}_n,$
$$
\overline{\underline{\rho}}_\lambda (\sigma) = [ d_{hk}^\lambda
(\underline{\rho}_\lambda (\sigma)) ].
%\qquad
%Y(S_h|\fbox{$S_h$}) \rho_\lambda (\sigma) Y(S_k|\fbox{$S_k$}) =
%d_{hk}^\lambda (\rho_\lambda (\sigma)) Y(S_h|\fbox{$S_k$}).
$$

From Proposition~\ref{proposition: twofold approach to irreducible
matrix representations}, it follows that the irreducible
representations $\overline{\underline{\rho}}_\lambda$ and
$\overline{\underline{\nu}}_{\lambda,j}$ are equal.


We now specialize Theorem~\ref{theorem: natural form of
irreducible matrix representation} to the multilinear case,
thereby obtaining a simple combinatorial interpretation of the
coefficients
$$
c_{hk}^\lambda (\underline{\nu}_{\lambda j} (\sigma)) =
d_{hk}^\lambda (\underline{\rho}_\lambda (\sigma)).
$$
We have:
$$
C^{\lambda}(\sigma) = \Theta^{\lambda}(I)^{-1} \times
\Theta^{\lambda}(\sigma),
$$
where the entries $ \theta^{\lambda}_{ij}(\sigma) $ of the matrix
$\Theta^{\lambda}(\sigma)$ are the symmetry transition
coefficients $ \theta^{-+}_{S_i S_j}(\sigma), $ defined by the
relations
$$
DS\ \sigma \cdot \underline{TC} = \theta^{-+}_{ST}(\sigma)\
\underline{DC}.
$$
Notice that
$$
\theta^{-+}_{PQ}(\sigma) = \theta^{-+}_{P\ \sigma Q}.
$$
These coefficients admit a simple combinatorial description;
specifically
$$
\theta^{-+}_{PQ} = 
\begin{cases}
(-1)^{ |\beta |} & \text{if there exists a (unique)}\ \alpha\in C(P),\ \beta \in R(Q):\ \alpha P = \beta Q ,\\
0& \text{otherwise},
\end{cases}
$$
where $R(Q)$ and $C(P)$ are the subgroups of ${\bf S}_n$
which are the row-stabilizer of $Q$ and the column-stabilizer of $P$,
respectively.


The matrix $\Theta^{\lambda}(I)^{-1}$ is the same as the
transition matrix from the (normalized) generalized Young
symmetrizers $\frac 1 {h_\lambda}\ e_{ij}$ to the {\it Young
natural units} $\gamma_{ij}$ in ${\mathbb K}[{\bf S}_n],$ (see
e.g., \cite{JK, Rut}). As a matter of fact, we have
$$
\underline{\rho}(\gamma_{ij}) = \underline{\rho}
\bigg(\sum_h
\varrho_{ih}^\lambda \frac 1 {h_\lambda} e_{hj}\bigg) = \sum_h
\varrho_{ih}^\lambda \frac {(-1)^{(^n_2)}} {h_\lambda}\
\gamma\Big(S_h, \fbox{$S_j$}\Big) = Y\Big(S_i, \fbox{$S_j$}\Big).
$$
Therefore, $\overline{\underline{\rho}}_\lambda =
\overline{\underline{\nu}}_{\lambda,j}$ is indeed the {\it Young
natural form} of the irreducible matrix representations of ${\bf
S}_n,$ for every $\lambda \vdash n$ (see, e.g., \cite {JK, GML}).


\newpage

\part{A Glimpse of the General Representation Theory of Finite Dimensional Lie Superalgebras}



\section{A Brief Historical Outline of the Theory of Representations of Finite
Dimensional Lie Superalgebras over the Complex Field (after
V.~G.~Kac)}

The  classification of complex finite dimensional  simple Lie
superalgebras was published by Kac in 1977 \cite{KAC1}. Shortly
afterwards, Kac founded the representation theory of these Lie
superalgebras (cf.\ \cite{KAC2}, \cite{KAC3}).




The list of complex finite dimensional  simple Lie superalgebras
$L = L_0 \oplus L_1$ consists of two essentially different parts,
namely, {\it classical} and {\it Cartan} Lie superalgebras;
classical Lie superalgebras are those for which the Lie subalgebra
$L_0$ is {\it reductive}.

The list of classical Lie superalgebras is in turn divided into
two parts, {\it basic} and {\it strange} Lie superalgebras.



The basic classical Lie superalgebras are the classical Lie superalgebras
which admit a non degenerate  invariant bilinear form $(\ , )$;
they are from many points of view the closest to the ordinary
simple  Lie superalgebras (see, e.g., \cite{KAC3}).

A basic classical Lie superalgebra  $L = L_0 \oplus L_1$ is said to be
of {\it type I} whenever its $\BZ_2$ homogeneous component $L_1$,
regarded as an $L_0$-module with respect to the adjoint
representation, turns out to be the direct sum of two irreducible
representations.

Our work essentially deals with general linear Lie superalgebras
which are not simple ones; however, their  representation theory
is still ``essentially the same" as the representation theory of
the basic classical Lie superalgebras $\bf A(m,n)$, the
super-analogues of the special linear Lie algebras (for
details, see, e.g., \cite{KAC1}, \cite{KAC2}, \cite{KAC3}, \cite{FSS}).


In the following section, we will recall some of the basic results
of the general representation theory of finite dimensional Lie
superalgebras  in the special case of general linear Lie
superalgebras. We will follow along the lines of the recent
approaches of Brundan (\cite{BRU}, \cite{BRU1}) and Soergel
\cite{SOERG} which are, in turn, inspired by the work of
Bernstein, Gelfand and Gelfand (see \cite{BGG}, \cite{BGG1}).





\section{General Linear  Lie Superalgebras.\\ Highest Weight Modules, Kac Modules,
Typical Modules and Covariant Modules}

Over the past years, a substantial part of the work on the
representation theory of general linear Lie superalgebras was
concerned with the problem of computing the characters of the
finite dimensional irreducible representations. This problem was
raised originally by Kac in 1977 (cf.\ \cite{KAC2}, \cite{KAC3}), who
also introduced the two main classes of modules, namely, the  class of {\it highest weight representations}
and the class of modules known nowadays as {\it Kac modules}.

The highest weight representations ${\mathcal V}
(\Lambda)$ ({\it with integral weights}) are irreducible modules,
but they are not, in general, of finite dimension. Given a general linear
Lie superalgebra $\textbf{gl}(m|n)$,
let $\mathcal{O}_{m|n}$ be the category whose objects are
all $\BZ_2$-graded $\textbf{gl}(m|n)$-modules with integral
weights  which are finitely generated
$\textbf{gl}(m|n)$-modules and are locally finite dimensional over
a {\it distinguished} Borel subalgebra;
any irreducible  module $M \in \mathcal{O}_{m|n}$ is
isomorphic to an integral highest  weight module ${\mathcal V}
(\Lambda)$ (see, e.g., \cite{KAC2}, \cite{KAC3}, \cite{FSS},
\cite{BRU}, \cite{BRU1}, \cite{SOERG}).

The Kac modules $\overline{\mathcal V} (\Lambda)$, $\Lambda$ a
{\it dominant integral} weight, are finite dimensional,  but they
are not, in general, irreducible.

The main connection between highest weight modules and Kac modules
is that any finite dimensional integral highest weight module
${\mathcal V} (\Lambda)$ is a quotient module of the  Kac module
$\overline{\mathcal V} (\Lambda)$, $\Lambda$ a {\it dominant
integral} weight, and the module $\overline{\mathcal V} (\Lambda)$ is
isomorphic to the highest weight module ${\mathcal V} (\Lambda)$
(that is, $\overline{\mathcal V} (\Lambda)$ is an irreducible
module) if and only if its highest weight $\Lambda$ is  a {\it
typical}  weight.

Furthermore, any finite dimensional irreducible
$\textbf{gl}(m|n))$-module  is either typical
or it can be obtained from $\mathcal V(\Lambda)$ for some integral
dominant weight $\Lambda$ by tensoring with a one-dimensional
representation.

The characters of the modules $\overline{\mathcal V} (\Lambda)$
were computed by Kac (cf.\ \cite{KAC2}, \cite{KAC3}) in the case of
typical highest weights. After that, there were several
conjectures and partial results dealing with atypical highest
weights (see, e.g., \cite{BERN}, \cite{Berele2}, \cite{SERG1},
\cite{VANDER2}, \cite{HUGH}, \cite{KAC4}, \cite{PENFSERG2}),
before the complete solution of the problem was given by Serganova
(see \cite{SERGA1}, \cite{SERGA2}), using a mixture of algebraic and
geometric techniques (see, e.g., \cite{BRU}).

Inspired by some ideas of borrowed from the work of Lascoux,
Leclerc and Thibon  \cite{LLT}, Brundan  \cite{BRU} obtained a
purely algebraic version of Serganova's results; in addition, the
work of Brundan exploits a deep connection between the
representation theory of general linear Lie superalgebras and the
Kazhdan--Lusztig representation theory of Coxeter groups and Hecke
algebras \cite{KL}.


\subsection{$\BZ_2$-homogeneous bases of $V = V_0 \oplus V_1$ and consistent $\BZ$-graduations of a
general linear Lie superalgebra $\textbf{gl}(m|n) = pl(V)$}

In the remainder of this section, we will denote by the symbols $\overline 0$, $\overline 1$
the elements of the field $\BZ_2$.

Let $V = V_{\overline 0} \oplus V_{\overline 1}$ be a $\BZ_2$-graded vector space,
$\dim (V_{\overline 0}) = m$, $\dim (V_{\overline 1}) = n$, and let $\mathcal L = \{x_1,
\ldots, x_m, x_{m+1}, \ldots,x_{m+n} \}$ be a $\BZ_2$-homogeneous
basis of $V$, $|x_i| = {\overline 0}$ for every $i = 1, \ldots, m$, $|x_i| =
{\overline 1}$ for every $i = m+1, \ldots, m+n$.

In the following, we will write
$$
\textbf{gl}(m|n) = \textbf{gl}(m|n)_{\overline 0} \oplus \textbf{gl}(m|n)_{\overline 1}
$$ 
for the general
linear Lie superalgebra $pl(V)$ of $V = V_{\overline 0} \oplus V_{\overline 1}$.

In matrix notation (see Subsection~3.3) --- with respect to the choice
of the $\BZ_2$-homogeneous basis
$\mathcal L = \{x_1, \ldots, x_m, x_{m+1}, \ldots,x_{m+n} \}$ ---
the Lie superalgebra
$$
\textbf{gl}(m|n) = \textbf{gl}(m|n)_{\overline 0} \oplus \textbf{gl}(m|n)_{\overline 1}
$$ 
is
the algebra of all $(m+n) \times (m+n)$-square matrices, where the
set
$$ \{ E_{x_i,x_j}; i,j = 1, \ldots, m \} \cup \{ E_{x_i,x_j}; i,j = m+1, \ldots, m+n \}$$
is a basis of $\textbf{gl}(m|n)_{\overline 0}$, and
$$ \{ E_{x_i,x_j}; i = 1, \ldots, m, \ j = m+1, \ldots, m+n \} \cup \{ E_{x_i,x_j}; i =  m+1,
\ldots, m+n, \    j = 1, \ldots, m \}$$ 
is a basis of $\textbf{gl}(m|n)_{\overline 1}.$


The choice of the  basis $\mathcal L = \{x_1, \ldots, x_m,
x_{m+1}, \ldots,x_{m+n} \}$ induces a {\it consistent}
$\BZ$-graduation on $\textbf{gl}(m|n)$ (see, e.g., \cite{Scheu}).
Specifically, we have the following direct sum decomposition:
$$
\textbf{gl}(m|n) = \bigoplus_{k \in \BZ} \
\textbf{gl}(m|n)_{k}, \quad {k} \in \BZ,
$$
where
\begin{itemize}
\item $\textbf{gl}(m|n)_{- 1}$ is the span of the set
$ \{ E_{x_i,x_j}; i = 1, \ldots, m, \ j = m+1, \ldots, m+n \},$ \item
$\textbf{gl}(m|n)_{0}$ is the span of the set $ \{
E_{x_i,x_j}; i,j = 1, \ldots, m \} \cup \{ E_{x_i,x_j}; i,j = m+1,
\ldots, m+n \},$ \item $\textbf{gl}(m|n)_{1}$ is the span of
the set $\{ E_{x_i,x_j}; i =  m+1,\ldots, m+n, \    j = 1, \ldots,
m \},$ \item and $\textbf{gl}(m|n)_{k} = (0)$, for every $k
\neq -1, 0, 1.$
\end{itemize}

\begin{remark}[\sc Scheunert, \cite{Scheu}]\label{remark: Consistent graduation}
One may define another consistent $\BZ$-graduation on
$\textbf{gl}(m|n)$ by inverting the roles of ${ -1}$ and ${1}$. As a matter of fact,
the two definitions are equivalent in the
following sense:
 consider the $\BZ$-graded algebra
$$
\textbf{g}' =   \textbf{g}'_{- 1} \oplus \textbf{g}'_{0}
\oplus \textbf{g}'_{1},
$$
where
$$
\textbf{g}'_{- 1} = \textbf{gl}(m|n)_{1}, \quad
\textbf{g}'_{0} = \textbf{gl}(m|n)_{0}, \quad
\textbf{g}'_{1} = \textbf{gl}(m|n)_{- 1}
$$
and the {\it supertransposition} map 
$$M \rightarrow M^T$$
 from $\textbf{gl}(m|n)$ to $\textbf{g}'$ such that
$$
E_{x_i,x_j}^T = (-1)^{(|x_i|+|x_j|)|x_i|} \ E_{x_j,x_i}, \quad i,j
= 1, 2, \ldots, m+n.
$$
The  map 
$$M \rightarrow -M^T$$ 
is an {\it isomorphism} of
$\BZ$-graded Lie superalgebras from $\textbf{gl}(m|n)$ to
$\textbf{g}'$.
\end{remark}


\subsection{The supertrace. A consistent, supersymmetric, invariant bilinear form on $\textbf{gl}(m|n)$}

Let $M = (m_{ij})_{i,j=1, \ldots, m+m}$ be a matrix in
$\textbf{gl}(m|n) = \textbf{gl}(m|n)_{\overline 0} \oplus \textbf{gl}(m|n)_{\overline 1}$.
The {\it supertrace} of $M$ is the number
$$
str(M) = \sum_{i=1}^m \ m_{ii} - \sum_{i=m+1}^{m+n} \ m_{ii}.
$$
Following Kac  \cite{KAC1}, we define a bilinear form on
$\textbf{gl}(m|n)$ by setting
$$
(M,N) = str(MN), \quad  M, N \in \textbf{gl}(m|n).
$$
This bilinear form $(\ ,\ )$ is
\begin{itemize}
\item {\it consistent}, that is
$$
(M,N) = 0, \quad for \ all \ M \in \textbf{gl}(m|n)_{\overline 0}, \ \ N \in
\textbf{gl}(m|n)_{\overline 1};
$$
\item {\it supersymmetric}, that is
$$
(M,N) = (-1)^{|M||N|} \ (N,M), \quad for \ all \ \BZ_2-homogeneous
\ M, N  \in \textbf{gl}(m|n);
$$
\item {\it invariant with respect to the adjoint action}, that is
$$
([M,P],N) = (M,[P,N]), \quad for \ all \ M, P, N \in
\textbf{gl}(m|n).
$$
\end{itemize}





\subsection{Distinguished triangular decompositions and roots of the general linear Lie superalgebra $\textbf{gl}(m|n)$}
\label{Distinguished triangular decompositions}

Let $\textbf{h} \subseteq \textbf{gl}(m|n)_{\overline 0} \subset
\textbf{gl}(m|n)$ be the {\it Cartan subalgebra} of all diagonal
matrices, and let
$$\{ \varepsilon_i; i = 1, \ldots, m+n \}$$
be the canonical basis of the dual space $\textbf{h}^*$ (in plain
words, the evaluation of $\varepsilon_i$ on a diagonal matrix in
$\textbf{h}$ equals its $i$-th diagonal entry).

Since the adjoint representation of $\textbf{h}$ in
$\textbf{gl}(m|n)$ is diagonalizable, we may consider the
so-called {\it distinguished choice} for a triangular
decomposition of $\textbf{gl}(m|n)$ (see, e.g., \cite{KAC1},
\cite{VANDER}). This decomposition is characterized by the
following conditions:

\begin{itemize}
\item $\textbf{gl}(m|n) = \textbf{n}^- \oplus  \textbf{h} \oplus
\textbf{n}^+,$ where $\textbf{n}^-$ and $\textbf{n}^+$ are
subalgebras such that $[\textbf{h},\textbf{n}^+ ] \subset
\textbf{n}^+$, $[\textbf{h},\textbf{n}^- ] \subset \textbf{n}^-$,
\item $\textbf{gl}(m|n)_{-1} \subset \textbf{n}^+$ and
$\textbf{gl}(m|n)_{1} \subset \textbf{n}^-$.
\end{itemize}

In plain words, the subalgebras $\textbf{n}^+$ and $\textbf{n}^-$
are the algebras of all strictly upper and lower matrices,
respectively.


The algebra $\textbf{b} = \textbf{h} \oplus \textbf{n}^+$ is the
distinguished Borel subalgebra and the set of {\it positive roots}
- with respect to the choice of the Borel subalgebra $\textbf{b}$
- is the set of linear functionals
$$
\Delta_+ = \Delta_{\overline 0,+} \cup \Delta_{\overline 1,+} \subset \textbf{h}^*,
$$
where
$$
\Delta_{\overline 0,+} = \{ \varepsilon_i - \varepsilon_j; 1 \leq i < j \leq
m \} \cup \{ \varepsilon_i - \varepsilon_j; m+1 \leq i < j \leq
m+n \}
$$
is the set of positive roots of $\BZ_2$-degree $\overline 0$, and
$$
\Delta_{\overline 1,+} = \{ \varepsilon_i - \varepsilon_j; i = 1, \ldots, m,
\ j = m+1, \ldots, m+n \},
$$
is the set of positive roots of $\BZ_2$-degree $\overline 1$.





The corresponding set of simple roots (the {\it distinguished
set}, see, e.g., \cite{VANDER}) is given by
$$
\Pi = \{ \alpha_1 = \varepsilon_1 - \varepsilon_2, \ldots,
\alpha_m = \varepsilon_m - \varepsilon_{m+1}, \dots,
\alpha_{m+n-1} =\varepsilon_{m+n-1} - \varepsilon_{m+n} \}.
$$
Thus, in the distinguished set there is only one simple  root of
$\BZ_2$-degree $\overline 1$, the root $\varepsilon_m - \varepsilon_{m+1}.$

As usual, we put
$$
\rho_{\overline 0} = \frac {1} {2} \ \sum_{\alpha \in \Delta_{\overline 0,+}}  \alpha,
\quad \rho_{\overline 1} = \frac {1} {2} \ \sum_{\alpha \in \Delta_{\overline 1,+}}
\alpha, \quad  \rho = \rho_{\overline 0} - \rho_{\overline 1}.
$$

Consider the symmetric bilinear form $(\ ,\ )$ on
$\textbf{h}^*$ induced by  the invariant supersymmetric bilinear
form on $\textbf{gl}(m|n)$; in the natural basis $\{
\varepsilon_i; i = 1, \ldots, m+n \}$ it takes the form
\begin{itemize}
\item $(\varepsilon_i,\varepsilon_j) = \delta_{ij}, \quad for \
all \ i,j = 1, \ldots, m,$ \item $(\varepsilon_i,\varepsilon_j) =
0, \quad for \ all \ i = 1, \ldots, m, \ j = m+1, \ldots, m+n,$
\item $(\varepsilon_i,\varepsilon_j) = -\delta_{ij}, \quad for \
all \ i,j = m+1, \ldots, m+n.$
\end{itemize}
Note that the positive roots of $\BZ_2$-degree $\overline 1$ are {\it isotropic},
that is $(\alpha,\alpha) = 0$, for all $\alpha \in \Delta_{\overline 1,+}.$

\begin{remark} \label{remark: the form rho}
$ \ $
\begin{enumerate}
\item
$$
(\rho, \alpha_i)
=
\frac 1 2 (\alpha_i, \alpha_i) = 
\begin{cases}
    1 &   \text{for }   1\leq i < m, \\
    0 &   \text{for  }        i= m, \\
  - 1 &   \text{for   } m< i   < m+n.  \\
\end{cases}
$$
\item
\begin{align*}
(\rho, \epsilon_i - \epsilon_j)
& = 
\sum_{h=i}^{j-1} (\rho, \alpha_h)
=
\sum_{h=i}^{j-1} \frac 1 2 (\alpha_h, \alpha_h)
\\
& = 
\begin{cases}
    j-i &   \text{for }   1\leq i < j \leq m ,\\
    2m + 1 - i - j   &   \text{for }   1\leq i \leq m < j \leq m+n,\\
    i-j &   \text{for }   m < i < j \leq m+n.  \\
\end{cases}
\end{align*}
\end{enumerate}
\end{remark}

\begin{remark}\label{remark: invariant bilinear form}
$ \ $

\begin{enumerate}
\item The supersymmetric bilinear form $(\ , )$ is non-degenerate on
$\textbf{gl}(m|n)$.
\item Regarded as a $\textbf{gl}(m|n)_{\overline 0}$-module
(with respect to the adjoint action), $\textbf{gl}(m|n)_{\overline 1}$ splits
into the direct sum of the irreducible submodules:
$$
\textbf{gl}(m|n)_{\overline 1} = \textbf{gl}(m|n)_{-1} \oplus
\textbf{gl}(m|n)_{1}, \quad \overline 1 \in \BZ_2, \quad -1, 1 \in \BZ.
$$
\item
The irreducible $\textbf{gl}(m|n)_{\overline 0}$-modules
$\textbf{gl}(m|n)_{-1}$ and
$\textbf{gl}(m|n)_{1}$ are the
irreducible $\textbf{gl}(m|n)_{\overline 0}$-modules with highest weight
vectors $E_{x_1,x_{m+n}}$ and $E_{x_{m+1},x_m}$, respectively.

\end{enumerate}

\end{remark}

The representation theory of the general linear Lie superalgebras
$\textbf{gl}(m|n)$ looks like the representation theory of basic
classical simple Lie superalgebras of type I.

\subsection{Highest weight modules with integral weights and Verma modules}



Let  $\textbf{h} \subset \textbf{gl}(m|n)$ be the  Cartan
subalgebra of all diagonal matrices, and let   $\textbf{b} =
\textbf{h} \oplus \textbf{n}^+$ be the distinguished Borel
subalgebra of all upper triangular
matrices.

Let
$$
X(m|n) = \{ \mu \in \textbf{h}^*; \ \mu = \sum_{i=1}^{m+n} \ \mu_i
\ \varepsilon_i, \ \mu_i \in \BZ \}
$$
be the set of all linear functionals on $\textbf{h}$ with integral
coefficients with respect to the standard basis $\{ \varepsilon_i;
\ i = 1, \ldots, m+n \}$.

Given a $\textbf{gl}(m|n)$-supermodule $M$ and an {\it integral
weight} $\mu \in X(m|n)$, define the $\mu$-weight subspace
$M(\mu)$ of $M$ with respect to $\textbf{h}$ as usual:
$$M(\mu) = \{ m \in M; h \cdot m = \mu(h)m \ for \ all \ h \in \textbf{h} \}.$$


 A $\textbf{gl}(m|n)$-supermodule $M$ is said to be a {\it module with integral weights}
 whenever it satisfies the following condition:
$$
M = \bigoplus_{\mu \in X(m|n)} \ M(\mu).
$$

Following Brundan (\cite{BRU}, \cite{BRU1})  and  Soergel
\cite{SOERG}, the category $\mathcal{O}_{m|n}$ is the category of
all $\BZ_2$-graded $\textbf{gl}(m|n)$-modules $M$ with integral
weights  which are finitely generated
$\textbf{gl}(m|n)$-modules and are locally finite dimensional over
$\textbf{b}$ (that is, all finitely generated
$\textbf{b}$-submodules of $M$ are finite dimensional $\BC$-vector
spaces). The category $\mathcal{O}_{m|n}$ is the (integral weight)
analogue of the category $\mathcal{O}$ for semisimple Lie
algebras, introduced by Bernstein, Gelfand and Gelfand in
\cite{BGG1}.


For every $\Lambda = \sum_{i=1}^{m+n}  \Lambda_i  \varepsilon_i
\in X(m|n)$, define a  $\BZ_2$-graded one-dimensional
$\textbf{b}$-module $<v_\Lambda>$ by setting
\begin{align*}
h(v_\Lambda) &= \Lambda(h)v_\Lambda, \ \ h \in \textbf{h} \\
\textbf{n}^+(v_\Lambda) &= 0,
\end{align*}
where $v_\Lambda$ is a $\BZ_2$-homogeneous vector of degree
$\sum_{i=m+1}^{m+n} \ \Lambda_i \ (\text{mod}\ 2).$

Define the {\it Verma module} ${\tilde {\mathcal V}}(\Lambda)$  as
the induced $\textbf{gl}(m|n)$-module
$$
{\tilde {\mathcal V}}(\Lambda) = {\mathcal U}(\textbf{gl}(m|n)) \
\otimes_{{\mathcal U}(\textbf{b})} \ <v_\Lambda> \ \in
\mathcal{O}_{m|n},
$$
where ${\mathcal U}(\textbf{gl}(m|n))$ and ${\mathcal
U}(\textbf{b})$ are the universal enveloping algebras of
$\textbf{gl}(m|n)$ and $\textbf{b}$, respectively.

\begin{remark}\label{remark: Verma Modules}
The fact that the Verma modules are objects in the category
$\mathcal{O}_{m|n}$ follows from the Poincar\'{e}--Birkhoff--Witt
theorem (see, e.g., \cite{KAC1}, \cite{Scheu}).
\end{remark}

The $\textbf{gl}(m|n)$-module ${\tilde {\mathcal V}}(\Lambda)$
contains a unique maximal submodule $ I(\Lambda)$ and, therefore,
the quotient module
$$
\mathcal V(\Lambda) = {\tilde {\mathcal V}}(\Lambda) / I(\Lambda)
$$
is an irreducible $\textbf{gl}(m|n)$-module.


The modules $\mathcal V(\Lambda)$ are usually called {\it highest
weight modules (with integral weights)} (see, e.g., \cite{FSS});
the highest weight of the module $\mathcal V(\Lambda)$ is clearly
the linear functional $\Lambda \in \textbf{h}$ and its unique --- up
to a scalar factor --- highest weight vector is the class vector of
the vector $\textbf{1}\ \otimes_{{\mathcal U}(\textbf{b})}
<v_\Lambda> \ \in {\tilde {\mathcal V}}(\Lambda).$

The following result is essentially due to Kac (cf.\ \cite{KAC2},
\cite{KAC3}).

\begin{theorem}[\rm see, e.g., \cite{BRU}, \cite{BRU1}]
The set
$$
\{ \mathcal V(\Lambda); \ \Lambda \in X(m|n) \}
$$
is a complete set of pairwise non-isomorphic irreducibles in
$\mathcal{O}_{m|n}.$
\end{theorem}







\subsection{Kac modules and typical   modules}

 The category $\mathcal{F}_{m|n}$ is the category of all finite dimensional
$\textbf{gl}(m|n)$-modules with integral weights.

We recall that an integral weight $
 \Lambda \in \textbf{h}^*,$  $\Lambda = \sum_{i=1}^{m+n} \ \Lambda_i \ \varepsilon_i \ \in X(m|n)$
is said to be an {\it  integral dominant weight} whenever it
satisfies the condition
$$
a_i = \frac {2(\Lambda,\alpha_i)} {(\alpha_i,\alpha_i)} \geq 0,
$$
for every  simple root $\alpha_i \in \Pi$, $i \neq m,$
that is,
$$
 \Lambda_1 \geq  \Lambda_2 \geq \cdots  \geq \Lambda_m,  \quad and \quad
 \Lambda_{m+1} \geq  \Lambda_{m+2} \geq \cdots \geq \Lambda_{m+n}.
$$

The set of all integral dominant  weights in $X(m|n)$ is denoted
by the symbol $X^+(m|n).$







Let $\mathcal V_{\overline 0}(\Lambda)$ be the irreducible
$\textbf{gl}(m|n)_{\overline 0}$-module with highest integral dominant weight
$\Lambda \in X^+(m|n);$ $\mathcal V_{\overline 0}(\Lambda)$ is turned into a
$\BZ_2$-graded module by specifying that its highest weight is a
$\BZ_2$-homogeneous vector of degree $\sum_{i=m+1}^{m+n} \
\Lambda_i \ (\text{mod}\ 2).$

Consider the  subalgebra $\textbf{p} =
\textbf{gl}(m|n)_{0} \oplus
\textbf{gl}(m|n)_{-1} \subset \textbf{gl}(m|n).$

The $\textbf{gl}(m|n)_{\overline 0}$-module $\mathcal V_{\overline 0}(\Lambda)$ is
extended to a $\textbf{p}$-module by setting
$$
\textbf{gl}(m|n)_{-1} \cdot \mathcal V_{\overline 0}(\Lambda) = 0.$$
The induced $\textbf{gl}(m|n)$-module
$$
\overline{\mathcal V} (\Lambda) = {\mathcal U}(\textbf{gl}(m|n)) \
\otimes_{{\mathcal U}(\textbf{p})} \ \mathcal V_{\overline 0}(\Lambda)
$$
is said to be a {\it Kac module}.

\begin{remark}\label{remark: finite dimensional}
The Kac module $\overline{\mathcal V} (\Lambda)$ is a finite
dimensional vector space isomorphic to the tensor product
$\bigwedge(\textbf{gl}(m|n)_{1}) \bigotimes \mathcal
V_{\overline 0}(\Lambda).$
\end{remark}

The Kac modules are not, in general, irreducible
$\textbf{gl}(m|n)$-modules; however, if $\overline{\mathcal V}
(\Lambda)$ is not an irreducible module, then it contains a unique
maximal submodule $\overline{\mathcal I}(\Lambda)$ and the
quotient module $ \overline{\mathcal V} (\Lambda) \diagup
\overline{\mathcal I}(\Lambda)$ is an irreducible module
isomorphic to the module $\mathcal V(\Lambda)$ defined in the
preceding subsection.

In general, we have the following result.

\begin{proposition}[\cite{KAC2}, \cite{KAC3}, \cite{FSS}, \cite{BRU}]
\label{proposition: typical modules}
Let $\Lambda \in X^+(m|n)$ be an integral dominant   weight. Then,
the following statements are equivalent.
\begin{enumerate}
\item $\mathcal V(\Lambda) = \overline{\mathcal V} (\Lambda)$.
\item The Kac module $\overline{\mathcal V} (\Lambda)$ is an
irreducible $\textbf{gl}(m|n)$-module.
\item $(\Lambda + \rho, \alpha) \neq 0$, for all positive roots $\alpha \in \Delta_+$.
\end{enumerate}
\end{proposition}

A finite dimensional irreducible $\textbf{gl}(m|n)$-module
$\mathcal V(\Lambda)$ with highest weight $\Lambda$ is called {\it
typical} whenever one of the equivalent conditions 1--3 of
Proposition~\ref{proposition: typical modules} is satisfied.









The following result is essentially due to Kac (cf.\ \cite{KAC2},
\cite{KAC3}).


\begin{theorem}[\rm see, e.g., \cite{BRU}, \cite{BRU1}]
\begin{itemize}
\item The set
$$
\{ \mathcal V(\Lambda); \ \Lambda \in X^+(m|n) \}
$$
is a complete set of pairwise non-isomorphic irreducibles in
$\mathcal{F}_{m|n}.$
\item Any finite dimensional irreducible
$\textbf{gl}(m|n)$-module is either typical or it can be obtained
from $\mathcal V(\Lambda)$ for some integral dominant weight
$\Lambda \in X^+(m|n)$ by tensoring with a one dimensional
representation.
\end{itemize}
\end{theorem}



\subsection{Covariant modules, Schur modules and letterplace superalgebras}
Let $V = V_{\overline 0} \oplus V_{\overline 1}$ be a $\BZ_2$-graded vector space,
$\dim (V_{\overline 0}) = m$, $\dim (V_{\overline 1}) = n$, and let $\mathcal L = \{x_1,
\ldots, x_m, x_{m+1}, \ldots,x_{m+n} \}$ be a {\it distinguished}
$\BZ_2$-homogeneous basis of $V$, $|x_i| = \overline 0$ for every $i = 1,
\ldots, m$, $|x_i| = \overline 1$ for every $i = m+1, \ldots, m+n$.

Let $\lambda \vdash N$ be a partition, and let $\lambda \in
H(\mathcal L)$, where
$$
H(\mathcal L) = \{ \lambda = ( \lambda_1 \geq \lambda_2 \geq
\cdots); \lambda_{m+1} < n+1 \}
$$
is the hook set of $\mathcal L$ (see Subsection~5.4).

The {\it covariant} modules ${\bf V}_\lambda$ are usually defined
as the finite dimensional irreducible $\textbf{gl}(m|n)$-modules
that appear in the Berele--Regev complete decomposition of the\break
$\textbf{gl}(m|n)$-module $V^{\otimes N}$ (see also Sergeev
\cite{SERG1}).

We know that the covariant modules ${\bf V}_\lambda$ admit a quite
explicit description as submodules of a letterplace algebra, and
Theorem~\ref{theorem: decomposition of Super} implies that all the
$pl(V)$-irreducible submodules of a letterplace algebra are indeed
covariant modules (up to isomorphism).

Furthermore, from Section~11 we infer that the covariant modules
admit an even more manageable combinatorial description. To be
precise, we have the following result.
\begin{proposition}
Let $\lambda$ be a partition, and let $\lambda \in H(\mathcal L)$.
The covariant module ${\bf V}_\lambda$ is
$\textbf{gl}(m|n)$-isomorphic to the module
$$
{\cal S}_\lambda = \langle  (S\,|\,D_\lambda); S \in Tab_\lambda(
\mathcal L ) \rangle_\BK \subseteq Super[\mathcal L|\mathcal P],
$$
where $\mathcal P = \mathcal P_{\overline 1} = \{1, 2, \ldots, t \}$, $t \geq
\lambda_1$ and $D_\lambda$ is a Deruyts tableau $\in Tab_\lambda(
\mathcal P ).$
\end{proposition}


\begin{remark}\label{remark: covariant modules}

\begin{enumerate}
\item The covariant module ${\cal S}_\lambda$ has a standard basis
(see Subsection~8.3):
$$
\{  (S\,|\,D_\lambda); S \in Stab_\lambda( \mathcal L ) \}.
$$
Since the place tableau $D_\lambda$ is of Deruyts type, the
bitableaux $(S\,|\,D_\lambda)$ are skew-symmetric in the rows of
$D_\lambda$. From Subsection~8.3, it follows that the process of
expanding the generators $(S\,|\,D_\lambda)$,  $S \in Tab_\lambda(
\mathcal L )$, of ${\cal S}_\lambda$ into  linear combinations of
the standard basis elements is ruled by the following special form
of the Straightening Law (see Theorem~\ref{theorem: Straightening Law}):
$$
\sum_{(v)} \left(
\begin{array} {l|l}
u v_{(1)} & x \\
v_{(2)} w & y
\end{array}
\right) = 0,
$$
where $x = 12 \cdots h$, $y = 12 \cdots k$, $h \geq k$, and $v$ is
a word of length greater than $h$. The above identities are known
as the ``{\it exchange rules}" (see, e.g., \cite{GRS}). \item The
covariant module ${\cal S}_\lambda$ has a Clebsch--Gordan--Capelli
basis (see Subsection~9.4); namely,
$$
\Big\{ \Big(\fbox{$S$}\,\Big|\,D_\lambda\Big); S \in Stab_\lambda( \mathcal L ) \Big\}.
$$
\end{enumerate}
\end{remark}

Let $S \in Tab_\lambda(\mathcal L)$. We recall that the content
$c(S)$ of the tableau $S$ is the vector
$$
c(S) = (c(S,x_1), \ldots, c(S,x_m), c(S,x_{m+1}), \ldots,
c(S,x_{m+n})),
$$
where $c(S,x_i)$ is the number of occurrences of the letter $x_i$
in the tableau $S$, for $i = 1, 2, \ldots, m+n.$

Let $\lambda \vdash N$ be a partition, and let $\lambda \in
H(\mathcal L)$. An element
$$
\varphi = \sum_{i=1}^{m+n} \ \varphi_i \ \varepsilon_i, \quad
\varphi_i \in \BZ
$$
of the dual space $\textbf{h}^{*}$ is called a {\it standard
weight} of the covariant module ${\cal S}_\lambda$ if there exists
a standard tableau $S \in Stab_\lambda(\mathcal L)$ such that
$$
(\varphi_1, \varphi_2, \ldots, \varphi_{m+n}) = c(S).
$$
From Remark~\ref{remark: finite dimensional}, it follows that the
module ${\cal S}_\lambda$ is $\textbf{h}$-diagonalizable with
weight decomposition
$$
{\cal S}_\lambda = \bigoplus_\varphi \ {\cal S}_\lambda(\varphi),
$$
where the direct sum is over the set of all standard weights
$\varphi$ of the covariant module ${\cal S}_\lambda.$



We come now to the main combinatorial definition of the present
subsection.

The standard tableau
$$
F_\lambda = (w_1, w_2, \ldots, w_p) \in   Tab_\lambda(\mathcal L)
$$
is the tableau  defined by the following conditions:
\begin{itemize}
\item $w_i = x_i^{\lambda_i}$, for all  $i \in \BZ^+, \ i \leq m$,
\item $w_i = x_{m+1} x_{m+2} \cdots x_{m+\lambda_i}$, for all $i
\in \BZ^+, \ m < i \leq m+n$.
\end{itemize}

Note that $F_\lambda$ is a tableau of Co-Deruyts type if $n = 0$,
and $F_\lambda$ is a tableau of Deruyts type if $m = 0$.



The bitableau $(F_\lambda\,|\,D_\lambda)$ plays a crucial role in the
theory of the covariant module ${\cal S}_\lambda$. We notice that
the bitableau $(F_\lambda\,|\,D_\lambda)$ is (up to a scalar factor)
the unique weight vector of ${\cal S}_\lambda$ of  weight
$$
\Lambda_\lambda = \sum_{i=1}^m \ \lambda_i \varepsilon_i \ + \
\sum_{j=1}^{n} \ \nu_{m+j} \varepsilon_{m+j},
$$
where $\nu_{m+j} = \max \{ 0, \tilde \lambda_j - m \}$,
and $\tilde \lambda = (\tilde \lambda_1 \geq
\tilde \lambda_2 \geq \cdots)$ denotes the conjugate partition of
the partition $\lambda.$

We have the following results.

\begin{lemma}\label{lemma: highets contents}
Consider the  set of ordered $(m+n)$-tuples in $\BN^{m+n}$
$$
 \{ c(S); \ S \in Stab_\lambda( \mathcal L ) \}
$$
as a linearly ordered set with respect to the lexicographic order.
Then
$$
c( F_\lambda ) = \max \{ c(S); \ S \in Stab_\lambda( \mathcal L )
\}.
$$
\end{lemma}

\begin{proof}
The assertion immediately follows from Remark~\ref{remark: finite
dimensional}.
\end{proof}

\begin{proposition}\label{proposition: main fact on covariant modules}
\begin{enumerate}
\item The bitableau $(F_\lambda\,|\,D_\lambda)$   is  annihilated  by
all strictly upper  polarizations   ${\mathcal D}_{x_i x_j},$
 $i < j$, $i, j = 1, 2, \ldots,m+n.$
\item The elements of the basis $\big\{\big(\fbox{$S$}\,\big|\,D_\lambda\big); S \in
Stab_\lambda(\mathcal L) \big\}$ of ${\cal S}_\lambda$ may be obtained
from $(F_\lambda\,|\,D_\lambda)$ by iterated actions of strictly lower
polarization ${\mathcal D}_{x_i x_j}$, $i > j$, $i, j = 1, 2,
\ldots,m+n.$
\item The bitableau $(F_\lambda\,|\,D_\lambda)$ is (up to
a scalar factor) the unique element of ${\cal S}_\lambda$ which is
annihilated  by all   strictly upper  polarizations   ${\mathcal
D}_{x_i x_j},$
 $i < j$, $i, j = 1, 2, \ldots,m+n.$
\end{enumerate}
\end{proposition}

\begin{proof}
1) Since ${\mathcal D}_{x_i x_j} (F_\lambda\,|\,D_\lambda)$, $i < j$,
is a linear combination of bitableaux $(T\,|\,D_\lambda)$ with $c(T) >
c(F_\lambda)$ in the lexicographic order, the assertion follows
from Remark~\ref{remark: finite dimensional} and 
Lemma~\ref{lemma: highets contents}.

2) We know that the action of the Capelli operator $[\tilde
S\,|\,\tilde F_\lambda]_-$ on the bitableau $(F_\lambda\,|\,D_\lambda)$
yields the following result:
$$
[\tilde S\,|\,\tilde F_\lambda]_- \left( (F_\lambda\,|\,D_\lambda) \right)
= c_{F_\lambda}  \Big(\fbox{$S$}\,\Big|\,D_\lambda\Big),
$$
where $c_{F_\lambda}$ is a non-zero integer (see Subsection~8.2).

When we devirtualize the Capelli operator $[\tilde S\,|\,\tilde
F_\lambda]_-$, we may write it as a linear combination of products
$$
{\mathcal D}_1 \ {\mathcal D}_2 \ {\mathcal D}_3,
$$
where $ {\mathcal D}_1 $ is a product of strictly lower
polarizations ${\mathcal D}_{x_i x_j}$, $i > j$, $i, j = 1, 2,
\ldots, m+n$,

$ {\mathcal D}_2 $ is a product of diagonal polarizations
${\mathcal D}_{x_i x_i}$,  $i = 1, 2, \ldots,m+n$, and $
{\mathcal D}_3 $ is a product of strictly upper polarizations
${\mathcal D}_{x_h x_k}$, $h < k$, $h, k = 1, 2, \ldots,m+n,$ by
the ``easy" part of the Poincar\'{e}--Birkhoff--Witt theorem (see,
e.g., \cite{KAC1}, \cite{Scheu}, \cite{KAP}).

Then, by assertion 1), the action of the operator $[\tilde
S\,|\,\tilde F_\lambda]_-$ on $(F_\lambda\,|\,D_\lambda)$ is the same as
the action  of an operator which may be written as a linear
combination of products of strictly lower polarization ${\mathcal
D}_{x_i x_j}$, $i > j$, $i, j = 1, 2, \ldots,m+n.$

3)  If an element of $ \cal F \in {\cal S}_\lambda$ satisfies the
conditions of assertion 1), then the cyclic module generated by $
\cal F$ (with respect to the actions of the strictly lower
polarization ${\mathcal D}_{x_i x_j}$, $i > j$, $i, j = 1, 2,
\ldots,m+n$) equals the $\textbf{gl}(m|n)$-irreducible module
${\cal S}_\lambda$. Thus, the assertion immediately follows from
Lemma~\ref{lemma: highets contents}.
\end{proof}



 Proposition~\ref{proposition: main fact on covariant modules} implies the following result.
\begin{corollary}\label{corollary: weights in covariant}
The  vector $(F_\lambda\,|\,D_\lambda)$ is the highest weight vector of
the covariant module ${\cal S}_\lambda$, and, thus, ${\cal S}_\lambda$ is isomorphic to the
highest weight module $\mathcal V(\Lambda_\lambda)$
 with highest weight $\Lambda_\lambda$
(with respect to the distinguished set of positive roots defined
in Subsection~20.3).
\end{corollary}


Covariant modules are not, in general, typical modules.

\begin{proposition} \label{proposition: typical covariant modules}
The covariant module ${\cal S}_\lambda$ is a typical module if and only if
    $\lambda_m \geq n.$
\end{proposition}
\begin{proof}
From Remark~\ref{remark: the form rho}, it follows that:

\begin{itemize}
\item
\begin{align*}
(\Lambda_\lambda + \rho, \epsilon_i - \epsilon_j)
& = 
(\Lambda_\lambda, \epsilon_i -\epsilon_j) + (\rho, \epsilon_i - \epsilon_j)
\\
& = 
\begin{cases}
    \lambda_i - \lambda_j + j-i > 0         &   \text{for }   1\leq i < j \leq m ,\\
    \lambda_i + \nu_j + 2m +1 - i - j   &   \text{for }   1\leq i \leq m < j \leq m+n,\\
   -\nu_i + \nu_j + i-j < 0         &   \text{for }   m < i < j \leq m+n.  \\
\end{cases}
\end{align*}


\item
$$
\lambda_i + \nu_j + 2m + 1 - i - j \neq 0,
\quad \text{for} \quad
1\leq i \leq m < j \leq m+n.
$$
In particular, for $i=m$ we have the set of conditions
\begin{align*}
\lambda_m + \nu_{m+1} &\neq 0,
\\
\lambda_m + \nu_{m+2} &\neq 1,
\\
\lambda_m + \nu_{m+3} &\neq 2,
\\
\vdots
\\
\lambda_m + \nu_{m+n} &\neq n-1,
\end{align*}
which is equivalent to the single condition
$$
\lambda_m \geq n.
$$
In turn, this condition implies
\begin{multline*}
\lambda_i + \nu_j + 2m + 1 - i - j
\geq
n + 0 + 2m + 1 - m - (m+n) = 1 ,
\\ \text{for} \quad
1\leq i \leq m < j \leq m+n.
\end{multline*}
\end{itemize}

\end{proof}

\begin{remark}\label{remark: Highest weight in covariant}
\begin{enumerate}
\item The highest weights $\Lambda_\lambda$ were  computed  by Van
der Jeugt et al. in \cite{VANDER2} and by Cheng and Wang in
\cite{CW}.



\item The covariant modules are  ``tame" modules in the sense of
Kac and Wakimoto \cite{KAC4}.

Th recent result \cite{VANDER} by Moens and Van der Jeugt
 has a significant consequence in the theory of supersymmetric Schur polynomials $s_\lambda(x/y)$.
These polynomials appeared in the work of Berele and Regev
\cite{Berele2} and turn out to be the characters of the covariant
modules, regarded as irreducible $\textbf{gl}(m|n)$-modules. Since
the covariant modules are tame, one can apply the character
formula of Kac and Wakimoto \cite{KAC4}, and a nice determinantal
formula holds for the polynomials $s_\lambda(x/y)$ (cf.\
\cite[Formula~(1.17)]{VANDER}).

\end{enumerate}
\end{remark}

\subsection{The basic plethystic superalgebras $ \textbf S( \textbf S^k(V))$ and $ \bigwedge( \textbf S^k(V))$}

In a series of rather recent papers, Cheng and Wang (\cite{CW},
\cite{CW1}) and Sergeev (\cite{SERG2}, \cite{SERG3}) independently
rediscovered our complete decomposition Theorem~\ref{theorem: decomposition of Super} from \cite{Brini1}
for the letterplace algebra (regarded as the supersymmetric
algebra of the tensor product of a pair of finite dimensional
$\BZ_2$-graded vector spaces, see Subsection~5.3) and found
explicit formulas for the highest weight vectors. Their method is
essentially based on the notion of {\it Howe duality} (see, e.g
\cite{Howe}, \cite{Howe1} and Theorem~\ref{theorem: double centralizer}).



Furthermore, they describe the complete decompositions of the
supersymmetric and superexterior algebras $\textbf S (\textbf S^2
(V))$ and $\bigwedge (\textbf S^2 (V))$ of the supersymmetric
square of the natural representations of $\textbf{gl}(m|n),$ and
ask (see \cite[Introduction]{CW}) ``whether the results concerning the
decomposition of $\textbf S (\textbf S^2 (V))$ (respectively
$\bigwedge (\textbf S^2 (V))$) and the highest weight vectors in
these modules may also be obtained with extra insights from the
combinatorial approach in \cite{Brini1} as well."

In this subsection, we briefly describe the main connections
between the letterplace algebra approach and the above-mentioned
work.

First of all, we remark that the highest weight vectors in
$Super[\mathcal L|\mathcal P]$ can be explicitly described by
combining Corollary~\ref{corollary: weights in covariant}  and
Theorem~\ref{theorem: decomposition of Super}.

With reference to the modules $\textbf S (\textbf S^2 (V))$ and
$\bigwedge (\textbf S^2 (V))$, we mention that they have been
systematically studied by Brini, Huang and Teolis  \cite{BHT} and
Grosshans \cite{Gr1}, in the more
general setting of ``plethystic superalgebras".

For the convenience of the reader, we recall some basic
definitions from  \cite{BHT}.


Let $W = W_{\overline 0} \oplus W_{\overline 1}$ be a finite dimensional $\BZ_2$-graded
vector space.

In order to simplify  notations, in the remainder of this
subsection we write $ \textbf S(W)$ for the supersymmetric algebra
$Super[W] = Sym(W_{\overline 0}) \otimes \Lambda(W_{\overline 1})$ and $ \bigwedge(V)$ for
the superexterior algebra $Super[W] = \Lambda(W_{\overline 0}) \otimes
Sym(W_{\overline 1}).$

We recall that $ \textbf S^k(W)$ is a $\BZ_2$-graded vector space,
where
$$
 \textbf S^k(W)_{\overline 0} = \bigoplus_m (Sym^{k-2m}(W_{\overline 0}) \otimes \Lambda^{2m}(W_{\overline 1})),
$$
and
$$
 \textbf S^k(W)_{\overline 1} = \bigoplus_m (Sym^{k-2m-1}(W_{\overline 0}) \otimes \Lambda^{2m+1}(W_{\overline 1})).
$$

Let $V = V_{\overline 0} \oplus V_{\overline 1}$ be a finite dimensional $\BZ_2$-graded vector space,
$\dim (V_{\overline 0}) = m$, $\dim (V_{\overline 1}) = n.$

The {\it basic k-th plethystic (super)symmetric superalgebra} of $V = V_{\overline 0} \oplus
V_{\overline 1}$ is the superalgebra
$$
\textbf S(\textbf S^k(V)) = Sym( \textbf S^k(V)_{\overline 0}) \otimes
\Lambda( \textbf S^k(V)_{\overline 1}),
$$
and the {\it basic k-th plethystic (super)exterior superalgebra}
of   $V = V_{\overline 0}
\oplus V_{\overline 1}$ is the superalgebra
$$
\bigwedge(\textbf S^k(V)) = \Lambda( \textbf S^k(V)_{\overline 0}) \otimes
Sym( \textbf S^k(V)_{\overline 1}).
$$

Let $\textbf S^h(\textbf S^k(V))$ be the $\BZ$-homogeneous
component of degree $h$ of $\textbf S(\textbf S^k(V))$,
$$
\textbf S^h(\textbf S^k(V)) = \bigoplus_p (Sym^p(\textbf S^k(V)_{\overline 0})
\otimes \Lambda^{h-p}(\textbf S^k(V)_{\overline 1}),
$$
and let $\bigwedge^h(\textbf S^k(V))$ be the $\BZ$-homogeneous
component of degree $h$ of $\bigwedge(\textbf S^k(V)$,
$$
\bigwedge^h(\textbf S^k(V)) = \bigoplus_p (\Lambda^p(\textbf
S^k(V)_{\overline 0}) \otimes Sym^{h-p}(\textbf S^k(V)_{\overline 1}).
$$
The  $\BZ_2$-graded vector spaces $\textbf S^h(\textbf S^k(V))$
and $\bigwedge^h(\textbf S^k(V))$ are in a
natural way $pl(V)$-modules.

In  \cite{BHT} and \cite{Gr1}, the modules $\textbf S^h(\textbf
S^k(V))$ and $\bigwedge^h(\textbf S^k(V))$ are shown to be
epimorphic images of suitable letterplace algebras under the so-called {\it umbral operator} $U$ (see also
\cite{GRS}). The operator $U$ is a $pl(V))$-equivariant operator.

Since the letterplace algebras are semisimple modules 
(Theorem~\ref{theorem: decomposition of Super}),
it follows that the modules $\textbf S^h(\textbf S^k(V))$ and
$\bigwedge^h(\textbf S^k(V))$ are semisimple modules and all their
irreducible submodules are explicitly constructed; these
irreducible submodules are  covariant modules. The multiplicities
of the covariant modules ${\cal S}_\lambda$ in a complete
decomposition is described in a rather implicit way; these
multiplicities are shown to be equal to the $\BK$-linear
dimensions of suitable representations of the symmetric group
${\mathcal S}_h$ (\cite[Theorem~12]{BHT},  and 
\cite[Theorems~13 and 17]{Gr1}).


In the case $k = 2$ (i.e., the case of {\it supersymmetric
matrices}), the situation is much more satisfactory: the
$pl(V)$-modules $\textbf S^h(\textbf S^2(V))$ and
$\bigwedge^h(\textbf S^2(V))$ are described in detail in 
Sections~4 and 6 of \cite{BHT}. In particular,
Straightening Laws for both the modules of type $\textbf
S^h(\textbf S^2(V))$ and for  the modules of type
$\bigwedge^h(\textbf S^2(V))$ are provided; these Straightening
Laws are closely related to the work of De Concini and Procesi  \cite{DP} 
and Rota and Stein
(cf.\ \cite{RS1}, \cite{RS2}). Furthermore, two classes of
Clebsch--Gordan--Capelli bases are exhibited for both  classes of
modules in \cite[Sections~3 and 4]{BHT}.

We have the following  structure theorems.

\begin{theorem}[see \cite{BHT}, \cite{Gr1}, \cite{CW}, \cite{SERG3}]

\begin{enumerate}
\item The $pl(V)$-module $\textbf S^h(\textbf S^2(V))$ is a
mul\-ti\-pli\-ci\-ty-free module. We have the following complete
decomposition result:
$$
\textbf S^h(\textbf S^2(V)) \cong \bigoplus_\lambda \ {\cal
S}_\lambda,
$$
where the direct sum ranges over all the partitions $\lambda =
(\lambda_1, \lambda_2, \ldots ) \vdash 2h$, $\lambda_i$ even for
every $i$, $\lambda_{m+1} \leq n.$
\item The $pl(V)$-module
$\bigwedge^h(\textbf S^2(V))$ is a multiplicity-free module. We
have the following complete decomposition result:
$$
\bigwedge^h(\textbf S^2(V)) \cong \bigoplus_\lambda \ {\cal
S}_\lambda,
$$
where the direct sum ranges over all the partitions $\lambda =
(\lambda_1, \lambda_2, \ldots ) \vdash 2h$ which are obtained by
nesting $(q+1,q)$-hooks, $\lambda_{m+1} \leq n.$

\end{enumerate}

\end{theorem}






\newpage


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\def\topsep{0pt}
\def\parsep{0pt plus 5pt minus 1pt}
\def\itemsep{-0.5ex} %seems to be a nice small skip between items
\small               %seems to be the best


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\end{thebibliography}

\small \vskip 1pc

{\obeylines \noindent Andrea  Brini

\noindent Dipartimento di Matematica, Universit\`a di Bologna

\noindent 40126 Bologna, Italy

\noindent E-mail: brini@dm.unibo.it \vskip 1pc}
\end{document}





LISTA RIFERIMENTI INTERNI

T1) \label{theorem: action of Capelli-type operator}

P7) \label{proposition: Capelli bitableaux and Capelli rows}

T3) \label{theorem: Capelli bitableaux and products of Capelli
Rows}

T4) \label{theorem: Laplace expansion type identities}

P4) \label{proposition: polarization of biproduct}

P5) \label{proposition: polarization of bitableaux}

S \label{theorem: Straightening Law})

L1) \label{lemma: -+ triangulariy}

L2) \label{lemma: symmetry coefficients}

L3) \label{lemma: nondegeneracy}

L4) \label{lemma: +- triangularity}

L5) \label{lemma: -+ operator triangulariy}

T8) \label{theorem: Clebsch--Gordan--Capelli basis}

T4) \label{theorem: triangularity of Young--Capelli symmetrizers
and symmetrized bitableaux}

T5) \label{theorem: action of orthorormal generators on
symmetrized bitableaux}

P3) \label{proposition: Schur modules}

T2) \label{theorem: decomposition of Super}

T6) \label{theorem: decomposition of B}

T7) \label{theorem: double centralizer}

T9) \label{theorem: natural form of irreducible matrix
representation}

P6) \label{proposition: $gl$ and $sl$ representation}

T7) \label{theorem: characterization of elements annihilated by
strictly upper polarizations}

C1) \label{corollary: characterization of highest weight vectors}

C2) \label{corollary: modules generated by highest weight vectors}

P8) \label{proposition: maximal sets of linearly independent
highest weight vectors}

P2) \label{proposition: typical modules}
