Séminaire Lotharingien de Combinatoire 55 (2007), Article B55g

COMBINATORICS, SUPERALGEBRAS, INVARIANT THEORY AND REPRESENTATION

THEORY

A. Brini

Dipartimento di Matematica

“Alma Mater Studiorum” Università degli Studi di Bologna

Abstract

We provide an elementary introduction to the (characteristic zero) theory of Letterplace Superalgebras, regarded as bimodules with respect to the superderivation actions of a pair of general linear Lie superalgebras, and discuss some applications.

Contents

1 Introduction

2 Synopsis

I The General Setting

3 Superalgebras 3.1 The supersymmetric superalgebra of a ℤ_{2}-graded vector space . . . . . 3.2 Lie superalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Basic example: the general linear Lie superalgebra of a ℤ_{2}-graded vector

space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 The supersymmetric superalgebra Super[A] of a signed set A . . . . . 3.5 ℤ_{2}-graded bialgebras. Basic definitions and the Sweedler notation … 3.6 The superalgebra Super[A] as a ℤ_{2}-graded bialgebra. Left and right

derivations, coderivations and polarization operators . . . . . . . . . . .

4 The Letterplace Superalgebra as a Bimodule 4.1 Letterplace superalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Superpolarization operators . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Letterplace superalgebras and supersymmetric algebras: the classical

description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 General linear Lie superalgebras, representations and polarization oper-

ators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 General linear groups and even polarization operators . . . . . . . . . .

5 Tableaux 5.1 Young tableaux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Co-Deruyts and Deruyts tableaux . . . . . . . . . . . . . . . . . . . . . 5.3 Standard Young tableaux . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 The Berele–Regev hook property . . . . . . . . . . . . . . . . . . . . . 5.5 Orders on tableaux . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

II The General Theory

6 The Method of Virtual Variables 6.1 The metatheoretic significance of Capelli’s idea of virtual variables … 6.2 Tableau polarization monomials . . . . . . . . . . . . . . . . . . . . . . 6.3 Capelli bitableaux and Capelli rows . . . . . . . . . . . . . . . . . . . . 6.4 Devirtualization of Capelli rows and Laplace expansion type identities . 6.5 Basic examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7 Biproducts and Bitableaux in Super[L|P] 7.1 Capelli rows and supersymmetries in Super[L|P] . . . . . . . . . . . . 7.2 Biproducts as basic symmetrized elements in Super[L|P] . . . . . . . . 7.3 Bitableau monomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Bitableaux in Super[L|P] . . . . . . . . . . . . . . . . . . . . . . . . .

8 The Standard Basis 8.1 The Straightening Law of Grosshans, Rota and Stein. . . . . . . . . . . 8.2 Triangularity and nondegeneracy results . . . . . . . . . . . . . . . . . 8.3 The standard basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 An invariant filtration associated to the standard basis . . . . . . . . .

9 Clebsch–Gordan–Capelli Expansions 9.1 Right symmetrized bitableaux . . . . . . . . . . . . . . . . . . . . . . . 9.2 Left symmetrized bitableaux . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Doubly symmetrized bitableaux . . . . . . . . . . . . . . . . . . . . . . 9.4 Clebsch–Gordan–Capelli bases of Super_{n}[L|P] . . . . . . . . . . . . . .

10 Young–Capelli Symmetrizers and Orthonormal Generators 10.1 Young–Capelli symmetrizers . . . . . . . . . . . . . . . . . . . . . . . . 10.2 The Triangularity Theorem . . . . . . . . . . . . . . . . . . . . . . . . 10.3 Orthonormal generators . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4 Factorization properties . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5 Place operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11 Schur and Weyl Modules 11.1 Schur modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Weyl modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 The Schur–Weyl correspondence . . . . . . . . . . . . . . . . . . . . . .

12 Decomposition Theorems

13 The Natural Form of Irreducible Matrix Representations of Schur Superalgebras

III Applications

14 Decomposition of Tensor Products of Spaces of Symmetric and Skew-Symmetric Tensors

15 Letterplace Algebras, Highest Weight Vectors and sl_{m}(ℂ)-Irreducible Modules 15.1 Representations of sl_{m}(ℂ): basic definitions and results . . . . . . . . . 15.2 sl_{m}(ℂ)-irreducible modules and gl_{m}(ℂ)-irreducible modules . . . . . . . 15.3 Letterplace algebras and representations of sl_{m}(ℂ) . . . . . . . . . . . . 15.4 The action of upper polarizations . . . . . . . . . . . . . . . . . . . . . 15.5 The action of lower polarizations . . . . . . . . . . . . . . . . . . . . . 15.6 Highest weight vectors and complete decompositions . . . . . . . . . . 15.7 Letterplace algebras and complete sets of pairwise non-isomorphic irre-

ducible sl_{m}(ℂ)-representations . . . . . . . . . . . . . . . . . . . . . . .

16 Deruyts’ Theory of Covariants (after J. A. Green) 16.1 Left and right actions of GL_{n}(𝕂) on 𝕂[A] and 𝕂[X] . . . . . . . . . . . 16.2 Invariants and covariants . . . . . . . . . . . . . . . . . . . . . . . . . . 16.3 Isobaric elements and semiinvariants. Weights . . . . . . . . . . . . . . 16.4 The map σ : 𝕂[A] → 𝕂[A] ⊗ 𝕂[X] . . . . . . . . . . . . . . . . . . . . 16.5 Left and right spans of covariants of weight zero . . . . . . . . . . . . . 16.6 Primary covariants and irreducible GL_{n}(𝕂)-representations . . . . . . . 16.7 The Deruyts–Capelli expansion . . . . . . . . . . . . . . . . . . . . . .

16.8 On the complete reducibility of the span of covariants of weight zero. A

“pre-Schur” description of polynomial GL_{n}(𝕂)- irreducible representations 16.9 Weyl’s First Fundamental Theorem . . . . . . . . . . . . . . . . . . . . 16.10 The Capelli identities . . . . . . . . . . . . . . . . . . . . . . . . . . .

17 ℤ_{2}-Graded Tensor Representations: the Berele–Regev Theory 17.1 The letter multilinear subspace as a 𝕂[S_{n}]-module . . . . . . . . . . . . 17.2 The ℤ_{2}-graded tensor representation theory . . . . . . . . . . . . . . .

18 The Symmetric Group 18.1 The doubly multilinear subspace as a 𝕂[S_{n}]-module . . . . . . . . . . . 18.2 Complete decompositions of the group algebra 𝕂[S_{n}] as a left regular

module. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.3 On the coefficients h_{λ} (Lemma 8.3 and Theorem 10.1). . . . . . . . . . 18.4 The Young natural form of irreducible representations . . . . . . . . . .

IV A Glimpse of the General Representation Theory of Finite Dimensional Lie Superalgebras

19 A Brief Historical Outline of the Theory of Representations of Finite Dimensional Lie Superalgebras over the Complex Field (after V. G. Kac)

20 General Linear Lie Superalgebras.

Highest Weight Modules, Kac Modules, Typical Modules and Covariant Modules 20.1 ℤ_{2}-homogeneous bases of V = V_{0} ⊕ V_{1} and consistent ℤ-graduations of

a general linear Lie superalgebra gl(m|n) = pl(V) . . . . . . . . . . . . 20.2 The supertrace. A consistent, supersymmetric, invariant bilinear form

on gl(m|n) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.3 Distinguished triangular decompositions and roots of the general linear

Lie superalgebra gl(m|n) . . . . . . . . . . . . . . . . . . . . . . . . . . 20.4 Highest weight modules with integral weights and Verma modules … 20.5 Kac modules and typical modules . . . . . . . . . . . . . . . . . . . . . 20.6 Covariant modules, Schur modules and letterplace superalgebras …. 20.7 The basic plethystic superalgebras S(S^{k}(V)) and ⋀(S^{k}(V)) . . . . . . .

1 Introduction

The purpose of the present work is to provide an elementary introduction to the (characteristic zero) theory of Letterplace Superalgebras, regarded as bimodules with respect to the superderivation actions of a pair of 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, supersymmetry (see, e.g., [52], [39], [88], [2], [29], [37], [60]). General linear Lie superalgebra actions on letterplace superalgebras yield a natural setting that allows Capelli’s method of 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 [11], to introduce the notion of Young–Capelli symmetrizers, and to provide a “natural matrix form” of Schur superalgebras [12]. 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:

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

• The classical representation theory of general linear and symmetric groups over tensor spaces, as well as its pioneering generalization to the ℤ_{2}-graded case due to Berele and Regev [4], [5].

• Vector invariant theory (see, e.g., [92], [38], [34], [31]).

• After the work of Grosshans, Rota and Stein [46], letterplace superalgebras provide a unified language for the symbolic representation of invariants of symmetric tensors (Aronhold symbols) and skew-symmetric tensors (Weitzenböck’s Komplex-Symbolik) (see, e.g., Weyl [92], Brini, Huang and Teolis [16], Grosshans [47]).

• The Deruyts theory of covariants of weight zero [33], 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}(ℂ). This work of Deruyts has been defined by Green “a pearl of nineteenth century algebra” ([45, page 249]).

• Letterplace superalgebras provide a natural setting to extend (in an effective and non-trivial way), the theory of transvectants from binary forms to n-ary forms, n an arbitrary positive integer (Brini, Regonati and Teolis [22]).

Two methodological remarks. Thanks to the systematic use of the superalgebraic version of Capelli’s method of 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 [46]) and the Triangularity Theorem for the action of (superstandard) Young–Capelli symmetrizers on the basis of (superstandard) symmetrized bitableaux (Brini and Teolis [12]).

The (super) Straightening Formula admits a few lines proof in terms of virtual variables (see, e.g., [17], [19]). A quite direct proof of the Triangularity Theorem has been recently derived from a handful of combinatorial lemmas on Young tableaux (see, e.g.,

[21] and Regonati [69]).

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.

2 Synopsis

The work is organized as follows.

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

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 combinatorial one (“signed alphabets”) and the more traditional language of multilinear algebra (“ℤ_{2}-graded vector spaces”).

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

In Section 6, we provide an introduction to the superalgebraic version of Capelli’s method of 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 Capellitype operators, can be defined, in a quite natural and simple way, by appealing to “extra” variables (the virtual variables).

The true meaning of Theorem 6.1 is that the action of these operators is the same as the action of operators induced by the action of the enveloping algebra U(pl(V) of the general linear Lie superalgebra of a ℤ_{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.

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 biproducts, bitableaux, and left, right and doubly symmetrized bitableaux.

Bitableaux provide the natural generalization of bideterminants of a pair of Young tableaux in the sense of [38] and [34].

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

• Images of highest weight vectors, under the action of the “negative” root spaces of sl_{n}(ℂ) (see, e.g., [40]).

• Generators of the irreducible symmetry classes of tensors (see, e.g., [92], [4], [5]).

• Generators of minimal left ideals of the group algebra 𝕂[S_{n}], S_{n} the symmetric group on n elements (see, e.g., [53]).

We provide an elementary proof of the (super)-Straightening Law of Grosshans, Rota and Stein [46] 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 complete decompositions of the letterplace superalgebra as a module, while the standard basis yields an invariant filtration (this is a characteristicfree 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).

In Section 10, we introduce the basic operators of the theory, the 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 Triangularity Theorem (Theorem 10.1, Subsection 10.2).

In Section 11, Schur and Weyl modules are described as subspaces spanned by right and 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 ℤ_{2}-graduation, they yield the two basic constructions of the irreducible representations of the general linear group; see, e.g., [1], [85]).

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 double centralizer theorem also follows from the same argument.

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

The matrix entries are strictly related (Theorem 13.1) to the symmetry transition coefficients discussed in Subsection 8.2 that are, in turn, a generalization of the Désarménien straightening coefficients [35].

Furthermore, these matrix representations yield, as a very special case, the Young natural form of the irreducible matrix representations of the symmetric group S_{n}.

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

In Section 14, we derive an explicit decomposition result for spaces obtained by performing tensor products of spaces of symmetric tensors and 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 Feynman–Rota idea of the entangling/disentangling operator in the language of letterplace superalgebras. Specifically, the components of the tensor product are faithfully encoded by “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, [49]) can be obtained from Corollary 14.1.

In Section 15, the classical representation theory of the special linear Lie algebra sl_{m}(ℂ) 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}[L|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 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 irreducible sl_{m}(ℂ)-modules, and provide a combinatorial description of two families of bases of these irreducible sl_{m}(ℂ)-modules in terms of left symmetrized bitableaux and of doubly symmetrized bitableaux (Subsection 15.3).

In Section 16, we discuss Deruyts’ theory of covariants, following along the lines of its admirable reconstruction provided by J.A. Green [45] (see also [30]).

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}(ℂ), as well as a proof of the complete reducibility of any polynomial representations of GL_{n}(ℂ), and, therefore, he anticipates by nearly a decade the main results of Schur’s Dissertation.

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 [45, p. 248]).

The starting point of Deruyts and Green’s approach is the construction of an algebra isomorphism σ from a commutative letterplace algebra to the algebra of covariants of weight zero and the fact that, given any such covariant φ, its left span L(φ) is a 𝕂[GL_{n}]-module that turns out to be equal to the cyclic 𝕂[GL_{n}]-module generated by the unique preimage γ of φ with respect to the isomorphism σ (γ is called the source of the covariant φ).

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 σ, we translate these results into Deruyts’ language of covariants.

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 [4], [80], [5] (special case: general linear groups [92]), as well as the theory of regular representations of symmetric groups (see, e.g., [53, 72]).

In Section 17, we consider the special case of a positively ℤ_{2}-trivially graded alphabet L = {1, …, n}, while P is any finite signed set.

Let Super_{n}[\underline{L}|P] be the subspace of the letterplace algebra spanned by all the multilinear monomials of degree n in the symbols of L; this subspace is isomorphic (via the so-called Feynman entangling/disentangling operators [38]) to the space T^{n}[W_{0} ⊕ W_{1}] of n-tensors over the ℤ_{2}-graded vector space W_{0} ⊕W_{1} whose basis is identified with the signed alphabet P. There is a natural action of the symmetric group S_{n} on the subspace Super_{n}[\underline{L}|P]; via the linear isomorphism mentioned above, this action corresponds to the Berele–Regev action [4, 5] of S_{n} on the tensor space T^{n}[W_{0} ⊕ W_{1}].

Here, the point of the specialization argument is the fact that the operator algebra induced by the action of the symmetric group S_{n} admits a simple description in terms of polarization operators (Proposition 17.1). 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 — in the case of trivial positive ℤ_{2}-graduation on P — the Berele–Regev theory reduces to the classical Schur–Weyl tensor representation theory of symmetric and general linear groups, see, e.g., [92]).

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{L}|\underline{P}] of the letterplace superalgebra spanned by doubly multilinear monomials (on the negatively signed set L = P = {1, …, n}). The space Super_{n}[\underline{L}|\underline{P}] is a 𝕂[S_{n}]-module in an obvious way; here, the point of the specialization argument is that the operator algebra \underline{\underline{B}}_{n} induced by the action of 𝕂[S_{n}] is an isomorphic copy of 𝕂[S_{n}] and, again, it admits a simple description in terms of polarization operators (Proposition 18.1).

We recall that the 𝕂[S_{n}]-modules Super_{n}[\underline{L}|\underline{P}] and 𝕂[S_{n}] (regarded as left regular module) are naturally isomorphic.

Therefore, the decomposition theory of 𝕂[S_{n}] as a left regular module is “the same” as the theory of the module 𝕂[S_{n}] · Super_{n}[\underline{L}|\underline{P}] = \underline{\underline{B}}_{n} · Super_{n}[\underline{L}|\underline{P}] that, in turn, immediately follows by specializing the general theory.

In the isomorphic structures

𝕂[S_{n}] ≃ Super_{n}[\underline{L}|\underline{P}] ≃ \underline{\underline{B}}_{n},

Young symmetrizers in 𝕂[S_{n}] correspond, up to a sign, to “doubly multilinear” symmetrized bitableaux in Super_{n}[\underline{L}|\underline{P}] and to “doubly multilinear” Young–Capelli symmetrizers in \underline{\underline{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., [72], [53], [41]).

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 ℂ) 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., [54], [55], [56]), 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 basic classical simple Lie superalgebras of type I (see, e.g [54], [23], [24]).

The main constructions are, in this context, those of the integral highest weight modules and of the Kac modules relative to dominant integral highest weights (see Subsections 20.4 and 20.5). A deep result of Kac (see, e.g., [55], [56], [39]) states that highest weight modules and Kac modules coincide if and only if the highest weight Λ of the representation is a typical one (Subsection 20.5, Proposition 20.1).

It follows from Theorem 12.1 and from the results in Section 17 that the irreducible modules that appear in the theory of letterplace superalgebra representations are covariant modules, in the sense of [26], [66].

Covariant modules are finite dimensional highest weight representations but they are not, in general, Kac modules (since their highest weights can be atypical ones, see Remark 20.7).

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., [46], [16], [47] — the decomposition theory of the super-symmetric algebra S(S^{2}(V)) and of the super-antisymmetric algebra ⋀(S^{2}(V)), recently rediscovered by Cheng and Wang ([26], [27]) and Sergeev ([81], [82]).

We provide a rather extensive bibliography. Some items are not mentioned in the text; they are books and papers of general or historical interest ([3], [9], [36], [43], [44], [48], [63], [64], [65], [68], [75], [86], [89], [90], [91], [93]) and a couple of papers that deal with some aspects of the theory not treated in this work ([14], [15]).

Part I

The General Setting

Throughout the paper, 𝕂 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.

3 Superalgebras

A superalgebra A is simply a ℤ_{2}-graded algebra, in symbols

A = A_{0} ⊕ A_{1},

such that

A_{i}A_{j} ⊆ A_{i+j}, i, j ∈ ℤ_{2}.

Given a ℤ_{2}-homogeneous element a ∈ A, its ℤ_{2}-degree is denoted by |a|.

3.1 The supersymmetric superalgebra of a ℤ_{2}-graded vector space

Given a ℤ_{2}-graded vector space U = U_{0}⊕U_{1}, its supersymmetric superalgebra Super[U] is the superalgebra

Super[U] = Sym[U_{0}] ⊗ Λ[U_{1}],

where

Super[U] = Super[U]_{0} ⊕ Super[U]_{1},
Super[U]_{0} = Sym[U_{0}] ⊗ ( ⨁_{h∈ℕ}Λ^{2h}[U_{1}]),

Super[U]_{1} = Sym[U_{0}] ⊗ ( ⨁_{h∈ℕ}Λ^{2h+1}[U_{1}]).

The supersymmetric algebra Super[U] has a natural structure of a ℤ_{2}-graded bialgebra (see Subsection 3.5).

3.2 Lie superalgebras

A Lie superalgebra is a superalgebra L = L_{0} ⊕ L_{1} whose product (Lie superbracket) satisfies the following identities:

• [x, y] = −(−1)^{|x||y|}[y, x]

• (−1)^{|x||z|}[x, [y, z]] + (−1)^{|z||y|}[z, [x, y]] + (−1)^{|y||x|}[y, [z, x]] = 0 (ℤ_{2}-graded Jacobi identity.)

3.3 Basic example: the general linear Lie superalgebra of a ℤ_{2}-graded vector space

Given a ℤ_{2}-graded vector space U = U_{0} ⊕ U_{1}, its general linear Lie superalgebra pl(U) is the ℤ_{2}-graded vector space

End_{𝕂}[U] = End_{𝕂}[U]_{0} ⊕ End_{𝕂}[U]_{1},
End_{𝕂}[U]_{i} = {φ ∈ End_{𝕂}[U]; φ[U_{j}] ⊆ U_{i+j}, j ∈ ℤ_{2}}, i ∈ ℤ_{2},

endowed with the supercommutator

[φ, ψ] = φψ − (−1)^{|φ||ψ|}ψφ.

MATRIX VERSION: let L = {x_{1}, …, x_{n}} be a ℤ_{2}-homogeneous basis of U = U_{0} ⊕ U_{1}. A standard basis of pl(U) is given by the set of linear endomorphisms (elementary matrices)

E_{x_{i},x_{j}}, E_{x_{i},x_{j}}(x_{k}) = δ_{j,k}x_{i}, i, j, k = 1, …, n.

The E_{x_{i},x_{j}}’s are ℤ_{2}-homogeneous elements of End_{𝕂}[U] of degree |x_{i}| + |x_{j}| and satisfy the identities

[E_{x_{i},x_{j}}, E_{x_{h},x_{k}}] = δ_{j,h}E_{x_{i},x_{k}} − (−1)^{(|x_{i}|+|x_{j}|)(|x_{h}|+|x_{k}|)}δ_{i,k}E_{x_{h},x_{j}}.

3.4 The supersymmetric superalgebra Super[A] of a signed set A

A signed set (or, equivalently, a ℤ_{2}-graded set) is a set A endowed with a sign map | | : A → ℤ_{2}; the sets A_{0} = {a ∈ A; |a| = 0} and A_{1} = {a ∈ A; |a| = 1} are called the subsets of positive and negative symbols, respectively.

The supersymmetric 𝕂-superalgebra Super[A] is the quotient algebra of the free associative 𝕂-algebra with 1 generated by the signed set A modulo the bilateral ideal generated by the elements of the form:

xy − (−1)^{|x||y|}yx, x, y, ∈ A.

Remark 3.1. 1. Super[A] is a ℤ_{2}-graded algebra

Super[A] = (Super[A])_{0} ⊕ (Super[A])_{1},

where (Super[A])_{i} is the subspace of Super[A] spanned by the monomials m of ℤ_{2}-degree |m| = i, where, for m = a_{i_{1}} ⋯ a_{i_{n}}, |m| = |a_{i_{1}}| + ⋯ + |a_{i_{n}}|.

With respect to this grading, Super[A] is supersymmetric, i.e.:

mm′ = (−1)^{|m||m′|}m′m,

for every ℤ_{2}-homogeneous elements m, m′ ∈ Super[A].

2. Super[A] is an ℕ-graded algebra

Super[A] = ⨁_{n∈ℕ}Super_{n}[A]
Super_{n}[A] = ⟨a_{i_{1}}a_{i_{2}} ⋯ a_{i_{n}}; a_{i_{h}} ∈ A⟩_{𝕂};

3. the ℤ_{2}-graduation and the ℕ-graduation of Super[A] are coherent, that is

(Super[A])_{i} = ⨁_{n∈ℕ}(Super_{n}[A])_{i}, i ∈ ℤ_{2}.

Let A = A_{0} ∪ A_{1} be a finite signed set, and let U = U_{0} ⊕ U_{1}, where U_{0} = ⟨A_{0}⟩_{𝕂}, U_{1} = ⟨A_{1}⟩_{𝕂}. The superalgebra Super[A] is isomorphic to the supersymmetric algebra Super[U] of the ℤ_{2}-graded vector space U = U_{0} ⊕ U_{1}.

3.5 ℤ_{2}-graded bialgebras. Basic definitions and the Sweedler notation

Let us consider an algebraic structure (X, π, Δ, η, ε) where:

• X = X_{0} ⊕ X_{1} is a ℤ_{2}- graded 𝕂-vector space.

• π is an associative product on X, that is an even linear map

π : X ⊗ X → X

such that

π(I ⊗ π) = π(π ⊗ I).

• Δ is a coassociative coproduct on X, that is an even linear map

Δ : X → X ⊗ X

such that

(Δ ⊗ I)Δ = (I ⊗ Δ)Δ.

The Sweedler notation [84] is a way to write the coproduct of an element x ∈ X, namely

Δ(x) = ∑_{(x)} x_{(1)} ⊗ x_{(2)}.

For example, in the Sweedler notation the fact that Δ is a coassociative coproduct reads as follows:

∑_{(x)} x_{(1)} ⊗ ( ∑_{(x_{(2)})} x_{(2)_{(1)}} ⊗ x_{(2)_{(2)}} ) = ∑_{(x)} ( ∑_{(x_{(1)})} x_{(1)_{(1)}} ⊗ x_{(1)_{(2)}} ) ⊗ x_{(2)}.

• η is a unit map, that is an even linear map

η : 𝕂 → X

such that

η(k)x = kx, k ∈ 𝕂, x ∈ X.

• ε is a counit map, that is an even linear map

ε : X → 𝕂

such that

∑_{(x)} ε(x_{(1)})x_{(2)} = ∑_{(x)} x_{(1)}ε(x_{(2)}) = x, x ∈ X.

• (ε ∘ η)(1) = 1, 1 ∈ 𝕂.

An algebraic structure (X, π, Δ, η, ε is said to be a ℤ_{2}-graded 𝕂-bialgebra whenever the following conditions hold:

• The coproduct Δ is an algebra morphism. Notice that X ⊗ X is meant as the tensor product of ℤ_{2}-graded algebras, that is (x ⊗ x′)(y ⊗ y′) = (−1)^{|x′||y|}xy ⊗ x′y′, x, y, x′, y′ ℤ_{2}-homogeneous elements in X. In the Sweedler notation, the above condition reads as follows:

Δ(xy) = ∑_{(xy)} (xy)_{(1)} ⊗ (xy)_{(2)} = ∑_{(x),(y)} (−1)^{|x_{(2)}||y_{(1)}|} x_{(1)}y_{(1)} ⊗ x_{(2)}y_{(2)} = Δ(x)Δ(y),

x, y ∈ X.

• The unit map η is a coalgebra morphism, that is

Δ(η(1)) = η(1) ⊗ η(1), 1 ∈ 𝕂.

• The counit map ε is an algebra morphism, that is

ε(xy) = ε(x)ε(y), x, y ∈ X.

3.6 The superalgebra Super[A] as a ℤ_{2}-graded bialgebra. Left and right derivations, coderivations and polarization operators

Let A = A_{0} ∪ A_{1} be a finite signed set, A_{0} = {a_{1}, a_{2}, …, a_{m}}, A_{1} = {a_{m+1}, a_{m+2}, …, a_{m+n}}, and let U = U_{0} ⊕ U_{1}, where U_{0} = ⟨A_{0}⟩_{𝕂}, U_{1} = ⟨A_{1}⟩_{𝕂}.

The superalgebra Super[A] ≅ Super[U] is a ℤ_{2}-graded bialgebra, where the structure maps are defined in the following way:

• Δ(1) = 1 ⊗ 1, Δ(a_{i}) = a_{i} ⊗ 1 + 1 ⊗ a_{i}, a_{i} ∈ A;

• η(1) = 1;

• ε(1) = 1, ε(a_{i}) = 0, a_{i} ∈ A.

A linear map

D : Super[A] → Super[A],

homogeneous of degree d = |D| ∈ ℤ_{2}, i.e., such that D(Super[A])_{i} ⊆ (Super[A])_{i+d}, is a left superderivation if

D(m m′) = D(m) m′ + (−1)^{|D||m|}m D(m′),

for all monomials m, m′ ∈ Super[A].

Let a_{i}, a_{j} ∈ A. The left superpolarization D_{a_{i},a_{j}} of the letter a_{j} to the letter a_{i} is the unique left superderivation of ℤ_{2}-degree |a_{i}| + |a_{j}|, such that

D_{a_{i},a_{j}}(a_{h}) = δ_{j,h} a_{i},

for every a_{h} ∈ A.

Here and in the following the Greek letter δ will denote the Kronecker symbol.

Any linear map D : Super[A] → Super[A] may be extended to an operator

(D ⊕ D) : Super[A] ⊗ Super[A] → Super[A] ⊗ Super[A],

by the rule of “left superderivation”, that is by setting

(D ⊕ D)(m ⊗ m′) = D(m) ⊗ m′ + (−1)^{|D||m|}m ⊗ D(m′),

for every m, m′ ∈ Super[A]. If no confusion arises, we will frequently write D in place of D ⊕ D.

We notice that if (D is a left superderivation of ℤ_{2}-degree |D|, then (D ⊕ D) is a left superderivation of ℤ_{2}-degree |D|.

A linear map

D : Super[A] → Super[A],

homogeneous of degree |D| ∈ ℤ_{2}, is said to be a left coderivation if the following condition holds:

Δ(D(m)) = (D ⊕ D)(Δ(m)),

for every m ∈ Super[A]. In the Sweedler notation, the above condition reads as follows:

Δ(D(m)) = ∑_{(m)} [D(m_{(1)}) ⊗ m_{(2)} + (−1)^{|D||m_{(1)}|}m_{(1)} ⊗ D(m_{(2)})] .

Proposition 3.1. Any left superpolarization D_{a_{i},a_{j}} is a left coderivation of the bialgebra Super[A].

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

Corollary 3.1. Let m = a_{i_{1}}a_{i_{2}} ⋯ a_{i_{p}} ∈ Super[A] and let a_{j} ∈ A, |a_{j}| = 0, such that a_{j} ≠ a_{i_{h}}, h = 1, 2, …, p. Then

Δ(m) = \frac{1}{p!}D_{a_{i_{1}},a_{j}}D_{a_{i_{2}},a_{j}} ⋯ D_{a_{ip},a_{j}}(Δ((a_{j})^{p})).

Example 3.1. Let a_{1}, a_{2}, a_{3} ∈ A, |a_{1}| = 1, |a_{2}| = 1, |a_{3}| = 0 and let a ∈ A, |a| = 0, such that a ≠ a_{i}, i = 1, 2, 3. Let m = a_{1}a_{2}a_{3}. We have the following identity:

Δ(m) = Δ(a_{1})Δ(a_{2})Δ(a_{3}) =

(a_{1} ⊗ 1 + 1 ⊗ a_{1})(a_{2} ⊗ 1 + 1 ⊗ a_{2})(a_{3} ⊗ 1 + 1 ⊗ a_{3}) =

a_{1}a_{2}a_{3} ⊗ 1 + a_{1}a_{2} ⊗ a_{3} + a_{1}a_{3} ⊗ a_{2} − a_{2}a_{3} ⊗ a_{1}+
a_{1} ⊗ a_{2}a_{3} − a_{2} ⊗ a_{1}a_{3} + a_{3} ⊗ a_{1}a_{2} + 1 ⊗ a_{1}a_{2}a_{3} =

\frac{1}{3!}D_{a_{1},a}D_{a_{2},a}D_{a_{3},a}(a^{3} ⊗ 1 + 3a^{2} ⊗ a + 3a ⊗ a^{2} + 1 ⊗ a^{3}) = D_{a_{1},a}D_{a_{2},a}D_{a_{3},a}(\frac{1}{3!}Δ(a^{3})).

In the following sections, we also need the notion of a right superderivation: a linear map

Super[A] ← Super[A] : D̃,

homogeneous of degree |D̃|, is a right superderivation if

(−1)^{|D̃||m′|}(m)D̃ m′ + m (m′)D̃ = (m m′)D̃,

for all monomials m, m′ ∈ Super[A]. Let a_{h}, a_{k} ∈ A. The right superpolarization _{a_{h},a_{k}}D of the letter a_{h} to the letter a_{k} is the unique right superderivation of ℤ_{2}-degree |a_{h}| + |a_{k}|, such that

(a_{j})_{a_{h},a_{k}}D = δ_{j,h} a_{k},

for every a_{j} ∈ A. Consider linear automorphism

R : Super[A] → Super[A]

such that

R(a_{i_{1}}a_{i_{2}} ⋯ a_{i_{n}}) = a_{i_{n}}a_{i_{n−1}} ⋯ a_{i_{1}},

for every a_{i_{1}}a_{i_{2}} ⋯ a_{i_{n}} ∈ Super[A]. Clearly, R is an involutorial map, that is R^{2} = id, and R(mm′) = R(m′)R(m), for every m, m′ ∈ Super[A]. It follows that the map

D ↦ R ∘ D ∘ R = D̃

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}}D of the letter a_{h} to the letter a_{k} is the right superderivation

a_{h},a_{k}D = R ∘ D_{a_{k},a_{h}} ∘ R.

The next result follows from the definitions.

Proposition 3.2. 1. _{a_{h},a_{k}}D is a right coderivation of the bialgebra Super[A], that is

Δ((m)_{a_{h},a_{k}}D)

= ∑_{(m)} ((−1)^{(|a_{h}|+|a_{k}|)|m_{(2)}|} (m_{(1)})_{a_{h},a_{k}}D ⊗ m_{(2)} + m_{(1)} ⊗ (m_{(2)})_{a_{h},a_{k}}D) ,

for every m ∈ Super[A].

2. Let m = a_{i_{1}}a_{i_{2}} ⋯ a_{i_{p}} ∈ Super[A] and let a_{j} ∈ A, |a_{j}| = 0, such that a_{j} ≠ a_{i_{h}}, h = 1, 2, …, p. Then

Δ(m) = \frac{1}{p!}(Δ((a_{j})^{p})))_{a_{j},a_{i_{1}}}D_{a_{j},a_{i_{2}}}D ⋯_{a_{j},a_{ip}} D.

4 The Letterplace Superalgebra as a Bimodule

4.1 Letterplace superalgebras

In the following, we consider a pair of signed sets X = X_{0} ∪ X_{1} and Y = Y_{0} ∪ Y_{1}, that we call the letter set and the place set, respectively. The letterplace set

[X | Y] = {(x | y); x ∈ X , y ∈ Y}

inherits a sign by setting |(x | y)| = |x| + |y| ∈ ℤ_{2}.

The letterplace 𝕂-superalgebra Super[X |Y] is the quotient algebra of the free associative 𝕂-algebra with 1 generated by the letterplace alphabet [X | 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), x, z ∈ X , y, t ∈ Y.

In other words, the letterplace 𝕂-superalgebra Super[X |Y] is the supersymmetric superalgebra of the Z_{2}-graded set A = [X | Y] (see Subsection 3.4).

Remark 4.1. 1. Super[X |Y] is a ℤ_{2}-graded algebra

Super[X |Y] = (Super[X |Y])_{0} ⊕ (Super[X |Y])_{1},

where (Super[X |Y])_{i} is the subspace of Super[X |Y] spanned by the letterplace monomials M of ℤ_{2}-degree |M| = i, where, for M = (x_{i_{1}} | y_{i_{1}}) ⋯ (x_{i_{n}} | y_{i_{n}}), |M| = |(x_{i_{1}} | y_{i_{1}})| + ⋯ + |(x_{i_{n}} | y_{i_{n}})|.

With respect to this grading, Super[X |Y] is supersymmetric, i.e.:

MN = (−1)^{|M||N|}NM,

for every ℤ_{2}-homogeneous elements M, N ∈ Super[X |Y].

2. Super[X |Y] is an ℕ-graded algebra

Super[X |Y] = ⨁_{n∈ℕ}Super_{n}[X |Y]

Super_{n}[X |Y] = ⟨(x_{i_{1}} | y_{i_{1}})(x_{i_{2}} | y_{i_{2}}) ⋯ (x_{i_{n}} | y_{i_{n}}), x_{i_{h}} ∈ X , y_{j_{k}} ∈ Y⟩_{𝕂};

3. the ℤ_{2}-graduation and the ℕ-graduation of Super[X |Y] are coherent, that is

(Super[X |Y])_{i} = ⨁_{n∈ℕ}(Super_{n}[X |Y])_{i}, i ∈ ℤ_{2}.

4.2 Superpolarization operators

Let x′, x ∈ X . The superpolarization D_{x′,x} of the letter x to the letter x′ is the unique left superderivation

D_{x′,x} : Super[X |Y] → Super[X |Y]

of ℤ_{2}-degree |x′| + |x|, such that

D_{x′,x}(z | t) = δ_{x,z}(x′ | t),

for every (z | t) ∈ [X | Y].

Let y, y′ ∈ Y. The superpolarization _{y,y′}D of the place y to the place y′ is the unique right superderivation

Super[X |Y] ← Super[X |Y] : _{y,y′}D

of ℤ_{2}-degree |y| + |y′|, such that

(z | t) _{y,y′}D = δ_{t,y}(z | y′),

for every (z | t) ∈ [X | Y].

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

4.3 Letterplace superalgebras and supersymmetric algebras: the classical description

Given a pair of finite alphabets L = L_{0} ∪ L_{1} and P = P_{0} ∪ P_{1}, L_{0} ⊆ X_{0}, L_{1} ⊆ X_{1}, P_{0} ⊆ Y_{0}, P_{1} ⊆ Y_{1}, consider the ℤ_{2}-graded vector spaces

V = V_{0} ⊕ V_{1} = ⟨L_{0}⟩_{𝕂} ⊕ ⟨L_{1}⟩_{𝕂}

and

W = W_{0} ⊕ W_{1} = ⟨P_{0}⟩_{𝕂} ⊕ ⟨P_{1}⟩_{𝕂}.

The tensor product V ⊗ W has a natural ℤ_{2}-grading

V ⊗ W = [(V_{0} ⊗ W_{0}) ⊕ (V_{1} ⊗ W_{1})] ⊕ [(V_{0} ⊗ W_{1}) ⊕ (V_{1} ⊗ W_{0})]).

The supersymmetric algebra of the tensor product V ⊗ W is the superalgebra

Super[V ⊗ W] = Sym[(V_{0} ⊗ W_{0}) ⊕ (V_{1} ⊗ W_{1})] ⊗ Λ[(V_{0} ⊗ W_{1}) ⊕ (V_{1} ⊗ W_{0})].

Clearly, one has a natural isomorphism

Super[L|P] ≅ Super[V ⊗ W].

4.4 General linear Lie superalgebras, representations and polarization operators

Given a pair of finite alphabets L = L_{0} ∪ L_{1} and P = P_{0} ∪ P_{1}, regard them as homogeneous bases of the pair of vector spaces V = V_{0} ⊕ V_{1} and W = W_{0} ⊕ W_{1}.

The Lie superalgebras pl(L) and pl(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′ ∈ L}

and

{E_{y,y′}; y, y′ ∈ P}

of pl(L) = pl(V) and pl(P) = pl(W).

We recall the canonical isomorphism

Super[L|P] ≅ Super[V ⊗ W].

The (even) mappings

E_{x′x} ↦ D_{x′,x}, x, x′ ∈ L, E_{y′y} ↦ _{y,y′}D, y, y′ ∈ P

induce Lie superalgebra actions of pl(L) and pl(P) over any ℕ-homogeneous component Super_{n}[L|P] of the letterplace algebra.

In the following, we will denote by

B_{n}, _{n}B

the (finite dimensional) homomorphic images in End_{𝕂}(Super_{n}[L|P]) of the universal enveloping algebras U(pl(L)) and U(pl(P)), induced by the actions of pl(L) and pl(P), respectively.

The operator algebras B_{n}, _{n}B are therefore the algebras generated by the proper letter and place polarization operators (restricted to Super_{n}[L|P]), respectively.

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

4.5 General linear groups and even polarization operators

Let L = L_{1} = {x_{1}, …, x_{m}} be a finite alphabet of letters, let P be a finite alphabet of places; since L is trivially ℤ_{2}-graded, the general linear Lie superalgebra pl(L) reduces to the usual general linear Lie algebra gl_{m}(𝕂) of all square matrices of order m over 𝕂, and the letterplace superalgebra Super[L|P] is a (left) gl_{m}(𝕂)-module via the usual action, that is, a matrix S = [s_{ij}] acts on a letterplace variable as

S(x_{i} | y) = ∑_{j=1}^{m}(x_{j} | y) s_{ji}

and is extended as a derivation (the action of any elementary matrix E_{hk} is implemented by the polarization operator D_{x_{h},x_{k}}, |D_{x_{h},x_{k}}| = 0 ∈ ℤ_{2}). We denote by σ_{n} the corresponding representation of the universal enveloping algebra U[gl_{m}(𝕂)] over the homogeneous component Super_{n}[L|P].

Let Gl_{m}(𝕂) be the general linear group of nonsingular matrices of order m over 𝕂; the letterplace superalgebra Super[L|P] is a (left) GL_{m}(𝕂)-module via the usual action, that is, a matrix S = [s_{ij}] acts on a letterplace variable as

S(x_{i} | y) = ∑_{j=1}^{m}(x_{j} | y) s_{ji}

and is extended as an algebra automorphism. We denote by ρ_{n} the corresponding representation of the group algebra 𝕂[GL_{m}(𝕂)] over the homogeneous component Super_{n}[L|P].

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

Proposition 4.1. The algebra ρ_{n} [𝕂[GL_{m}(𝕂)]] generated by the action of the general linear group coincides with the algebra σ_{n} [U[gl_{m}(𝕂)]] generated by the action of the general linear Lie algebra.

Therefore, the algebra ρ_{n} [𝕂[GL_{m}(𝕂)]] is the algebra B_{n} generated by the letter polarization operators D_{x_{i}x_{j}}, x_{i}, x_{j} ∈ L.

Proof. Recall that the group GL_{m}(𝕂) is generated by the transvections, namely:

• T_{ij}(λ) = I + λ E_{ij}, with i ≠ j;

• T_{ii}(λ) = I + λ E_{ii}, with λ ≠ −1.

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

• The image under the representation ρ_{n} of any transvection T_{ij}(λ) = I + λ E_{ij}, with i ≠ j, is a polynomial in the image under the representation σ_{n} of the elementary matrix E_{ij}. Specifically, we have:

ρ_{n}(T_{ij}(λ)) = ∑_{h=0}^{n} λ^{h} \frac{σ_{n}(E_{ij})^{h}}{h!} ;

by evaluating this relation in n + 1 different values λ = λ_{1}, …, λ_{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}.

• The same argument applies to the representation of any transvection T_{ii}(λ). Specifically, we have:

ρ_{n}(T_{ii}(λ)) = ∑_{h=0}^{n} λ^{h} ( \begin{matrix}σ_{n}(E_{ii}) \\ h \end{matrix} );

by evaluating this relation in n + 1 different values λ = λ_{1}, …, λ_{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}.

5 Tableaux

5.1 Young tableaux

We recall that signed set is a set A endowed with a sign map | | : A → ℤ_{2}; the sets A_{0} = {a ∈ A; |a| = 0} and A_{1} = {a ∈ A; |a| = 1} are called the subsets of positive and negative symbols, respectively. A signed alphabet is a linearly ordered signed set.

A Young tableau over a signed alphabet A is a sequence

S = (w_{1}, w_{2}, …, w_{p})

of words w_{i} = a_{i1}a_{i2} … a_{iλ_{i}}, a_{ij} ∈ A, whose lengths form a weakly decreasing sequence, i.e., a partition

λ = (λ_{1} ≥ λ_{2} ≥ … ≥ λ_{p}) = sh(S),

called the shape of S. The concatenation of the words w_{i}

w = w_{1}w_{2} … w_{p} = w(S)

is called the word of S. If n is the length of w, then λ is a partition of n:

λ ⊢ n.

The 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{matrix}a & b & b \\ b & a & e \\ c \end{matrix}

The set of all the tableaux over A is denoted by T ab(A).

5.2 Co-Deruyts and Deruyts tableaux

A tableau C is said to be of 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{matrix}a & a & a & a & a \\ b & b & b \\ c \end{matrix}

A tableau D is said to be of 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{matrix}a & b & c & d & e \\ a & b & c \\ a \end{matrix}

In the following, the symbol C will denote a co-Deruyts tableau filled with positive symbols, and the symbol D will denote a Deruyts tableau filled with 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.

5.3 Standard Young tableaux

Following Grosshans, Rota and Stein [46], a Young tableau S over a (linearly ordered) signed alphabet A is called (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{matrix}a & a & c & c & d \\ b & c & d \\ b \end{matrix} a, c ∈ A_{0}, b, d ∈ A_{1}

is a standard tableau. The set of all the standard tableaux over A is denoted by Stab(A).

5.4 The Berele–Regev hook property

Assume now that the linearly ordered signed alphabet A is finite, with |A_{0}| = r and |A_{1}| = s. The hook set of A is

H(A) = {(λ_{1} ≥ λ_{2} ≥ …); λ_{r+1} < s + 1}.

Proposition 5.1. There are some standard tableaux of shape λ over A if and only if λ ∈ H(A). Furthermore, the number p_{λ}(A) of standard tableaux of any given shape λ over A is independent of the linear order defined on A.

Proof. Let A_{0} = {x_{1}, x_{2}, …, x_{m}}, A_{1} = {x_{m+1}, x_{m+2}, …, x_{m+n}}. If A is endowed with a linear order such that x_{i} < x_{j} for every i = 1, 2, …, m and j = m + 1, m + 2, …, m+n, the proof follows from a straightforward argument (see, e.g., [4], [5]). If 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 [46] and Proposition 11.1).

5.5 Orders on tableaux

Let L be a finite signed alphabet. We define a partial order on the set of all standard tableaux over 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, …, 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, …, 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 ≤ T ⇔ sh(S^{(p)}) ⊴ sh(T^{(p)}), p = 1, 2, …,

where ⊴ stands for the dominance order on partitions. We recall that the dominance order on partitions is defined as follows: λ = (λ_{1} ≥ λ_{2} ≥ …) ⊴ μ = (μ_{1} ≥ μ_{2} ≥ …) if and only if λ_{1} + ⋯ + λ_{i} ≤ μ_{1} + ⋯ + μ_{i}, for every i = 1, 2, …

We extend this partial order to the set of all standard tableaux on 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) ≠ c(T) are incomparable.

We define a linear order on the set of all tableaux over L which contain a total number (taking into account multiplicities) of n symbols, by setting Q < Q′ if and only if

sh(Q) <_{l} sh(Q′)
or sh(Q) = sh(Q′) and w(Q) >_{l} w(Q′),

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.

Part II

The General Theory

6 The Method of Virtual Variables

6.1 The metatheoretic significance of Capelli’s idea of virtual variables

Let L = L_{0} ∪ L_{1} ⊂ X and P = P_{0} ∪ P_{1} ⊂ Y be finite signed subsets of the “universal” signed letter and place alphabets X and Y, respectively. The elements x ∈ L (y ∈ P) are called proper letters (proper places), and the elements x ∈ X \ L (y ∈ Y \ P) are called virtual letters (places). Usually we denote virtual symbols by Greek letters.

The signed subset [L | P] = {(x | y); x ∈ L, y ∈ P} ⊂ [X | Y] is called a proper letterplace alphabet.

Consider an operator of the form:

D_{x_{i_{1}},α_{i_{1}}} ⋯ D_{x_{in},α_{in}} · D_{α_{i_{1}},x_{j_{1}}} ⋯ D_{α_{in},x_{jn}}
(x_{i_{1}}, …, x_{i_{n}}, x_{j_{1}}, …, x_{j_{n}} ∈ L, i.e., proper letters)

that is an operator that creates some virtual letters α_{i_{1}}, …, α_{i_{n}} (with prescribed multiplicities) times an operator that annihilates the same virtual letters (with the same prescribed multiplicities).

Such an operator will be called a (letter) Capelli-type operator.

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

Theorem 6.1 ([12, 20]). The action of a Capelli-type operator over the proper letterplace superalgebra Super[L|P] is the same as the action of a “polynomial” operator in the proper polarizations D_{x_{i_{h}},x_{i_{k}}}, x_{i_{h}}, x_{i_{k}} ∈ L, that is an operator that does not involve virtual variables.

Informally speaking, a Capelli-type operator is of pl(L)-representation theoretic meaning, and, in general is much more manageable than its “non-virtual companion”.

In the following, we will write T_{1} ≅ T_{2} to mean that two operators T_{1}, T_{2} on Super[X |Y] are the same when restricted to the proper letterplace algebra Super[L|P] and say that the operators T_{1}, T_{2} are [L | P] − equivalent.

Example 6.1. Let x, y ∈ L_{1}, with x ≠ y, and α ∈ X \ L, with |α| = 0. Then

D_{yα}D_{xα}D_{αx}D_{αy} = −D_{yα}D_{αx}D_{xα}D_{αy} + D_{yα}D_{xx}D_{αy} + D_{yα}D_{αα}D_{αy}
= +D_{yα}D_{αx}D_{αy}D_{xα} − D_{yα}D_{αx}D_{xy}
− D_{xx}D_{αy}D_{yα} + D_{xx}D_{yy} − D_{xx}D_{αα}
+ D_{yα}D_{αy}D_{αα} + D_{yα}D_{αy}
≅ −D_{yα}D_{αx}D_{xy} + D_{xx}D_{yy} + D_{yα}D_{αy}
≅ …
≅ −D_{yx}D_{xy} + D_{xx}D_{yy} + D_{yy}.

Here the identities = are obtained by applying the commutator identity, and the [L | P]- equivalences ≅ 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[L|P].

6.2 Tableau polarization monomials

Let u′ and u be words of the same length m on the alphabet X, say u′ = x′_{1}x′_{2} ⋯ x′_{m}, u = x_{1}x_{2} ⋯ x_{m}. The letter polarization monomial of the word u to the word u′ is defined to be the 𝕂-linear operator

D_{u′u} = D_{x′_{1}x_{1}}D_{x′_{2}x_{2}} … D_{x′_{m}x_{m}} ∈ End_{𝕂}[Super_{n}[X |Y]].

Let S′, S ∈ T ab(X), with sh(S′) = sh(S) ⊢ n. The letter polarization monomial of the tableau S to the tableau S′ is defined to be the 𝕂-linear operator

D_{S′S} = D_{x′_{1}x_{1}}D_{x′_{2}x_{2}} … D_{x′_{n}x_{n}} ∈ End_{𝕂}[Super_{n}[X |Y]],

where x′_{1} … x′_{n} = w(S′) and x_{1} … x_{n} = w(S).

In passing, we point out that, if S′ = (u′_{1}, …, u′_{p}) and S = (u_{1}, …, u_{p}), then

D_{S′S} = D_{u′_{1}u_{1}}D_{u′_{2}u_{2}} … D_{u′_{p}u_{p}}.

Example 6.2. Let

S′ = \begin{matrix}x & y \\ x & z \end{matrix} , S = \begin{matrix}x & z \\ z & t \end{matrix} ;

then

D_{S′S} = D _{\begin{matrix}x & y \\ x & z \end{matrix} , \begin{matrix}x & z \\ z & t \end{matrix}} = D_{xx}D_{yz}D_{xz}D_{zt}

Let v and v′ be words of the same length m on the alphabet Y, say v = y_{1}y_{2} ⋯ y_{m}, v′ = y′_{1}y′_{2} ⋯ y′_{m}. The place polarization monomial of the word v to the word v′ is defined to be the 𝕂-linear operator

vv′D =_{y_{1}y′_{1}} D _{y_{2}y′_{2}}D … _{y_{m}y′_{m}}D ∈ End_{𝕂}[Super_{n}[X |Y]].

Let V, V′ ∈ T ab(Y), with sh(V) = sh(V′) ⊢ n. The place polarization monomial of the tableau V to the tableau V′ is defined to be the 𝕂-linear operator

V V′D =_{y_{1}y′_{1}} D _{y_{2}y′_{2}}D … _{y_{n}y′_{n}}D ∈ End_{𝕂}[Super_{n}[X |Y]],

where y_{1} … y_{n} = w(V) and y′_{1} … y′_{n} = w(V′).

Clearly, if V = (v_{1}, …, v_{p}) and V′ = (v′_{1}, …, v′_{p}), then

V V′D =_{v_{1}v′_{1}} D _{v_{2}v′_{2}}D … _{v_{p}v′_{p}}D.

Example 6.3. Let

V = \begin{matrix}a & c \\ b & c \end{matrix} , V′ = \begin{matrix}b & b \\ c & d \end{matrix} ;

then

V V′D = _{\begin{matrix}a & c \\ b & c \end{matrix} , \begin{matrix}b & b \\ c & d \end{matrix}} D = _{ab}D_{cb}D_{bc}D_{cd}D

6.3 Capelli bitableaux and Capelli rows

Among Capelli-type operators, a distinguished role is played by those involving virtual letters of the same sign. Specifically, let U = (u_{1}, …, u_{p}) and V = (v_{1}, …, v_{p}) be tableaux on the alphabet of proper letters L, sh(U) = sh(V) = λ = (λ_{1}, …, λ_{p}), and let α_{1}, …, …, α_{p} be p distinct virtual letters of the same sign.

The Capelli-type operator

[U | V] = [ u_{1} ; u_{2} ; ⋮ ; u_{p} | v_{1} ; v_{2} ; ⋮ ; v_{p} ] = D_{u_{1},α_{1}^{λ_{1}}} ⋯ D_{u_{p},α_{p}^{λp}}D_{α_{1}^{λ_{1}},v_{1}} ⋯ D_{α_{p}^{λp},v_{p}}

will be called a Capelli bitableau. A Capelli bitableau is said to be positive (negative) if the virtual letters α_{1}, …, …, α_{p} have positive (negative) sign.

Example 6.4. 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 α_{1}, α_{2}, the operator

[U | V] = [ \begin{matrix}u_{1} \\ u_{2} \end{matrix} | \begin{matrix}v_{1} \\ v_{2} \end{matrix} ] = D_{u_{1},α_{1}^{3}}D_{u_{2},α_{2}^{2}}D_{α_{1}^{3},v_{1}}D_{α_{2}^{2},v_{p}},

where

D_{u_{1},α_{1}^{3}} = D_{xα_{1}}D_{yα_{1}}D_{zα_{1}} D_{u_{2},α_{2}^{2}} = D_{yα_{2}}D_{zα_{2}}
D_{α_{1}^{3},v_{1}} = D_{α_{1}x}D_{α_{1}z}D_{α_{1}z} D_{α_{2}^{2},v_{2}} = D_{α_{2}x}D_{α_{2}y},

is a positive Capelli bitableau.

If U, V are tableaux with just one row, the Capelli bitableau [U | V] is said to be a 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 [13]; it provides the basic identity that relates Capelli bitableaux and products of Capelli rows.

Proposition 6.1. Let

u_{1}, u_{2}, … u_{p}, u_{p+1}; v_{1}, v_{2}, … v_{p}, v_{p+1}

be words over the proper letter alphabet L, where v_{i} has the same length as u_{i}.

We have the following [L | P]-equivalence involving positive (negative) Capelli bitableaux:

[ u_{1} ; u_{2} ; ⋮ ; u_{p} | v_{1} ; v_{2} ; ⋮ ; v_{p} ] [u_{p+1} | v_{p+1}] ≅ ∑ c_{∗} [ u_{1} ; u_{2} ; ⋮ ; u_{p} ; u′_{p+1} | v′_{1} v_{p+1,(1)} ; v′_{2} v_{p+1,(2)} ; ⋮ ; v′_{p} v_{p+1,(p)} ; v_{p+1,(p+1)} ]

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}, … v′_{p}

of the words u_{p+1}, v_{1}, v_{2}, … 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

∑_{i=1}^{p} (c(v_{i}) − c(v′_{i})) ,

and over all the splittings

v_{p+1,(1)}, v_{p+1,(2)}, …, v_{p+1,(p+1)}

of the word v_{p+1}, such that

l(v′_{i} v_{p+1,(i)}) = l(u_{i}), i = 1, …, p.

It is a simple fact that, in the above summation, the Capelli tableau

[ u_{1} ; u_{2} ; ⋮ ; u_{p} ; u_{p+1} | v_{1} ; v_{2} ; ⋮ ; v_{p} ; v_{p+1} ]

appears with coefficient ±1. Therefore, the identity in the previous proposition can be rewritten as

± [ u_{1} ; u_{2} ; ⋮ ; u_{p} ; u_{p+1} | v_{1} ; v_{2} ; ⋮ ; v_{p} ; v_{p+1} ] ≅ [ u_{1} ; u_{2} ; ⋮ ; u_{p} | v_{1} ; v_{2} ; ⋮ ; v_{p} ] [u_{p+1} | v_{p+1}] − ∑ c_{∗} [ u_{1} ; u_{2} ; ⋮ ; u_{p} ; u′_{p+1} | v′_{1} v_{p+1,(1)} ; v′_{2} v_{p+1,(2)} ; ⋮ ; v′_{p} v_{p+1,(p)} ; v_{p+1,(p+1)} ] ,

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.

Theorem 6.2. • Every Capelli bitableau is [L | P]-equivalent to a linear combination of products of Capelli rows, with rational coefficients.

• Every product of Capelli rows is [L | P]-equivalent to a linear combination of Capelli bitableaux, with rational coefficients.

6.4 Devirtualization of Capelli rows and Laplace expansion type identities

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

Theorem 6.3 ([13, 20]). Let α ∈ X \ L be a virtual letter, |α| = 0; then

D_{y_{1}α}D_{y_{2}α} … D_{y_{n}α} · D_{αz_{1}}D_{αz_{2}} … D_{αz_{n}}

≅ ∑_{i=1}^{n} ±(D_{y_{i}z_{1}} − (−1)^{|y_{i}||z_{1}|}(n − 1)δ_{y_{i}z_{1}}I)D_{y_{1}α} … D̂_{y_{i}α} … D_{y_{n}α} · D_{αz_{2}} … D_{αz_{n}}

where ± is the sign associated to the pair of words

y_{1} … y_{i} … y_{n}z_{1} … z_{n} and y_{i}z_{1}y_{1} … ŷ_{i} … y_{n}z_{2} … z_{n},

that is

(−1)^{|z_{1}|(|y_{1}|+...+ ˆ|y_{i}|+...+|y_{n}|)+|y_{i}|(|y_{1}|+...+|y_{i−1}|)}.

A similar result holds in the case |α| = 1.

By Theorem 6.2 and iterating the identity of Theorem 6.3, one can eliminate all the virtual variables α_{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.

6.5 Basic examples

• Let x ≠ y be proper letters, α a virtual letter, |α| = |x| = |y| = 0; then

D_{xα}^{n}D_{αy}^{n} ≅ n!D_{xy}^{n}.

• Let x be a proper letter, α a virtual letter, |α| = |x| = 0; then

D_{xα}^{n}D_{αx}^{n} ≅ n!D_{xx}(D_{xx} − I) … (D_{xx} − (n − 1)I) = (n!)^{2}\binom{D_{xx}}{n}.

• Let y_{i_{1}}, …, y_{i_{n}}, x_{j_{1}}, …, 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, …, n. Assume that the two n-tuples above have no letters in common. Then

D_{y_{i_{1}}α} … D_{y_{in}α}D_{αx_{jn}} … D_{αx_{jn}} = \begin{cases}per[D_{y_{i_{h}}x_{j_{k}}}]_{h,k} & if |α| = 0, \\ det[D_{y_{i_{h}}x_{j_{k}}}]_{h,k} & if |α| = 1. \end{cases}

• Let L = L^{−} = {x_{1}, … x_{m}} be a linearly ordered set of (distinct) negative proper letters; then, we get the classical Capelli operator:

D_{x_{m}α} … D_{x_{1}α} · D_{αx_{1}} … D_{αx_{m}}

= det [ \begin{matrix}D_{x_{1}x_{1}} + (m − 1)I & D_{x_{1}x_{2}} & … & D_{x_{1}x_{m}} \\ D_{x_{2}x_{1}} & D_{x_{2}x_{2}} + (m − 2)I & … & D_{x_{2}x_{m}} \\ ⋮ & ⋮ & ... & ⋮ \\ D_{x_{m}x_{1}} & D_{x_{m}x_{2}} & … & D_{x_{m}x_{m}} \end{matrix} ] = H_{m},

where the expansion of the “determinant” is by column from left to right (Weyl [92]).

7 Biproducts and Bitableaux in Super[L|P]

7.1 Capelli rows and supersymmetries in Super[L|P]

Let x_{1}, …, x_{n} ∈ L and y_{1}, …, y_{n} ∈ P be (not necessarily distinct) proper letters and places, respectively. Let α ∈ X \L be a virtual letter. By applying the Capelli operator D_{x_{1}α}D_{x_{2}α} … D_{x_{n}α} · D_{αx_{n}} … D_{αx_{2}}D_{αx_{1}} to the monomial (x_{1} | y_{1})(x_{2} | y_{2}) … (x_{n} | y_{n}) ∈ Super_{n}[L|P], we get

D_{x_{1}α}D_{x_{2}α} … D_{x_{n}α} · D_{αx_{n}} … D_{αx_{2}}D_{αx_{1}}(x_{1} | y_{1})(x_{2} | y_{2}) … (x_{n} | y_{n}) = k · D_{x_{1}α}D_{x_{2}α} … D_{x_{n}α}(α | y_{1})(α | y_{2}) … (α | y_{n}), k ∈ ℤ

which is

• supersymmetric in the x’s and the y’s if |α| = 0 (that is equals zero whenever the word x_{1} ⋯ x_{n} or the word y_{1} ⋯ y_{n} contain repeated negatively signed symbols);

• “dual” supersymmetric in the x’s and the y’s if |α| = 1 (that is equals zero whenever the word x_{1} ⋯ x_{n} or the word y_{1} ⋯ y_{n} contain repeated positively signed symbols).

Informally speaking, Capelli rows are supersymmetrization operators in disguise.

7.2 Biproducts as basic symmetrized elements in Super[L|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}[L|P], both associated to pairs of sequences of the same length in L and P. These objects, here presented in their three different virtual forms [11], are called biproducts and *– biproducts, respectively; in particular, the biproducts coincide, in characteristic 0, with the Grosshans–Rota–Stein biproducts [46].

Let x_{1}, …, x_{n} ∈ L and y_{1}, …, y_{n} ∈ P be proper letters and places, respectively. Let α ∈ X \ L be a virtual letter and β ∈ Y \ P be a virtual place, with |α| = |β|.

The element of Super_{n}[L|P]

D_{x_{1}α}D_{x_{2}α} … D_{x_{n}α}(α | y_{1})(α | y_{2}) … (α | y_{n})

= D_{x_{1}α}D_{x_{2}α} … D_{x_{n}α}(\frac{(α | β)^{n}}{n!} ) _{βy_{1}}D_{βy_{2}}D …_{βy_{n}} D

= (x_{1} | β)(x_{2} | β) … (x_{n} | β)_{βy_{1}}D_{βy_{2}}D …_{βy_{n}} D

is called, for |α| = |β| = 0, the biproduct of x_{1} … x_{n} and y_{1} … y_{n}, denoted by

(x_{1}x_{2} … x_{n} | y_{1}y_{2} … y_{n}),  (1)

and, for |α| = |β| = 1, the *-biproduct of x_{1} … x_{n} and y_{1} … y_{n}, denoted by

(x_{1}x_{2} … x_{n} | y_{1}y_{2} … y_{n})^{∗}.  (2)

Remark 7.1. The Laplace expansions of biproducts and *–biproducts correspond to the Leibniz rule for superderivations.

Example 7.1. 1. Let x_{1}, …, x_{n} ∈ L_{0} = L and y_{1}, …, y_{n} ∈ P_{0} = P be proper letters and places, respectively; then Super_{n}[L|P] = Sym_{n}[L|P]. We have

(x_{1}x_{2} … x_{n} | y_{1}y_{2} … y_{n}) = per ((x_{i} | y_{j}))_{i,j=1,...,n}
(x_{1}x_{2} … x_{n} | y_{1}y_{2} … y_{n})^{∗} = (−1)^{(_{2}^{n})} det ((x_{i} | y_{j}))_{i,j=1,...,n}

2. Let x_{1}, x_{2} ∈ L_{0} = L, y_{1}, y_{2} ∈ P_{1} = P; then Super[L_{0}|P_{1}] = Λ[L|P]. We have

(x_{1}x_{2} | y_{1}y_{2}) = D_{x_{1}α}D_{x_{2}α} ((α | y_{1})(α | y_{2}))
= (x_{2} | y_{1})(x_{1} | y_{2}) + (x_{1} | y_{1})(x_{2} | y_{2});

on the other hand,

(x_{1}x_{2} | y_{1}y_{2}) = ((x_{1} | β)(x_{2} | β)) _{βy_{1}}D _{βy_{2}}D
= −(x_{1} | y_{2})(x_{2} | y_{1}) + (x_{1} | y_{1})(x_{2} | y_{2}).

Note that (x_{1}x_{2} | y_{1}y_{2}) is symmetric in the x′s and skew-symmetric in the y′s.

The notion of biproduct is extended to a bilinear map

Super[L] × Super[P] → Super[L|P]

by setting

• If ω = x_{i_{1}}x_{i_{2}} ⋯ x_{i_{p}} is a monomial of ℤ-degree p in Super[L], and ϖ = y_{j_{1}}y_{j_{2}} ⋯ y_{q} is a monomial of ℤ-degree q in Super[P], with p ≠ q, then (ω | ϖ) = 0.

• ( ∑_{h} c_{h}ω_{h} | ∑_{k} d_{k}ϖ_{k}) = ∑_{h,k} c_{h}d_{k} (ω_{h} | ϖ_{k}).

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

Proposition 7.1. Let x_{i}, x_{j} ∈ L, y_{h}, y_{k} ∈ P, ω ∈ Super[L], ϖ ∈ Super[P]. We have the following identities:

D_{x_{i}x_{j}}(ω | ϖ)_{y_{h}y_{k}}D = (D_{x_{i}x_{j}}(ω) | ϖ)_{y_{h}y_{k}}D
= D_{x_{i}x_{j}}(ω | (ϖ)_{y_{h}y_{k}}D)
= (D_{x_{i}x_{j}}(ω) | (ϖ)_{y_{h}y_{k}}D),

where D_{x_{i}x_{j}} and _{y_{h}y_{k}}D are left and right superpolarizations on Super[L] and Super[P], respectively (see Subsection 3.6).

Example 7.2. Let x_{1}, x_{2}, x_{3} ∈ L, |x_{1}| = |x_{3}| = 1, |x_{2}| = 0, y_{1}, y_{2}, y_{3} ∈ L, |y_{1}| = |y_{2}| = 1, |y_{3}| = 0. We have the following identities:

D_{x_{3}x_{2}}(x_{1}x_{2} | y_{1}y_{2})_{y_{1}y_{3}}D = (D_{x_{3}x_{2}}(x_{1}x_{2}) | y_{1}y_{2})_{y_{1}y_{3}}D
= −(x_{1}x_{3} | y_{1}y_{2})_{y_{1}y_{3}}D
= −(x_{1}x_{3} | (y_{1}y_{2})_{y_{1}y_{3}}D)
= (x_{1}x_{3} | y_{3}y_{2}).

Proposition 7.1 may be rephrased as a “representation-theoretical” result.

Recall that Super[L] is pl(L)-module and Super[P] is pl(P)-module, where the actions of pl(L) and pl(P) are implemented by left and right superpolarizations, respectively. Thus, the tensor product Super[L] ⊗ Super[P] is a (pl(L), pl(P))-bimodule

pl(L) · Super[L] ⊗ Super[P] · pl(P),

since the actions of pl(L) and pl(P) clearly commute.

Corollary 7.1. The biproduct induces a (pl((L), pl((P))-equivariant linear map

Super[L] ⊗ Super[P] → Super[L|P],
ω ⊗ ϖ ↦ (ω | ϖ),

ω ∈ Super[L], ϖ ∈ Super[P].

7.3 Bitableau monomials

For any words x′ = x′_{1}x′_{2} ⋯ x′_{n} and x = x_{1}x_{2} ⋯ x_{n} on the letter alphabet X , any words y = y_{1}y_{2} ⋯ y_{n} and y′ = y′_{1}y′_{2} ⋯ y′_{n} on the place alphabet Y, all of the same length, we set

D_{x′x} = D_{x′_{1}x_{1}} D_{x′_{2}x_{2}} ⋯ D_{x′_{n}x_{n}},
⟨x | y⟩ = (x_{1} | y_{1}) (x_{2} | y_{2}) ⋯ (x_{n} | y_{n}),
yy′D = _{y_{1}y′_{1}}D _{y_{2}y′_{2}}D ⋯_{y_{n}y′_{n}} D.

For any pair of tableaux S′ and S on the letter alphabet X , and any pair of tableaux T and T′ on the place alphabet Y, all of the same shape, we set

D_{S′S} = D_{x′x},  x′ = w(S′), x = w(S),
⟨S | T ⟩ = ⟨x | y⟩,  x = w(S), y = w(T),
T T′D = _{yy′}D,  y = w(T), y′ = w(T′),

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

S′S for D_{S′S},
\underline{ST} for ⟨S | T ⟩,
T T′ for _{T T′}D.

The definition of biproduct can also be written in the following form:

D_{xα^{n}} ⟨α^{n} | y⟩ = D_{xα^{n}} \frac{⟨α^{n} | β^{n}⟩}{n!} _{β^{n}y}D = ⟨x | β^{n}⟩ _{β^{n}y}D,

or in an even shorter form:

xα^{n} \underline{α^{n}y} = \frac{1}{n!} xα^{n} \underline{α^{n}β^{n}} β^{n}y = \underline{xβ^{n}} β^{n}y;

in these formulas x and y denote words over the proper alphabets L and P, while α and β denote constant words of positive virtual symbols.

7.4 Bitableaux in Super[L|P]

For every partition λ = (λ_{1}, λ_{2}, …, λ_{p}) ⊢ n, and every pair of tableaux T ∈ T ab(L), U ∈ T ab(P), with sh(T) = λ = sh(U), we define the bitableau [46, 11]

(T | U) ∈ Super_{n}[L|P]

as the common value of the expressions

T C_{1} \underline{C_{1}U} = \frac{1}{λ!} T C_{1} \underline{C_{1}C_{2}} C_{2}U, = \underline{T C_{2}} C_{2}U,

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 λ, and λ! = ∏_{i} λ_{i}!.

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

Example 7.3. In the following, let |α_{i}| = 0 and |β_{i}| = 0 be any virtual positive symbols.

( \begin{matrix}x & y \\ x & z \\ y \end{matrix} | \begin{matrix}a & b \\ a & c \\ c \end{matrix} ) = D _{\begin{matrix}x & y \\ x & z \\ y \end{matrix} \begin{matrix}α_{1} & α_{1} \\ α_{2} & α_{2} \\ α_{3} \end{matrix}} ⟨ \begin{matrix}α_{1} & α_{1} \\ α_{2} & α_{2} \\ α_{3} \end{matrix} | \begin{matrix}a & b \\ a & c \\ c \end{matrix} ⟩

= \frac{1}{2!2!1!} D _{\begin{matrix}x & y \\ x & z \\ y \end{matrix} \begin{matrix}α_{1} & α_{1} \\ α_{2} & α_{2} \\ α_{3} \end{matrix}} ⟨ \begin{matrix}α_{1} & α_{1} \\ α_{2} & α_{2} \\ α_{3} \end{matrix} | \begin{matrix}β_{1} & β_{1} \\ β_{2} & β_{2} \\ β_{3} \end{matrix} ⟩ _{\begin{matrix}β_{1} & β_{1} \\ β_{2} & β_{2} \\ β_{3} \end{matrix} \begin{matrix}a & b \\ a & c \\ c \end{matrix}} D

= ⟨ \begin{matrix}x & y \\ x & z \\ y \end{matrix} | \begin{matrix}β_{1} & β_{1} \\ β_{2} & β_{2} \\ β_{3} \end{matrix} ⟩ _{\begin{matrix}β_{1} & β_{1} \\ β_{2} & β_{2} \\ β_{3} \end{matrix} \begin{matrix}a & b \\ a & c \\ c \end{matrix}} D.

Proposition 7.2. Let T = (ω_{1}, …, ω_{p}), U = (ω′_{1}, …, ω′_{p}) be tableaux of the same shape. Then the bitableau (T | U) equals, up to a sign, the product of the biproducts (ω_{1} | ω′_{1}), …, (ω_{p} | ω′_{p}). In symbols

(T | U) = ( ω_{1} ; ω_{2} ; ⋮ ; ω_{p} | ω′_{1} ; ω′_{2} ; ⋮ ; ω′_{p} ) = (−1)^{|ω_{2}||ω′_{1}|+···+|ω_{p}|(|ω′_{1}|+···+|ω′_{p−1}|)}(ω_{1}|ω′_{1}) ⋯ (ω_{p}|ω′_{p}).

Example 7.4.

( \begin{matrix}x & y \\ x & z \\ y \end{matrix} | \begin{matrix}a & b \\ a & c \\ c \end{matrix} ) = ±(xy | ab) (xz | ac) (y | c)

where the sign is given by the parity of (|xz|)(|ab|) + |y|(|ab| + |ac|).

Remark 7.2. The bitableaux of shape (1, 1, …, 1), i.e., column-bitableaux, are monomials and, conversely, monomials can be written, up to a sign, as column bitableaux. Bitableaux of shape (n), i.e., row-bitableaux, are biproducts.

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

Proposition 7.3. Let x_{i}, x_{j} ∈ L, y_{h}, y_{k} ∈ P, and let S = (ω_{1}, …, ω_{p}) ∈ T ab(L), T = (ω′_{1}, …, ω′_{p}) ∈ T ab(P). We have the following identity:

D_{x_{i}x_{j}}(T | U)_{y_{h}y_{k}}D = D_{x_{i}x_{j}} ( ω_{1} ; ω_{2} ; ⋮ ; ω_{p} | ω′_{1} ; ω′_{2} ; ⋮ ; ω′_{p} ) _{y_{h}y_{k}}D

= ∑_{s,t=1}^{p} (−1)^{(|x_{i}|+|x_{j}|)ϵ_{s}}(−1)^{(|y_{h}|+|y_{k}|)ϵ′_{t}} ( ω_{1} ; ω_{2} ; ⋮ ; ⋮ ; D_{x_{i}x_{j}}(ω_{s}) ; ⋮ ; ω_{p} | ω′_{1} ; ω′_{2} ; ⋮ ; (ω′_{t})_{y_{h}y_{k}}D ; ⋮ ; ⋮ ; ω′_{p} ) ,

where

ϵ_{s} = |ω_{1}| + ⋯ + |ω_{s−1}|, s = 2, …, p,
ϵ′_{t} = |ω′_{t+1}| + ⋯ + |ω′_{p}|, t = 1, 2, …, p − 1,

and D_{x_{i}x_{j}}, _{y_{h}y_{k}}D are left and right superpolarizations on Super[L] and Super[P], respectively (see Subsection 3.6).

Proposition 7.3 may be rephrased as a “representation-theoretical” result.

Let λ = (λ_{1}, λ_{2}, …, λ_{p}) ⊢ n be a partition. Set

Super^{λ}[L] = Super_{λ_{1}}[L] ⊗ Super_{λ_{2}}[L] ⊗ ⋯ ⊗ Super_{λ_{p}}[L],

and

Super^{λ}[P] = Super_{λ_{1}}[P] ⊗ Super_{λ_{2}}[P] ⊗ ⋯ ⊗ Super_{λ_{p}}[P].

The tensor product Super^{λ}[L] is pl(L)-module and the tensor product Super^{λ}[P] is pl(P)-module, where the actions of pl(L) and pl(P) are implemented by left and right superpolarizations and extended as left and right superderivations , respectively. Thus, the tensor product Super^{λ}[L] ⊗ Super^{λ}[P] is a (pl(L), pl(P))-bimodule

pl(L) · Super^{λ}[L] ⊗ Super^{λ}[P] · pl(P),

since the actions of pl(L) and pl(P) clearly commute.

Corollary 7.2. The map

ω_{1} ⊗ ω_{2} ⊗ ⋯ ⊗ ω_{p} ⊗ ω′_{1} ⊗ ω′_{2} ⊗ ⋯ ⊗ ω′_{p} ↦ ( ω_{1} ; ω_{2} ; ⋮ ; ω_{p} | ω′_{1} ; ω′_{2} ; ⋮ ; ω′_{p} ) ,

induces a (pl((L), pl((P))-equivariant linear map

Super^{λ}[L] ⊗ Super^{λ}[P] → Super_{n}[L|P],

(ω_{1}, ω_{2}, ⋯, ω_{p}) ∈ T ab_{λ}((L), (ω′_{1}, ω′_{2}, ⋯, ω′_{p}) ∈ T ab_{λ}((P).

8 The Standard Basis

8.1 The Straightening Law of Grosshans, Rota and Stein.

Theorem 8.1 (Straightening Law, general form [46]). For all monomials u, v, w ∈ Super(L), x, y ∈ Super(P),

∑_{(v)} ( \begin{matrix}uv_{(1)} \\ v_{(2)}w \end{matrix} | \begin{matrix}x \\ y \end{matrix} ) = (−1)^{|u||v|} ∑_{(u) (y)}(−1)^{l(u_{(2)})} ( \begin{matrix}vu_{(1)} \\ u_{(2)}w \end{matrix} | \begin{matrix}xy_{(1)} \\ y_{(2)} \end{matrix} ) .

In the statement above, the summations are meant with respect to the coproducts of the supersymmetric algebras Super(L) and Super(P), regarded as ℤ_{2}-bialgebras (Subsection 3.6).

In the following, for any positive symbol ξ and any natural number n, we set ξ^{(n)} = \frac{ξ^{n}}{n!} ; notice that

ξ^{(n)}ξ^{(m)} = \binom{n + m}{n}ξ^{(n+m)}, Δ (ξ^{(n)}) = ∑_{k=0}^{n} ξ^{(k)} ⊗ ξ^{(n−k)}.

Proof. [17] First we prove the identity in the case in which the words u, v, w, x, y are powers of positive symbols: let α, β, γ, δ, ϵ be positive virtual symbols, and a, b, d, e natural numbers such that b ≥ d; then

( \begin{matrix}α^{(a)}β^{(b−d)} \\ β^{(d)}ϵ^{(e)} \end{matrix} | \begin{matrix}γ^{(a+b−d)} \\ δ^{(d+e)} \end{matrix} ) = ∑_{k=0}^{min(a,d)}(−1)^{k} ( \begin{matrix}β^{(b)}α^{(a−k)} \\ α^{(k)}ϵ^{(e)} \end{matrix} | \begin{matrix}γ^{(a+b−d)}δ^{(d−k)} \\ δ^{(k+e)} \end{matrix} ) . (∗)

Indeed, starting from the right-hand side

∑_{k=0}^{min(a,d)}(−1)^{k} (β^{(b)}α^{(a−k)} | γ^{(a+b−d)}δ^{(d−k)}) (α^{(k)}ϵ^{(e)} | δ^{(k+e)})

we get

∑_{k=0}^{min(a,d)}(−1)^{k} ∑_{s=0}^{min(a−k,d−k)}(β | γ)^{(b−d+k+s)}(β | δ)^{(d−k−s)}(α | γ)^{(a−k−s)}(α | δ)^{(s)}(α | δ)^{(k)}(ϵ | δ)^{(e)}

= ∑_{k=0}^{min(a,d)}(−1)^{k} ∑_{s=0}^{min(a−k,d−k)} \binom{s + k}{k}(β | γ)^{(b−d+k+s)}(β | δ)^{(d−k−s)}

· (α | γ)^{(a−k−s)}(α | δ)^{(s+k)}(ϵ | δ)^{(e)}

= ∑_{k=0}^{min(a,d)}(−1)^{k} ∑_{t=k}^{min(a,d)} \binom{t}{k}(β | γ)^{(b−d+t)}(β | δ)^{(d−t)}(α | γ)^{(a−t)}(α | δ)^{(t)}(ϵ | δ)^{(e)}

= ∑_{t=0}^{min(a,d)} ∑_{k=0}^{t}(−1)^{k}\binom{t}{k}(β | γ)^{(b−d+t)}(β | δ)^{(d−t)}(α | γ)^{(a−t)}(α | δ)^{(t)}(ϵ | δ)^{(e)}

= (β | γ)^{(b−d)}(β | δ)^{(d)}(α | γ)^{(a)}(ϵ | δ)^{(e)},

which equals the left-hand side

( \begin{matrix}α^{(a)}β^{(b−d)} \\ β^{(d)}ϵ^{(e)} \end{matrix} | \begin{matrix}γ^{(a+b−d)} \\ δ^{(d+e)} \end{matrix} ) = (α^{(a)}β^{(b−d)} | γ^{(a+b−d)}) (β^{(d)}ϵ^{(e)} | δ^{(d+e)}) .

Consider the letter polarization monomial

D_{uα^{(a)}}D_{vβ^{(b)}}D_{wϵ^{(e)}} = (−1)^{|u||v|}D_{vβ^{(b)}}D_{uα^{(a)}}D_{wϵ^{(e)}}

and the place polarization monomial

γ^{(a+b−d)}xD_{δ^{(d+e)}y}D.

and apply these operators on both sides of the positive identity (∗).

Since letter and place polarizations operators are coderivations (see Subsection 3.6, and, for further details, [18] and [19]), we get:

( \begin{matrix}uv_{(1)} \\ v_{(2)}w \end{matrix} | \begin{matrix}x \\ y \end{matrix} ) = D_{uα^{(a)}}D_{vβ^{(b)}}D_{wϵ^{(e)}} ( \begin{matrix}α^{(a)}β^{(b−d)} \\ β^{(d)}ϵ^{(e)} \end{matrix} | \begin{matrix}γ^{(a+b−d)} \\ δ^{(d+e)} \end{matrix} ) _{γ^{(a+b−d)}x}D _{δ^{(d+e)}y}D
= (−1)^{|u||v|}D_{vβ^{(b)}}D_{uα^{(a)}}D_{wϵ^{(e)}}

× ∑_{k=0}^{min(a,d)}(−1)^{k} ( \begin{matrix}β^{(b)}α^{(a−k)} \\ α^{(k)}ϵ^{(e)} \end{matrix} | \begin{matrix}γ^{(a+b−d)}δ^{(d−k)} \\ δ^{(k+e)} \end{matrix} ) _{γ^{(a+b−d)}x}D _{δ^{(d+e)}y}D

= (−1)^{|u||v|} ∑_{(u) (y)}(−1)^{l(u_{(2)})} ( \begin{matrix}vu_{(1)} \\ u_{(2)}w \end{matrix} | \begin{matrix}xy_{(1)} \\ y_{(2)} \end{matrix} ) .

8.2 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 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, …, 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, … 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 ≤ T ⇔ sh(S^{(p)}) ⊴ sh(T^{(p)}), p = 1, 2, …,

where ⊴ stands for the dominance order on partitions. We recall that the dominance order on partitions is defined as follows: λ = (λ_{1} ≥ λ_{2} ≥ …) ≤ μ = (μ_{1} ≥ μ_{2} ≥ …) if and only if

λ_{1} + ⋯ + λ_{i} ≤ μ_{1} + ⋯ + μ_{i},

for every i = 1, 2, . . ..

Lemma 8.1 ([21]). 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).

• DS \underline{T C} = 0, for S ≱ T;

• DS \underline{T C} = θ_{ST}^{−+} \underline{DC}, for sh(S) = sh(T).

Notice that θ_{ST}^{−+} are uniquely determined integer coefficient, since \underline{DC} ≠ 0.

For the proof see [69].

The coefficients θ_{ST}^{−+} are called symmetry transition coefficients and turn out to be a generalization of the “ Désarménien coefficients” [35].

Lemma 8.2 ([21]). For every pair of standard letter tableaux S, T of the same content, with sh(S) = sh(T), we have

θ_{ST}^{−+} \begin{cases}= 0 & if S ≱ T, \\ ≠ 0 & if S = T. \end{cases}

Moreover, each diagonal coefficient θ_{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.

For the proof see [69].

We have also the following nondegeneracy result.

Lemma 8.3 ([21]). Let C, C_{1} be co-Deruyts tableaux and D_{1}, D be Deruyts tableaux, all of the same shape λ ⊢ n. Then

CD_{1} \underline{D_{1}C_{1}} C_{1}D = (−1)^{(_{2}^{n})}h_{λ} \underline{CD},

where h_{λ} is a positive coefficient; indeed, it equals the product of the hook lengths of the shape λ.

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 8.1.

Lemma 8.4. 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).

• CS \underline{T D} = 0, for S ≰ T;

• CS \underline{T D} = θ_{ST}^{+−} \underline{CD}, for sh(S) = sh(T).

Notice that the two types of symmetry transition coefficients are related by

θ_{ST}^{+−} = θ_{T S}^{−+}.

As another example, we have the following operator analogue of Lemma 8.1.

Lemma 8.5 ([21]). 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).

• DS T C = 0, for S ≱ T;

• DS T C = θ_{ST}^{−+} DC, for sh(S) = sh(T).

The coefficients θ_{ST}^{−+} are indeed the symmetry transition coefficients, as before.

8.3 The standard basis

On the set of all the letter tableaux of a given content, we consider the linear order in which

P ≤ P′ ⇔ \begin{matrix}sh(P) ≤_{l} sh(P′) \\ w(P) ≥_{l} w(P′) \end{matrix} ,

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) ≤ (P′, Q′) ⇔ sh(P) = sh(Q) ≤_{l} sh(P′) = sh(Q′) w(P) ≥_{l} w(P′) w(Q) ≥_{l} w(Q′) .

Theorem 8.2 ([46]). Every bitableau (P | Q) ∈ Super_{n}[L|P] can be written as a linear combination, with rational coefficients,

(P | Q) = ∑_{S,T standard} c_{S,T} (S | T)

of standard bitableaux (S | T) ∈ Super_{n}[L|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) ≤ (S, T).

Proof. (Sketch) We consider the following rewriting rules on a bitableau (A | B):

1. if the bitableau (A | B) is not row-ordered, rewrite it as a row-ordered bitableau ±(A′ | B′);

2. 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{matrix}u_{1} & … & u_{p} & v_{p+1} & v_{p+2} & … & v_{m} \\ v_{1} & … & v_{p} & v_{p+1}^{∗} & w_{p+2} & … & w_{n} \end{matrix} | \begin{matrix}x_{1} & … & x_{m} \\ y_{1} & … & y_{n} \end{matrix} ) ,  (∗)

where

v_{1} ≤ … ≤ v_{p} ≤ v_{p+1}^{∗} ≤ v_{p+1} ≤ v_{p+2} ≤ … 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

∑_{(v)} ( \begin{matrix}uv_{(1)} \\ v_{(2)}w \end{matrix} | \begin{matrix}x \\ y \end{matrix} ) = (−1)^{|u||v|} ∑_{(u) (y)}(−1)^{l(u_{(2)})} ( \begin{matrix}vu_{(1)} \\ u_{(2)}w \end{matrix} | \begin{matrix}xy_{(1)} \\ y_{(2)} \end{matrix} ) ,

where

u = u_{1} ⋯ u_{p},
v = v_{1} ⋯ v_{p}v_{p+1}^{∗}v_{p+1}v_{p+2} ⋯ v_{m},
w = w_{p+2} ⋯ w_{n},
x = x_{1} ⋯ x_{m},
y = y_{1} ⋯ y_{n};

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.

3. 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.

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) ≤ (S, T′).

Theorem 8.3 (The Grosshans, Rota, Stein standard basis theorem for Super_{n}[L|P], [46]). The following set is a basis of Super_{n}[L|P]:

{(S | T); S ∈ Stab_{λ}(L), T ∈ Stab_{λ}(P), λ ⊢ n, λ ∈ H(L) ∩ H(P)},

where H(L) and H(P) are the hook sets defined in Subsection 5.4.

Proof. The standard bitableaux whose shape are partitions of n span Super_{n}[L|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, …, 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

∑ c_{ST} (S | T) = 0

among standard bitableaux. Let S_{0} be minimal among the letter tableaux S such that c_{ST} ≠ 0 for some place tableaux T, and let T_{0} be minimal among the place tableaux T such that c_{S_{0}T} ≠ 0. Denote by λ_{0} the shape of S_{0} and T_{0}, let D_{1} be a letter Deruyts tableau of shape λ_{0}, and let D_{2} be a place Deruyts tableau of shape λ_{0}.

By applying the polarization monomials D_{1}S_{0} and T_{0}D_{2} on both sides of the nontrivial relation, we get the contradiction

0 = D_{1}S_{0} (∑_{S,T} c_{ST} (S | T)) T_{0}D_{2}

= ∑_{S,T} c_{ST} D_{1}S_{0} \underline{SC} CT T_{0}D_{2}

= ∑_{S≤S_{0}, T ≤T_{0}} c_{ST} D_{1}S_{0} \underline{SC} CT T_{0}D_{2}

= θ_{S_{0}S_{0}}^{−+} c_{S_{0}T_{0}} θ_{T_{0}T_{0}}^{+−} D_{1}C \underline{CD_{2}} = θ_{S_{0}S_{0}}^{−+} c_{S_{0}T_{0}} θ_{T_{0}T_{0}}^{+−} (D_{1} | D_{2}) ≠ 0.

An alternative proof of the linear independence of standard bitableaux follows from the superalgebraic version of the Robinson–Schensted correspondence [10], [61].

8.4 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

sh(Q) <_{l} sh(Q′)
or sh(Q) = sh(Q′) and w(Q) >_{l} w(Q′).

Let

T_{1}, T_{2}, …, T_{f}

be the list of the standard place tableaux, whose shapes are partitions of n, in ascending order, and let

Super_{n}[L|P] = V_{0} ⊃ V_{1} … ⊃ V_{f} = (0)

be the chain of the subspaces

V_{i} = ⟨(S | T); S standard, T > T_{i}⟩.

Each subspace V_{i} is a pl(L)-submodule; indeed for any letter polarization D, and any basis element (S | T), with S, T standard and T > T_{i}, we have

D (S | T) = ∑_{P}(P | T) = ∑_{P} ∑_{U,V standard} c_{UV}^{P} (U | V),

where, by the standard expansion theorem, V ≥ T > T_{i}.

Thus, the chain Super_{n}[L|P] = V_{0} ⊃ V_{1} … ⊃ V_{f} = (0) is indeed a pl(L)-invariant filtration of Super_{n}[L|P]. In the associated graded module

V_{0}/V_{1} ⊕ V_{1}/V_{2} ⊕ ⋯ ⊕ V_{f−1}/V_{f}

each summand V_{i−1}/V_{i} has a basis {(S | T_{i}) + V_{i}; S standard, sh(S) = sh(T_{i})}.

An analogous argument gives a pl(P)-invariant filtration of Super_{n}[L|P].

9 Clebsch–Gordan–Capelli Expansions

9.1 Right symmetrized bitableaux

For every λ ⊢ n and every T ∈ T ab(L), U ∈ T ab(P), with sh(T) = λ = sh(U), we define the right symmetrized bitableau (T | U ) ∈ Super_{n}[L|P] by setting

(T | U ) = D_{T C} ⟨C | D⟩ _{DU}D
= T C \underline{CD} DU

where C is any virtual tableau of co-Deruyts type and D is any virtual tableau of Deruyts type, all of shape λ.

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

Proposition 9.1 ([11, 20, 21]). For every λ ⊢ n and every T ∈ T ab(L), U ∈ T ab(P), with sh(T) = λ = sh(U), we have

(T | U ) = \underline{T C_{1}} C_{1}D DU
= T C \underline{CD} DU
= T C CD_{1} \underline{D_{1}U},

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 λ.

Example 9.1. In the following, let |α_{i}| = |α′_{i}| = 0 and |β_{i}| = |β′_{i}| = 1 be virtual symbols.

( \begin{matrix}x & y \\ x & z \\ y \end{matrix} | \begin{matrix}a & b \\ a & c \\ c \end{matrix} )

= ⟨ \begin{matrix}x & y \\ x & z \\ y \end{matrix} | \begin{matrix}α′_{1} & α′_{1} \\ α′_{2} & α′_{2} \\ α′_{3} \end{matrix} ⟩ _{\begin{matrix}α′_{1} & α′_{1} \\ α′_{2} & α′_{2} \\ α′_{3} \end{matrix} \begin{matrix}β_{1} & β_{2} \\ β_{1} & β_{2} \\ β_{1} \end{matrix}} D _{\begin{matrix}β_{1} & β_{2} \\ β_{1} & β_{2} \\ β_{1} \end{matrix} \begin{matrix}a & b \\ a & c \\ c \end{matrix}} D

= D _{\begin{matrix}x & y \\ x & z \\ y \end{matrix} \begin{matrix}α_{1} & α_{1} \\ α_{2} & α_{2} \\ α_{3} \end{matrix}} ⟨ \begin{matrix}α_{1} & α_{1} \\ α_{2} & α_{2} \\ α_{3} \end{matrix} | \begin{matrix}β_{1} & β_{2} \\ β_{1} & β_{2} \\ β_{1} \end{matrix} ⟩ _{\begin{matrix}β_{1} & β_{2} \\ β_{1} & β_{2} \\ β_{1} \end{matrix} \begin{matrix}a & b \\ a & c \\ c \end{matrix}} D

= D _{\begin{matrix}x & y \\ x & z \\ y \end{matrix} \begin{matrix}α_{1} & α_{1} \\ α_{2} & α_{2} \\ α_{3} \end{matrix}} D _{\begin{matrix}α_{1} & α_{1} \\ α_{2} & α_{2} \\ α_{3} \end{matrix} \begin{matrix}β′_{1} & β′_{2} \\ β′_{1} & β′_{2} \\ β′_{1} \end{matrix}} ⟨ \begin{matrix}β′_{1} & β′_{2} \\ β′_{1} & β′_{2} \\ β′_{1} \end{matrix} | \begin{matrix}a & b \\ a & c \\ c \end{matrix} ⟩ .

Informally speaking, the right symmetrized bitableau (T | U ) is supersymmetric in the rows of T and dual supersymmetric in the columns of U: as a matter of fact, (T | U ) 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 (T | U ) can be also regarded as the result of applying a place polarization operator to a bitableau: more specifically,

(T | U ) = (T | D)_{DU}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.

Example 9.2. In the following all the proper symbols are negative and |β_{i}| = 1.

( \begin{matrix}x & y \\ x & z \\ y \end{matrix} | \begin{matrix}a & b \\ a & c \\ c \end{matrix} ) = ( \begin{matrix}x & y \\ x & z \\ y \end{matrix} | \begin{matrix}β_{1} & β_{2} \\ β_{1} & β_{2} \\ β_{1} \end{matrix} ) _{\begin{matrix}β_{1} & β_{2} \\ β_{1} & β_{2} \\ β_{1} \end{matrix} , \begin{matrix}a & b \\ a & c \\ c \end{matrix}} D

= 2 ( \begin{matrix}x & y \\ x & z \\ y \end{matrix} | \begin{matrix}a & b \\ a & c \\ c \end{matrix} ) + 2 ( \begin{matrix}x & y \\ x & z \\ y \end{matrix} | \begin{matrix}a & b \\ c & c \\ a \end{matrix} ) + 2 ( \begin{matrix}x & y \\ x & z \\ y \end{matrix} | \begin{matrix}c & b \\ a & c \\ a \end{matrix} )

+ 2 ( \begin{matrix}x & y \\ x & z \\ y \end{matrix} | \begin{matrix}a & c \\ a & b \\ c \end{matrix} ) + 2 ( \begin{matrix}x & y \\ x & z \\ y \end{matrix} | \begin{matrix}a & c \\ c & b \\ a \end{matrix} ) + 2 ( \begin{matrix}x & y \\ x & z \\ y \end{matrix} | \begin{matrix}c & c \\ a & b \\ a \end{matrix} ) .

Indeed, in the case of negatively signed alphabets, our definition coincides with the classical one [28].

Remark 9.1. Let D be a Deruyts tableau of shape λ on the proper place alphabet P.

The right symmetrized bitableau (T | D ) satisfies the identity:

(T | D ) = λ̃_{1}! ⋯ λ̃_{q}! (T | D),

for every T ∈ T ab(L).

9.2 Left symmetrized bitableaux

For every λ ⊢ n and every T ∈ T ab(L), U ∈ T ab(P), with sh(T) = λ = sh(U), we define the left symmetrized bitableau ( T | U) ∈ Super_{n}[L|P] by setting

( T | U) = D_{T D} ⟨D | C⟩ _{CU}D
= T D \underline{DC} CU,

where D is any virtual tableau of Deruyts type and C is any virtual tableau of co-Deruyts type, all of shape λ.

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

Proposition 9.2 ([11, 20, 21]). For every λ ⊢ n and every T ∈ T ab(L), U ∈ T ab(P), with sh(T) = λ = sh(U), we have

( T | U) = \underline{T D_{1}} D_{1}C CU
= T D \underline{DC} CU
= T D DC_{1} \underline{C_{1}U}

where D, D_{1} are virtual tableaux of Deruyts type and C, C_{1} are virtual tableaux of co-Deruyts type, all of shape λ.

Informally speaking, the left symmetrized bitableau ( T | U) is dual supersymmetric in the columns of T and supersymmetric in the rows of U: as a matter of fact, ( T | U) 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 ( T | U)) can be also regarded as the result of applying a letter polarization operator to a bitableau: more specifically,

( T | U) = D_{T D}(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.

Remark 9.2. Let D be a Deruyts tableau of shape λ on the proper letter alphabet L.

The left symmetrized bitableau ( D | T ) satisfies the identity:

( D | T ) = λ̃_{1}! ⋯ λ̃_{q}! (D | T),

for every T ∈ T ab(P).

9.3 Doubly symmetrized bitableaux

For every λ ⊢ n and every T ∈ T ab(L), U ∈ T ab(P), with sh(T) = λ = sh(U), we define the doubly symmetrized bitableau ( T | U ) ∈ Super_{n}[L|P] by setting

( T | U ) = D_{T D_{1}} (D_{1} | D_{2}) _{D_{2}U}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 λ.

9.4 Clebsch–Gordan–Capelli bases of Super_{n}[L|P]

Theorem 9.1 ([11, 20, 21]). The following sets are bases for Super_{n}[L|P]:

1. {(S | T ); S ∈ Stab_{λ}(L), T ∈ Stab_{λ}(P), λ ⊢ n, λ ∈ H(L) ∩ H(P)};

2. {( S | T ); S ∈ Stab_{λ}(L), T ∈ Stab_{λ}(P), λ ⊢ n, λ ∈ H(L) ∩ H(P)};

3. {( S | T ); S ∈ Stab_{λ}(L), T ∈ Stab_{λ}(P), λ ⊢ n, λ ∈ H(L) ∩ H(P)};

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

∑ c_{ST} (S | T ) = 0

among standard right symmetrized bitableaux. Let S_{0} be minimal among the letter tableaux S such that c_{ST} ≠ 0 for some place tableau T, and let T_{0} be maximal among the place tableaux T such that c_{S_{0}T} ≠ 0. Denote by λ_{0} the shape of S_{0} and T_{0}, let D_{0} be a letter Deruyts tableau of shape λ_{0}, and let C_{0} be a place co-Deruyts tableau of shape λ_{0}.

By applying the polarization monomials D_{0}S_{0} and T_{0}C_{0} on both sides of the nontrivial relation, we get the contradiction

0 = D_{0}S_{0} (∑_{S,T} c_{ST} (S | T )) T_{0}C_{0}

= ∑_{S,T} c_{ST} D_{0}S_{0} SC \underline{CD} DT T_{0}C_{0}

= ∑_{S≤S_{0}, T ≥T_{0}} c_{ST} D_{0}S_{0} SC \underline{CD} DT T_{0}C_{0}

= θ_{S_{0}S_{0}}^{−+} c_{S_{0}T_{0}} θ_{T_{0}T_{0}}^{−+} D_{0}C \underline{CD} DC_{0}

= (−1)^{(_{2}^{n})}h_{λ_{0}} θ_{T_{0}T_{0}}^{−+} c_{S_{0}T_{0}} θ_{S_{0}S_{0}}^{−+} \underline{D_{0}C_{0}} ≠ 0.

The linearly independent set of standard right symmetrized bitableaux spans Super_{n}[L|P], since it has the same cardinality as the basis of standard bitableaux.

10 Young–Capelli Symmetrizers and Orthonormal Generators

10.1 Young–Capelli symmetrizers

Let λ ⊢ n and let S′, S ∈ T ab(L), with sh(S′) = λ = sh(S). The product of letter bitableau polarization monomials D_{S′C} · D_{CD} · D_{DS} defines, by restriction, a linear operator

γ_{n}(S′, S ) = D_{S′C} · D_{CD} · D_{DS} ∈ End_{𝕂}[Super_{n}[L|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 λ. The operator γ_{n}(S′, S ) is called a right Young–Capelli symmetrizer [12, 20, 21].

Example 10.1. Let |α_{i}| = 0 and |β_{i}| = 1.

γ_{n} ( \begin{matrix}x & y \\ x & z \\ y \end{matrix} , \begin{matrix}x & y \\ y & z \\ z \end{matrix} )

= D _{\begin{matrix}x & y \\ x & z \\ y \end{matrix} , \begin{matrix}α_{1} & α_{1} \\ α_{2} & α_{2} \\ α_{3} \end{matrix}} · D _{\begin{matrix}α_{1} & α_{1} \\ α_{2} & α_{2} \\ α_{3} \end{matrix} , \begin{matrix}β_{1} & β_{2} \\ β_{1} & β_{2} \\ β_{1} \end{matrix}} · D _{\begin{matrix}β_{1} & β_{2} \\ β_{1} & β_{2} \\ β_{1} \end{matrix} , \begin{matrix}x & y \\ y & z \\ z \end{matrix}}

By the metatheoretic significance of the method of virtual variables (see, e.g., Subsection 6.1), the crucial fact is that γ_{n}(S′, S ) belongs to the subalgebra B_{n}, the algebra generated by proper letter polarizations. In plain words, even though the operator γ_{n}(S′, S ) is defined by using virtual variables, it admits presentations involving only superpolarizations between proper letters.

A right Young–Capelli symmetrizer is called standard when both its tableaux are standard.

In an analogous way, we define the left Young–Capelli symmetrizers. Let λ ⊢ n and let S′, S ∈ T ab(L), with sh(S′) = λ = sh(S). The product of letter bitableau polarization monomials D_{S′D} · D_{DC} · D_{CS} defines, by restriction, a linear operator

γ_{n}( S′ , S) = D_{S′D} · D_{DC} · D_{CS} ∈ End_{𝕂}[Super_{n}[L|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 λ.

10.2 The Triangularity Theorem

Theorem 10.1 ([12, 20, 21]). The action of standard right Young–Capelli symmetrizers on standard right symmetrized bitableaux is given by

γ_{n}(S′, S )(T | U ) = \begin{cases}(−1)^{(_{2}^{n})}h_{λ}θ_{ST}^{−+}(S′ | U ), & sh(S) = sh(T) = λ, \\ 0 & otherwise, \end{cases}

where θ_{ST}^{−+} are the symmetry transition coefficients from Lemma 8.1, thus integers satisfying the triangularity conditions θ_{ST}^{−+} = 0 unless S ≥ T, θ_{ST}^{−+} ≠ 0 for S = T, and h_{λ} is a positive integer which depends on the shape λ. Furthermore, the integer h_{λ} equals the product of the hook lengths of the shape λ.

Proof. (Sketch) [21] First of all, by definition, we have

γ_{n}(S′, S )(T | U ) = S′C_{1} (C_{1}D_{1} (D_{1}S \underline{T C_{2}} ) C_{2}D_{2}) D_{2}U.

We note that:

if D_{1}S \underline{T C_{2}} ≠ 0 then sh(D_{1}) ≥ sh(C_{2});
if C_{1}D_{1} D_{1}S \underline{T C_{2}} C_{2}D_{2} ≠ 0 then sh(C_{1}) ≤ sh(D_{2}).

Thus, the whole expression is nonzero only if S and T have the same shape, say sh(S) = sh(T) = λ. Under this condition, we have, using Lemmas 8.1, 8.2 and 8.3 from Subsection 8.2,

γ_{n}(S′, S )(T | U ) = S′C_{1} C_{1}D_{1} (D_{1}S \underline{T C_{2}}) C_{2}D_{2} D_{2}U

= θ_{ST}^{−+}S′C_{1} (C_{1}D_{1} \underline{D_{1}C_{2}} C_{2}D_{2}) D_{2}U

= (−1)^{(_{2}^{n})}h_{λ}θ_{ST}^{−+}S′C_{1} \underline{C_{1}D_{2}} D_{2}U

= (−1)^{(_{2}^{n})}h_{λ}θ_{ST}^{−+}(S′ | U ).

The coefficients θ_{ST}^{−+} and h_{λ} satisfy the triangularity and nondegeneracy conditions by Lemmas 8.2 and 8.3.

The last assertion will be proved in Subsection 18.3.

Given any linear extension of the partial order defined above, the matrix [θ_{S,T}^{−+}] is lower triangular with nonzero integral diagonal entries; the matrix

[ϱ_{S,T}^{−+}] = [θ_{S,T}^{−+}]^{−1}

is called the Rutherford matrix.

Remark 10.1. The action of standard left Young–Capelli symmetrizers on standard left symmetrized bitableaux is given by

γ_{n}( S′ , S)( T | U) = \begin{cases}(−1)^{(_{2}^{n})}h_{λ}θ_{ST}^{+−}( S′ | U), & sh(S) = sh(T) = λ, \\ 0 & otherwise, \end{cases}

where θ_{ST}^{+−} are the symmetry transition coefficients defined in Subsection 8.2, thus integers satisfying the triangularity conditions θ_{ST}^{+−} = 0 unless S ≤ T, θ_{ST}^{+−} ≠ 0 for S = T, and h_{λ} is the product of the hook lengths of the shape λ.

10.3 Orthonormal generators

For every S′, S ∈ Stab(L), with sh(S′) = sh(S) = λ ⊢ n, we define the right orthonormal generator Y_{n}(S′, S ) ∈ B_{n} by setting [12]

Y_{n}(S′, S ) = \frac{(−1)^{(_{2}^{n})}}{h_{λ}} ∑_{T ∈Stab(L)} ϱ_{ST}^{−+} γ_{n}(S′, T ).

From Theorem 10.1 and the definitions above, the next result follows immediately.

Theorem 10.2. The action of the right orthonormal generators on the standard right symmetrized bitableaux is given by

Y_{n}(S′, S )(T | U ) = δ_{S,T} (S′ | U ).

Therefore, the orthonormal generators

Y_{n}(S′, S ) S, S′ ∈ Stab_{λ}(L), λ ∈ H(L) ∩ H(P)

form a 𝕂-linear basis of the algebra B_{n}.

Remark 10.2. In an analogous way, we can define the left orthonormal generators Y_{n}( S′ , S), which act orthogonally on left symmetrized bitableaux.

10.4 Factorization properties

From the operator triangularity in Lemma 8.5, we have the following result.

Proposition 10.1 (Factorization theorem, [12, 20, 69]). For any standard tableau T, sh(T) = λ, with no letter in common with the tableau C and D, the Young– Capelli symmetrizer γ_{n}(S′, S ) can be factorized as follows:

γ_{n}(S′, S ) = θ_{T T}^{−+} S′C CT T D DS.

Remark 10.3. 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 T D DS can be expressed as the product of the Capelli operators relative to pairs of corresponding columns of T and S.

10.5 Place operators

Interchanging the roles of letters and places, we can define the place Young–Capelli symmetrizers

(T, T′ )_{n}γ, ( T , T′)_{n}γ,

and the corresponding place orthonormal generators

(T, T′ )_{n}Y, ( T , T′)_{n}Y,

which act orthogonally on the corresponding symmetrized bitableaux

(U, V ), ( U , V ).

11 Schur and Weyl Modules

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

11.1 Schur modules

Let λ ⊢ n be a partition of n, λ ∈ H(L) ∩ H(P). Given a standard tableau T ∈ Stab_{λ}(P), the subspace S_{λT} generated by the set

{(U | T ); U ∈ T ab_{λ}(L)}

is called the (letter-)Schur module parametrized by the place-tableau T. Proposition 11.1. The Schur modules have the following properties.

1. S_{λT} is a pl(L)-invariant subspace of Super_{n}[L|P];

2. the set {(S | T ); S ∈ Stab_{λ}(L)} is a 𝕂-linear basis of S_{λT};

3. the set {( S | T ); S ∈ Stab_{λ}(L)} is a 𝕂-linear basis of S_{λT};

4. S_{λT} is an irreducible submodule;

5. S_{λT} and S_{λ′T′} are isomorphic pl(L)-modules if and only if λ = λ′.

11.2 Weyl modules

Let λ ⊢ n be a partition of n, λ ∈ H(L) ∩ H(P). Given a standard tableau T ∈ Stab_{λ}(P), the subspace W_{λT} generated by the set

{( U | T ); U ∈ T ab_{λ}(L)}

is called the (letter-)Weyl module parametrized by the place-tableau T. Proposition 11.2. The Weyl modules have the following properties.

1. W_{λT} is a pl(L)-invariant subspace of Super_{n}[L|P];

2. the set {( S | T ); S ∈ Stab_{λ}(L)} is a 𝕂-linear basis of W_{λT};

3. W_{λT} is an irreducible submodule;

4. W_{λT} and W_{λ′T′} are isomorphic pl(L)-modules if and only if λ = λ′.

11.3 The Schur–Weyl correspondence

Proposition 11.3. Let T_{1}, T_{2} ∈ Stab_{λ}(P) and consider the (negative) place Capelli operator

S_{λT_{2}} ← W_{λT_{1}} : T_{1}D DT_{2}.

• ( U | T_{1}) T_{1}D DT_{2} = K_{T_{1}}( U | T_{2} ), K_{T_{1}} ∈ 𝕂^{∗}, for all U ∈ T ab_{λ}(L);

• T_{1}D DT_{2} is a pl(L)-equivariant isomorphism. Proposition 11.4. Let T_{1}, T_{2} ∈ Stab_{λ}(P) and consider the (positive) place Capelli operator

W_{λT_{1}} ← S_{λT_{2}} : T_{1}C CT_{2}.

• ( U | T_{2} ) T_{2}C CT_{1} = K_{T_{2}}( U | T_{1}), K_{T_{2}} ∈ 𝕂^{∗}, for all U ∈ T ab_{λ}(L);

• T_{2}C CT_{1} is a pl(L)-equivariant isomorphism.

12 Decomposition Theorems

Theorem 12.1 ([11, 20]). We have the following complete decompositions of the semisimple pl(L)-module Super_{n}[L|P]:

Super_{n}[L|P] = ⨁_{{λ∈H(L)∩H(P) \atop λ⊢n}} ⨁_{{T ∈Stab(P) \atop sh(T)=λ}} S_{λT}

= ⨁_{{λ∈H(L)∩H(P) \atop λ⊢n}} ⨁_{{T ∈Stab(P) \atop sh(T)=λ}} ⟨(S | T ); S ∈ Stab(L)⟩

= ⨁_{{λ∈H(L)∩H(P) \atop λ⊢n}} ⨁_{{T ∈Stab(P) \atop sh(T)=λ}} ⟨( S | T ); S ∈ Stab(L)⟩;

Super_{n}[L|P] = ⨁_{{λ∈H(L)∩H(P) \atop λ⊢n}} ⨁_{{T ∈Stab(P) \atop sh(T)=λ}} W_{λT}

= ⨁_{{λ∈H(L)∩H(P) \atop λ⊢n}} ⨁_{{T ∈Stab(P) \atop sh(T)=λ}} ⟨( S | T ); S ∈ Stab(L)⟩;

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.

Remark 12.1. We recall (see Subsection 8.4) that the basis of standard bitableaux gives rise to a pl(L)-invariant filtration

Super_{n}[L|P] = V_{0} ⊃ V_{1} … ⊃ V_{f} = (0),

where V_{i} = ⟨(S | T); S standard, T > T_{i}⟩. Notice that each term V_{i} in the invariant filtration admits the Weyl module W_{λT_{i}}, sh(T_{i}) = λ, as a complementary invariant subspace in the preceding term V_{i−1}:

V_{i−1} = V_{i} ⊕ W_{λT_{i}}.

Therefore, the standard basis theorem must be regarded as a “weak form” of the Clebsch–Gordan–Capelli basis theorem.

Theorem 12.2 ([12, 20]). We have the following complete decomposition for the operator algebra B_{n} generated by the letter polarization operators acting over Super_{n}[L|P]:

B_{n} = ⨁_{{λ∈H(L)∩H(P) \atop λ⊢n}} ⨁_{{S∈Stab(L) \atop sh(S)=λ}} ⟨Y_{n}(S′, S ); S′ ∈ Stab(L)⟩,

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.

Corollary 12.1 (Structure Theorem).

B_{n} ≅ ⨁_{{λ∈H(L)∩H(P) \atop λ⊢n}} M_{f_{λ}(L)}(𝕂),

where M_{f_{λ}(L)}(𝕂) is the full matrix algebra of square matrices of order f_{λ}(L).

A completely parallel theory holds for the operator algebra _{n}B generated by the proper place polarization operators acting over Super_{n}[L|P].

It should be clear from the preceding discussion how to define letter left orthonormal generators Y ( S′ , S) and place left and right orthonormal generators ( T′ , T )Y and

(T′, T )Y.

We have the following “symmetric” version of Theorem 10.2.

Proposition 12.1. Let S′, S, U ∈ Stab(L) and V, T, T′ ∈ Stab(P). Then:

Y (S′, S )(U, V )(T, T′ )Y = δ_{SU}δ_{V T} (S′, T′ ),
Y ( S′ , S)( U , V )( T , T′)Y = δ_{SU}δ_{V T} ( S′ , T′).

As a consequence, we get the double centralizer theorem.

Theorem 12.3 (The Double Centralizer Theorem [12]). The algebras

B_{n}, _{n}B ⊂ End_{𝕂}(Super_{n}[L|P])

are the centralizers of each other.

13 The Natural Form of Irreducible Matrix Representations of Schur Superalgebras

Let L, P be arbitrary finite signed alphabets, and λ ⊢ n a partition of n. In the following we will write S_{1}, S_{2}, …, S_{p_{λ}} to mean the list of all standard tableaux of shape λ over L, sorted with respect to the linear order defined in Subsection 5.5. For sake of simplicity, we will write

θ_{ij}^{λ} in place of θ_{S_{i}S_{j}}^{−+}, and ϱ_{ij}^{λ} in place of ϱ_{S_{i}S_{j}}^{−+}.

For every operator G ∈ B_{n} = ⊕_{λ⊢n}B_{λ}, we will denote by G_{λ} its component in the simple subalgebra B_{λ} (Theorem 12.2).

On the one hand, we have

Y_{n}(S_{h}, S_{h} ) G Y_{n}(S_{k}, S_{k} ) = d_{hk}^{λ}(G) Y_{n}(S_{h}, S_{k} ), d_{hk}^{λ}(G) ∈ 𝕂;

notice that the coefficients d_{hk}^{λ}(G) are precisely the coefficients that appear in the expansion of G_{λ} with respect to the basis {Y_{n}(S_{h}, S_{k} ); h, k = 1, 2, … p_{λ}} of B_{λ}.

On the other hand, given any standard tableau T of shape λ over P, we have

G(S_{k} | T ) = G_{λ}(S_{k} | T ) = ∑_{h} c_{hk}^{T}(G) (S_{h} | T ), c_{hk}^{T}(G) ∈ 𝕂.

Clearly, we have:

Proposition 13.1. For every standard tableau T over P with sh(T) = λ, we have

d_{hk}^{λ}(G_{λ}) = d_{hk}^{λ}(G) = c_{hk}^{T}(G) = c_{hk}^{T}(G_{λ}),

for every G ∈ B_{n}, and for every h, k = 1, 2, … p_{λ}.

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

We remark that, for every operator G ∈ B_{n}, and for every tableau S_{j} ∈ Stab(L) of shape λ, the following identity holds:

DS_{h} G \underline{S_{j}C} = θ_{hj}^{λ}(G) \underline{DC},

where θ_{hj}^{λ}(G) is a uniquely determined scalar coefficient (here, as usual, D denotes any virtual letter Deruyts tableau of shape λ and C denotes any virtual place co-Deruyts tableau of shape λ).

Notice that

[θ_{ij}^{λ}] = [θ_{ij}^{λ}(I)], [ϱ_{ij}^{λ}] = [θ_{ij}^{λ}(I)]^{−1},

where I denotes the identity in B_{n}.

Theorem 13.1 ([21]). We have:

c_{ij}^{λ}(G) = ∑_{h} ϱ_{ih}^{λ} θ_{hj}^{λ}(G),

or, in matrix form:

C^{λ}(G) = Θ^{λ}(I)^{−1} × Θ^{λ}(G).

Proof. (Sketch) We start from the definition of the coefficients c_{hk}^{λ}(G):

G(S_{k} | T ) = ∑_{h=1}^{p_{λ}} c_{hk}^{λ}(G) (S_{h} | T );

we apply the Young–Capelli symmetrizer γ_{n}(S_{l} | S_{l} ) to the left-hand side:

γ_{n}(S_{l} | S_{l} ) G (S_{k} | T ) = S_{l}C CD DS_{l} G \underline{S_{k}C} CD DT

= θ_{lk}^{λ}(G) S_{l}C CD \underline{DC} CD DT

= (−1)^{(_{2}^{n})}h_{λ} θ_{lk}^{λ}(G) S_{l}C \underline{CD} DT

= (−1)^{(_{2}^{n})}h_{λ} θ_{lk}^{λ}(G) (S_{l} | T );

we apply the Young–Capelli symmetrizer γ_{n}(S_{l} | S_{l} ) to the right-hand side:

γ_{n}(S_{l} | S_{l} ) ∑_{h=1}^{p_{λ}} c_{hk}^{λ}(G) (S_{h} | T ) = ∑_{h=1}^{p_{λ}} c_{hk}^{λ}(G) γ_{n}(S_{l} | S_{l} ) (S_{h} | T )

= (−1)^{(_{2}^{n})}h_{λ} ∑_{h=1}^{p_{λ}} θ_{lh}^{λ} c_{hk}^{λ}(G) (S_{l} | T ).

In the end, we have

θ_{lk}^{λ}(G) = ∑_{h=1}^{p_{λ}} θ_{lh}^{λ} c_{hk}^{λ}(G).

For every λ ⊢ n such that λ ∈ H(L) ∩ H(P) and every standard place tableau T of shape λ, the module structure U(pl(L))·S_{T} on the irreducible Schur module S_{T} induces a surjective algebra morphism

ν_{T} : U(pl(L)) → End_{𝕂}[S_{T}];

by choosing the basis of the standard symmetrized bitableaux (S_{i} | T ) in S_{T} , the morphism ν_{T} induces an irreducible matrix representation

\overline{ν}_{T} : U(pl(L)) → M_{p_{λ}}

where, for every G ∈ U(pl(L)),

\overline{ν}_{T} (G) = [c_{ij}^{λ}(ν_{T} (G))] = [c_{ij}^{λ}(G)],

where G = ν_{T} (G). Therefore, this irreducible representation has the matrix form described in the preceding theorem.

Part III

Applications

14 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}, …, x_{m}} be a basis of V. Consider a pair of multiindices

I = (i_{1}, …, i_{s}) ∈ ℤ^{+}^{s}, J = (j_{1}, …, j_{t}) ∈ ℤ^{+}^{t},

and set |I| = i_{1} + ⋯ + i_{s}, |J| = j_{1} + ⋯ + j_{t}. The space

W^{I,J} = Λ^{i_{1}}(V) ⊗ ⋯ ⊗ Λ^{i_{s}}(V) ⊗ Sym^{j_{1}}(V) ⊗ ⋯ ⊗ 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 L = L_{1} = {x_{1}, …, x_{m}} and the signed place alphabet P = P_{0} ∪ P_{1}, where

P_{0} = {φ_{1}, …, φ_{s}}, P_{1} = {ψ_{1}, …, ψ_{t}},

and the order is given by

φ_{1} < … < φ_{s} < ψ_{1} < … < ψ_{t}.

Let Super^{I,J}[L|P] be the “homogeneous” subspace of Super[L|P] spanned by all the monomials of content (I, J), that is, of content i_{h} in each positive place φ_{h}, for every h = 1, …, s, and of content j_{k′} in each negative place ψ_{k}, for every k = 1, …, t.

Clearly, Super^{I,J}[L|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} → Super^{I,J}[L|P]

be the map such that

F (x_{p_{11}} ∧ ⋯ ∧ x_{p_{1i_{1}}} ⊗ ⋯ ⊗ x_{p_{s1}} ∧ ⋯ ∧ x_{p_{sis}} ⊗ x_{q_{11}} ⋯ x_{q_{1j_{1}}} ⊗ ⋯ ⊗ x_{q_{t1}} ⋯ x_{q_{tj_{t}}})

= (x_{p_{11}} ⋯ x_{p_{1i_{1}}} | φ_{1}^{i_{1}}) ⋯ (x_{p_{s1}} ⋯ x_{p_{sis}} | φ_{s}^{i_{s}})(x_{q_{11}} | ψ_{1}) ⋯ (x_{q_{1j_{1}}} | ψ_{1})

⋯ (x_{q_{t1}} | ψ_{t}) ⋯ (x_{q_{tj_{t}}} | ψ_{t}).

Proposition 14.1. The map F : W^{I,J} → Super^{I,J}[L|P] is a GL(m)-equivariant isomorphism.

By specializing Theorem 12.1, we get:

Corollary 14.1. We have the following complete GL(m)-module decomposition

W^{I,J} = ⨁_{{λ⊢|I|+|J| \atop λ∈H(L)∩H(P)}} ⨁_{T ∈Stab_{λ}^{I,J} (P)} F^{−1}[S_{λT}],

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_{λ}^{I,J}(P) of all standard place tableaux of content (I, J).

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

15 Letterplace Algebras, Highest Weight Vectors and sl_{m}(ℂ)-Irreducible Modules

15.1 Representations of sl_{m}(ℂ): basic definitions and results

Let sl_{m}(ℂ) be the special linear Lie algebra of m × m traceless matrices with complex entries. Let h ⊆ sl_{m}(ℂ) be the Cartan subalgebra of all diagonal matrices.

For every i = 1, …, m, let ε_{i} be the linear function

ε_{i} : gl_{m}(ℂ) → ℂ

such that

ε_{i}(M) = m_{ii},

for every matrix M = (m_{ij})_{i,j=1,...,m} ∈ gl_{m}(ℂ).

Thus, h^{∗} = ℂ{ε_{1}, ε_{2}, …, ε_{m}}/⟨ε_{1} + ε_{2} + … + ε_{m}⟩. We often write ε_{i} for the image of ε_{i} in h^{∗}.

Consider the Cartan decomposition

sl_{m}(ℂ) = h ⊕ ⟨E_{i,j}; i ≠ j⟩;

the one-dimensional spaces ⟨E_{i,j}⟩ are invariant subspaces of sl_{m}(ℂ) with respect to the adjoint action of the subalgebra h, since [H, E_{i,j}] = (ε_{i} − ε_{j})(H)E_{i,j} for every H ∈ h.

The spaces ⟨E_{i,j}⟩ are called the root spaces and the elements ε_{i} − ε_{j} ∈ h^{∗} are called the roots of the Lie algebra sl_{m}(ℂ).

As usual, let

{ε_{i} − ϵ_{j}; i < j}

be the set of positive roots of sl_{m}(ℂ); thus, {⟨E_{i,j}⟩; i < j} is the set of positive root spaces, and {⟨E_{i,j}⟩; i > j} is the set of negative root spaces.

Let V be a finite-dimensional representation of sl_{m}(ℂ) and φ an element of h^{∗}; a nontrivial subspace V(φ) = {v ∈ V; H(v) = φ(H)v, for every H ∈ h} is called a weight subspace of the representation V, and φ is called the weight of V(φ).

It is easy to see that V decomposes into the (finite) direct sum of its weight spaces, in symbols V = ⊕_{φ} V(φ).

A weight vector is a vector v ∈ V which belongs to a weight subspace. A highest weight vector is a weight vector which is annihilated by the action of any positive root space.

Proposition 15.1 (see, e.g., [40]). 1. Every finite-dimensional representation V of sl_{m}(ℂ) possesses a highest weight vector.

2. The subspace W generated by the images of a highest weight vector v under successive applications of negative root spaces is an irreducible subrepresentation.

3. An irreducible representation possesses a unique highest weight vector, up to scalar factors.

4. Two irreducible representations W and W′ of sl_{m}(ℂ) are isomorphic if and only if they have the same highest weight (as an element of h^{∗}).

It follows from assertion 3) of the above proposition that the highest weight of an irreducible sl_{m}(ℂ)-representation may be unambiguously defined as the weight of its highest weight vector.

15.2 sl_{m}(ℂ)-irreducible modules and gl_{m}(ℂ)-irreducible modules

We recall that gl_{m}(ℂ) = sl_{m}(ℂ)⊕ℂI_{m}, I_{m} the identity m×m matrix (this is an instance of the so-called Levi decomposition theorem). The subalgebra ℂI_{m} is the radical ideal of gl_{m}(ℂ) and sl_{m}(ℂ) is its semisimple part. As a matter of fact, sl_{m}(ℂ) is a simple Lie algebra and it is also an ideal of gl_{m}(ℂ); furthermore, the non-trivial ideals of gl_{m}(ℂ) are precisely ℂI_{m} and sl_{m}(ℂ).

The following assertions are special instances of general results (see, e.g., [40], [42], [88]).

Proposition 15.2 (see, e.g., [40]). 1. Every irreducible representation V of gl_{m}(ℂ) is of the form V = V_{0} ⊗ L, where V_{0} is an irreducible representation of sl_{m}(ℂ) (i.e., a representation of gl_{m}(ℂ) that is trivial on the radical ℂI_{m}), and L is a one-dimensional representation of gl_{m}(ℂ).

2. Since the subalgebra sl_{m}(ℂ) is a simple one, it acts on L in the trivial way, and, then, each element of gl_{m}(ℂ) acts on L just by multiplying by a complex coefficient.

3. It follows from the preceding item that any irreducible representation of gl_{m}(ℂ) restricts to an irreducible representation of sl_{m}(ℂ), and any irreducible representation of sl_{m}(ℂ) extends to an irreducible representation of gl_{m}(ℂ).

Claim 15.1. From the preceding proposition, it follows that, if W and W′ are isomorphic sl_{m}(ℂ)-modules, it is not in general true that W and W′ are isomorphic gl_{m}(ℂ)-modules.

15.3 Letterplace algebras and representations of sl_{m}(ℂ)

Let L = L_{1} = {x_{1}, …, x_{m}} be a negatively signed alphabet of letters, and let P be any finite ℤ_{2}-graded alphabet of places. Set 𝕂 = ℂ. In the notation of Subsection 4.3, we have

Super[L|P] ≃ Sym[V ⊗ W_{1}] ⊗ Λ[V ⊗ W_{0}],

and

pl(L) = pl(V) = gl_{m}(ℂ),

the usual Lie algebra of m × m matrices. We will describe the connection between our combinatorial approach and the classical theory (highest weight vectors) of the Lie module

sl_{m}(ℂ) · Super[L|P] ≃ sl_{m}(ℂ) · Sym[V ⊗ W_{1}] ⊗ Λ[V ⊗ W_{0}].

Recall that any homogeneous component Super_{n}[L|P] is a gl_{m}(ℂ)-submodule, and that the action of gl_{m}(ℂ) is implemented by letter polarization operators; since L is a trivially ℤ_{2}-graded alphabet, these polarization operators are derivations in the usual sense (Subsection 4.4).

Claim 15.2. The identity matrix I_{m} acts on the homogeneous component Super_{n}[L|P] as the polarization operator

D_{x_{1},x_{1}} + D_{x_{2},x_{2}} + ⋯ + D_{x_{m},x_{m}},

and, then, by multiplying each element by the integer n. Thus, every sl_{m}(ℂ)-submodule of Super_{n}[L|P] is a gl_{m}(ℂ)-submodule.

Corollary 15.1. Every irreducible gl_{m}(ℂ)-submodule of Super_{n}[L|P] is an irreducible sl_{m}(ℂ)-submodule.

In the case of irreducible gl_{m}(ℂ)-submodules of Super_{n}[L|P], assertion 1) of Proposition 15.2 reduces to a quite simple construction.

Corollary 15.2. Every irreducible gl_{m}(ℂ)-submodule V of Super_{n}[L|P] is of the form V = V_{0} ⊗ ℂ, where:

• V_{0} is the irreducible sl_{m}(ℂ)-submodule obtained from V by restriction of the action of gl_{m}(ℂ) to sl_{m}(ℂ).

• ℂ is the one-dimensional gl_{m}(ℂ)-module such that the action of sl_{m}(ℂ) is trivial and I_{m} · 1 = n1.

Claim 15.3. Let \underline{α} = (α_{1}, α_{2}, …, α_{m}) ∈ ℕ^{m} be an m-multiindex such that α_{1} + α_{2} + … + α_{m} = n. Given a monomial

M = (x_{i_{1}} | y_{j_{1}})(x_{i_{2}} | y_{j_{2}}) ⋯ (x_{i_{n}} | y_{j_{n}}) ∈ Super_{n}[L|P],

we say that M has letter content \underline{α}, and write c(M) = \underline{α}, if

#{x_{i_{h}} = x_{k}; h = 1, 2, …, n} = α_{k},

for every k = 1, 2, …, m.

Let

Super_{\underline{α},n}[L|P]

be the subspace of Super_{n}[L|P] spanned by the set

\{M = (x_{i_{1}} | y_{j_{1}})(x_{i_{2}} | y_{j_{2}}) ⋯ (x_{i_{n}} | y_{j_{n}}); c(M) = \underline{α} = (α_{1}, α_{2}, …, α_{m})\}.

Note that Super_{\underline{α},n}[L|P] may be trivial.

A weight space the sl_{m}(ℂ)-module Super_{n}[L|P] is a non-trivial subspace of the form Super_{\underline{α},n}[L|P], and its weight is the element

α_{1}ε_{1} + α_{2}ε_{2} + ⋯ + α_{m}ε_{m} ∈ h^{∗}.

Thus, the weight space decomposition of Super_{n}[L|P] is given by the following formula:

Super_{n}[L|P] = ⨁_{\underline{α}} Super_{\underline{α},n}[L|P],

where the direct sum is over the set of all multiindices \underline{α} = (α_{1}, α_{2}, …, α_{m}) ∈ ℕ^{m} such that the subspaces Super_{\underline{α},n}[L|P] are non-trivial.

Furthermore, every weight space Super_{\underline{α},n}[L|P] possesses four classes of bases associated to pairs of standard Young tableaux. By specializing Theorem 9.1, we infer that the sets:

• \{(S | T); S ∈ Stab_{λ}(L), T ∈ Stab_{λ}(P), λ ⊢ n, c(S) = \underline{α}\},

• \{(S | T ); S ∈ Stab_{λ}(L), T ∈ Stab_{λ}(P), λ ⊢ n, c(S) = \underline{α}\},

• \{( S | T ); S ∈ Stab_{λ}(L), T ∈ Stab_{λ}(P), λ ⊢ n, c(S) = \underline{α}\},

• \{( S | T ); S ∈ Stab_{λ}(L), T ∈ Stab_{λ}(P), λ ⊢ n, c(S) = \underline{α}\},

are 𝕂-bases of Super_{\underline{α},n}[L|P], where c(S) denotes the content of S ∈ Stab_{λ}(L) (see Subsection 5.1).

15.4 The action of upper polarizations

We recall that the finite set of all the standard tableaux over 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 D_{x_{i}x_{j}} with 1 ≤ i ≤ j ≤ m will be called upper polarizations; if i < j, the operator D_{x_{i}x_{j}} will be said to be a strictly upper polarization. In the following, we will often use the following identity. For any Deruyts tableau of shape λ, we have

( D | T ) = ( ∏_{i} λ̃_{i}!) (D | T).

First of all, we establish a triangularity result.

Proposition 15.3. The action of the standard Capelli bitableaux of the type

[D̃_{λ} | S̃]_{−} = ±D_{λ}D DS

with sh(D) = sh(S) ⊢ n, on the standard bitableaux (T | U), with sh(T) = sh(U) ⊢ n, satisfies the nondegenerate triangularity conditions

[D̃ | S̃]_{−} (T | U) = \begin{cases}c_{S} (D | U) & for S = T, \\ 0 & for S ≱ T, \end{cases}

where c_{S} are nonzero integers.

Proof. By definition, we have

[D̃ | S̃]_{−} (T | U) = ±DD_{1} D_{1}S \underline{T C} 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 ≱ T, while for S = T becomes

DD_{1} D_{1}S \underline{SC} CU = θ_{SS}^{−+} DD_{1} \underline{D_{1}C} CU = ±(λ̃)! θ_{SS}^{−+} \underline{DC} CU = c_{S} (D | U),

where c_{S} ≠ 0.

In the following, given a shape λ = (λ_{1}, …, λ_{p}) ⊢ n, λ_{1} ≤ m, we denote by D_{λ} the Deruyts tableau of shape λ filled, in order, with the first λ_{1} symbols of the alphabet L: for example

D_{322} = \begin{matrix}x_{1} & x_{2} & x_{3} \\ x_{1} & x_{2} \\ x_{1} & x_{2} \end{matrix} .

Proposition 15.4. Let λ ⊢ n and S be any standard tableau of shape λ. Then the action of the Capelli bitableau

[D̃_{λ} | S̃]_{−} = ±D_{λ}D DS

on Super_{n}[L|P] is the same as the action of a linear combination of products of upper polarizations. Furthermore, if S ≠ D_{λ}, each term of this linear combination can be written as a product of upper polarizations, with a strictly upper polarization as rightmost factor.

Proof. (Sketch) The proof is based on three steps.

Step 1. By Proposition 6.1, we infer:

[D̃_{λ} | S̃]_{−} = [ \begin{matrix}xλ̃_{1} ; 1 & w̃_{1} \\ xλ̃_{2} ; 2 & w̃_{2} \\ ⋮ & ⋮ \\ xλ̃_{p−1} ; p−1 & w̃_{p−1} \\ xλ̃_{p} ; p & w̃_{p} \end{matrix} ]_{−} ≅ [ \begin{matrix}xλ̃_{1} ; 1 & w̃_{1} \\ xλ̃_{2} ; 2 & w̃_{2} \\ ⋮ & ⋮ \\ xλ̃_{p−1} ; p−1 & w̃_{p−1} \end{matrix} ]_{−} [xλ̃_{p} ; p | w̃_{p}]

− ∑ c_{∗} [ \begin{matrix}xλ̃_{1} ; 1 & w̃′_{1} w̃_{p,(1)} \\ xλ̃_{2} ; 2 & w̃′_{2} w̃_{p,(2)} \\ ⋮ & ⋮ \\ xλ̃_{p−1} ; p−1 & w̃′_{p−1} w̃_{p,(p−1)} \\ x_{p}^{m} & w̃_{p,(p)} \end{matrix} ]_{−} ,

where the sum is taken under the conditions of the same proposition, and m < λ̃_{p}.

Since S is a standard tableau, each column word w̃_{i} of S̃ contains only letters x_{j}, with j ≥ i. Furthermore, if S ≠ D_{λ}, there is at least one word w̃_{i} of S̃ 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 [D̃_{λ} | S̃] is [L | P]-equivalent to a linear combination of products of Capelli rows of the form

[x_{i} … x_{i} | w];

if S ≠ D_{λ}, 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 6.3, every Capelli row [x_{i} … x_{i} | w] produced by Step 1 is [L | P]- equivalent to a linear combination of products of upper polarizations. Thus [D̃_{λ} | S̃] is [L | P]-equivalent to a linear combination of products of upper polarizations. Furthermore, if S ≠ D_{λ}, 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 ≠ D_{λ}, by iterating the commutator identity, each of the above products can be turned into a linear combination of products of upper polarizations, with a strictly upper polarization as right-most factor.

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

Theorem 15.1. For any element F of Super_{n}[L|P], the following statements are equivalent.

1. F is annihilated by every strictly upper polarization.

2. The expansion of F as a linear combination of standard bitableaux is of the form

F = ∑_{λ⊢n} ∑_{U∈Stab_{λ}(P)} c_{U} (D_{λ} | U).

3. The expansion of F as a linear combination of standard right symmetrized bitableaux is of the form

F = ∑_{λ⊢n} ∑_{U∈Stab_{λ}(P)} d_{U} (D_{λ} | U ).

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) ⇒ 3) follows from a similar argument. By way of contradiction, we assume that there is an element F of Super_{n}[L|P] which is annihilated by all the upper polarizations and has a standard expansion of the form

F = ∑_{λ⊢n} ∑_{U} c_{U} (D_{λ} | U) + ∑_{T,U} c_{T U} (T | U),

where T ranges in a set T of standard tableaux different from any tableau D_{λ} and all the coefficients c_{T U} are nonzero. Let S be any minimal tableau in T , of shape μ, say, and consider the Capelli bitableau

[D̃_{μ} | S̃]_{−} = ±D_{μ}D DS.

By the preceding proposition, [D̃_{μ} | S̃]_{−} 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

0 = [D̃_{μ} | S̃]_{−} (∑_{λ⊢m} ∑_{U} c_{U} (D_{λ} | U) + ∑_{T,U} c_{T U} (T | U))

= [D̃_{μ} | S̃]_{−} (∑_{T,U} c_{T U} (T | U))

= [D̃_{μ} | S̃]_{−} ( ∑_{T,U: S≥T} c_{T U} (T | U))

= [D̃_{μ} | S̃]_{−} (∑_{U} c_{SU} (S | U))

= c_{S} (∑_{U} c_{SU} (D_{μ} | U)) ,

where c_{S} is a nonzero integer, in contradiction with the linear independence of standard bitableaux.

In the language of Subsection 15.1, Theorem 15.1 reads as follows:

Corollary 15.3. For any element F of Super_{n}[L|P], the following statements are equivalent.

1. F is a highest weight vector.

2. The expansion of F as a linear combination of standard bitableaux is of the form

F = ∑_{U∈Stab_{λ}(P)} c_{U} (D_{λ} | U),

where λ is some partition of n.

3. The expansion of F as a linear combination of standard right symmetrized bitabl-

eaux is of the form

F = ∑_{U∈Stab_{λ}(P)} d_{U} (D_{λ} | U ),

where λ is some partition of n.

15.5 The action of lower polarizations

The polarizations D_{x_{i}x_{j}} with m ≥ i ≥ j ≥ 1 will be called lower polarizations; if i > j, D_{ij} will be called a strictly lower polarization. Let λ = (λ_{1}, …, λ_{p}) ⊢ n be a partition of the integer n, and let λ̃ = (λ̃_{1}, …, λ̃_{q}) be its conjugate partition. Let S = (ω_{1}, …, ω_{p}) be a standard tableau of shape λ on the letter alphabet L = L_{1} = {x_{1}, …, x_{m}}, and let S̃ = (ω̃_{1}, …, ω̃_{q}) be its conjugate tableau. For every pair (i, j), with i = 1, …, q, and j = 1, …, m, denote by c_{j,i} the number of occurrences of x_{j} in the word ω̃_{i}. Let

𝔓_{S} = D_{x_{2},x_{1}}^{c_{2,1}} ⋯ D_{x_{m},x_{1}}^{c_{m,1}}D_{x_{3},x_{2}}^{c_{3,2}} ⋯ D_{x_{m},x_{2}}^{c_{m,2}} ⋯ D_{x_{q+1,q}x_{q}}^{c_{q+1,q}} ⋯ D_{x_{m},x_{q}}^{c_{m,q}}.

Proposition 15.5. For every standard letter tableau S and every standard place tableau T, both of shape λ, we have

𝔓_{S}( D_{λ} | T)) = ( ∏_{i=1}^{q} \frac{λ̃_{i}!}{c_{i,i}!}) ( S | T ).

Corollary 15.4. Denote by n_{m}^{−} be the maximal nilpotent subalgebra of sl_{m}(ℂ) spanned by all strictly lower triangular matrices.

1. Let F be a highest weight vector of Super_{n}[L|P] whose expansion as a linear combination of standard bitableaux is of the form

F = ∑_{U∈Stab_{λ}(P)} c_{U} (D_{λ} | U), λ ⊢ n.

Then, the cyclic submodule n_{m}^{−} · F is an irreducible sl_{m}(ℂ)-module, with basis

{ ∑_{U∈Stab_{λ}(P)} c_{U} ( S | U); S ∈ Stab_{λ}(L)}.

Furthermore, F is (up to a scalar factor) the unique highest weight vector of this irreducible sl_{m}(ℂ)-module.

2. Let F′ be a highest weight vector of Super_{n}[L|P] whose expansion as a linear combination of standard right symmetrized bitableaux is of the form

F′ = ∑_{U∈Stab_{λ}(P)} d_{U} (D_{λ} | U ), λ ⊢ n.

Then, the cyclic submodule n_{m}^{−} · F′ is an irreducible sl_{m}(ℂ)-module, with basis

{ ∑_{U∈Stab_{λ}(P)} d_{U} ( S | U ); S ∈ Stab_{λ}(L)}.

Furthermore, F′ is (up to a scalar factor) the unique highest weight vector of this irreducible sl_{m}(ℂ)-module.

Proof. We prove assertion 1. By the preceding proposition, we infer:

n_{m}^{−} · F ⊇ ⟨ ∑_{U∈Stab_{λ}(P)} c_{U} ( S | U); S ∈ Stab_{λ}(L)⟩_{𝕂}.

On the other hand, since F equals, up to a scalar factor, we have

∑_{U∈Stab_{λ}(P)} c_{U} ( D_{λ} | U) ∈ ⟨ ∑_{U∈Stab_{λ}(P)} c_{U} ( S | U); S ∈ Stab_{λ}(L)⟩_{𝕂}

and

b_{m}^{−} · F ⊆ ⟨ ∑_{U∈Stab_{λ}(P)} c_{U} ( S | U); S ∈ Stab_{λ}(L)⟩_{𝕂}.

Finally, the module

⟨ ∑_{U∈Stab_{λ}(P)} c_{U} ( S | U); S ∈ Stab_{λ}(L)⟩_{𝕂}

is a gl_{m}(ℂ)-irreducible module and, hence, an sl_{m}(ℂ)-irreducible module.

Suppose now that \overline{F} is another highest weight vector, \overline{F} ∈ n_{m}^{−} · F. On the one hand, by Corollary 15.3, \overline{F} is of the form

\overline{F} = ∑_{U∈Stab_{μ}(P)} \overline{c}_{U} (D_{μ} | U),

for some partition μ ⊢ n, and, on the other hand,

\overline{F} = ∑_{S∈Stab_{λ}(L)} a_{S} ∑_{U∈Stab_{λ}(P)} c_{U} ( S | U).

This implies λ = μ and a_{S} = 0 whenever S ≠ D_{λ}. Therefore, \overline{F} is a scalar multiple of F.

As a matter of fact, Corollary 15.4 describes all the irreducible sl_{m}(ℂ)-submodules of Super_{n}[L|P]. In the language of Subsection 15.1, we have:

Proposition 15.6. Let W be an irreducible sl_{m}(ℂ)-submodule of Super_{n}[L|P]. Then, there exists a highest weight vector F such that

W = n_{m}^{−} · F = sl_{m}(ℂ) · F.

Proof. Thanks to the claim at the beginning of this section, we can treat W as an irreducible gl_{m}-submodule. Let

f = ∑_{λ⊢n} ∑_{S,T} c_{S,T} (S | T ), S ∈ Stab_{λ}(L), T ∈ Stab_{λ}(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 (D_{λ}, S ) f = ∑_{T} c_{S,T} (D_{λ} | T ) ≠ 0

is a highest weight vector that belongs to W. The cyclic gl_{m}-submodule (equivalently, the cyclic sl_{m}(ℂ)-submodule) of Super_{n}[L|P] generated by this highest weight vector is a submodule of W. Since W is supposed to be irreducible, these modules coincide.

Claim 15.4. The set of all highest weight vectors of a given weight in Super_{n}[L|P] is a Schur irreducible pl(P)-module, namely, a vector subspace of the form

⟨(D_{λ} | U), U ∈ Stab(P)⟩_{𝕂} = ⟨(D_{λ} | U ), U ∈ Stab(P)⟩_{𝕂},

with λ ⊢ n, λ ∈ H(L) ∩ H(P). In particular, this means that each sl_{m}(ℂ)-irreducible submodule has a unique (up to a scalar factor) highest weight vector, but it can be expressed in two ways, namely, by using the basis {(D_{λ} | U), U ∈ Stab(P)} and by

using the basis {(D_{λ} | U ), U ∈ Stab(P)}.

15.6 Highest weight vectors and complete decompositions

By combining 1) and 3) of Theorem 9.1 (Clebsch–Gordan–Capelli bases) and Corollary 15.3, one immediately infers the following result.

Proposition 15.7. The following sets are maximal sets of linearly independent highest weight vectors in Super_{n}[L|P]:

1. {(D_{λ} | T); λ ⊢ n, T ∈ Stab_{λ}(P), λ ∈ H(L) ∩ H(P)};

2. {(D_{λ} | T ); λ ⊢ n, T ∈ Stab_{λ}(P), λ ∈ H(L) ∩ H(P)}.

The preceding facts are to be read as follows:

1. Given a highest weight vector in Super_{n}[L|P] of the form

(D_{λ} | T), T ∈ Stab_{λ}(P), λ ∈ H(L) ∩ H(P),

successive applications of the polarization operators D_{x_{i}x_{j}}, i > j, generate all the standard left symmetrized bitableaux

( S | T ), S ∈ Stab_{λ}(L), T ∈ Stab_{λ}(P), λ ∈ H(L) ∩ H(P);

2. Given a highest weight vector in Super_{n}[L|P] of the form

(D_{λ} | T ), T ∈ Stab_{λ}(P), λ ∈ H(L) ∩ H(P),

successive applications of the polarization operators D_{x_{i}x_{j}}, i > j, generate all the standard doubly symmetrized bitableaux

( S | T ), S ∈ Stab_{λ}(L), T ∈ Stab_{λ}(P), λ ∈ H(L) ∩ H(P);

Therefore:

1. For every T ∈ Stab_{λ}(P), λ ∈ H(L) ∩ H(P), the irreducible sl_{m}(ℂ)-submodule generated by the highest weight vector (D_{λ} | T) is the Weyl module W_{λT} parametrized by the place-tableau T.

2. For every T ∈ Stab_{λ}(P), λ ∈ H(L) ∩ H(P), the irreducible sl_{m}(ℂ)-submodule generated by the highest weight vector (D_{λ} | T ) is the Schur module S_{λT} parametrized by the place-tableau T (recall that the set {( S | T ), S ∈ Stab_{λ}(L), T ∈ Stab_{λ}(P), λ ∈ H(L) ∩ H(P)} is a basis of S_{λT}).

Finally, we have the following results (compare with Theorem 12.1 and Proposition 15.7).

Proposition 15.8. We have:

1. The complete decomposition of the semisimple sl_{m}(ℂ)-module Super_{n}[L|P]:

Super_{n}[L|P] = ⨁_{λ} ( ⨁_{T ∈Stab_{λ}(P)} W_{λT} )

is the complete decomposition corresponding to the maximal set of linearly independent highest weight vectors in Super_{n}[L|P]:

{(D_{λ} | T); T ∈ Stab_{λ}(P), λ ∈ H(L) ∩ H(P)};

2. The complete decomposition of the semisimple sl_{m}(ℂ)-module Super_{n}[L|P]:

Super_{n}[L|P] = ⨁_{λ} ( ⨁_{T ∈Stab_{λ}(P)} S_{λT} )

is the complete decomposition corresponding to the maximal set of linearly independent highest weight vectors in Super_{n}[L|P]:

{(D_{λ} | T ); T ∈ Stab_{λ}(P), λ ∈ H(L) ∩ H(P)}.

15.7 Letterplace algebras and complete sets of pairwise nonisomorphic irreducible sl_{m}(ℂ)-representations

We know from Subsection 15.1 that every irreducible representation of sl_{m}(ℂ) is uniquely determined by its highest weight vector, or, equivalently, by its highest weight.

From the “abstract” representations theory of sl_{m}(ℂ), we get the following result (see, e.g., [40], [42], [88]):

Proposition 15.9. Given an irreducible sl_{m}(ℂ)-module, its highest weight is of the form

a_{1}ε_{1} + a_{2}ε_{2} + ⋯ + a_{m}ε_{m},

with a_{i} ∈ ℤ and a_{1} ≥ a_{2} ≥ ⋯ ≥ a_{m}.

Since a weight in h^{∗} is defined up to a sum with a scalar multiple of the linear functional ε_{1} + ε_{2} + ⋯ + ε_{m}, the preceding proposition implies the following assertion.

Corollary 15.5. Given an irreducible sl_{m}(ℂ)-module, its highest weight is of the form

a_{1}ε_{1} + a_{2}ε_{2} + ⋯ + a_{m−1}ε_{m−1},

with a_{i} ∈ ℕ and a_{1} ≥ a_{2} ≥ ⋯ ≥ a_{m−1}.

Hence, any irreducible sl_{m}(ℂ)-representation can be realized as Weyl and Schur submodules of a suitable letterplace algebra.

More precisely, we have the following result.

Proposition 15.10. Let L = L_{1} = {x_{1}, …, x_{m}} be a negatively signed alphabet of proper letters, and let P = P_{1} = {y_{1}, …, y_{m}} be a negatively signed alphabet of proper places. Let λ = (λ_{1} ≥ λ_{2} ≥ ⋯ ≥ λ_{p}) be a partition such that λ_{1} ≤ m, and let λ̃ = (λ̃_{1} ≥ λ̃_{2} ≥ ⋯ ≥ λ̃_{q}) be the conjugate partition of λ.

Consider the (irreducible) Weyl submodule W_{λT} and the (irreducible) Schur submodule S_{λT} of Super[L|P], T ∈ Stab_{λ}(P).

The highest weight of both W_{λT} and S_{λT} is the element

(λ̃_{1} − λ̃_{m})ε_{1} + (λ̃_{2} − λ̃_{m})ε_{2} + ⋯ + (λ̃_{m} − λ̃_{m})ε_{m} ∈ h^{∗},

where λ̃_{s} is set to be zero whenever s > q.

16 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, …, r}.

Let A = \{a_{1}, …, a_{m}\}, X = \{x_{1}, …, x_{n}\}, and P = \underline{n} = \{1, 2, …, n\} be negatively signed alphabets. Consider the (positively signed) letterplace alphabets

[A | \underline{n}] = \{(a_{i} | μ); i = 1, …, m, μ = 1, …, n\},
[\underline{n} | X] = \{(μ | x_{j}); μ, j = 1, …, n\},

and the tensor product of commutative letterplace superalgebras

Super[A| \underline{n}] ⊗ Super[\underline{n}| X].

In the following we write 𝕂[A] for Super[A| \underline{n}], and 𝕂[X] for Super[\underline{n}| X], respectively, and 𝕂[A] ⊗ 𝕂[X] for the tensor product Super[A| \underline{n}] ⊗ Super[\underline{n}| X].

16.1 Left and right actions of GL_{n}(𝕂) on 𝕂[A] and 𝕂[X]

Let S = (s_{μν}) be a matrix in GL_{n}(𝕂). We define a left action GL_{n}(𝕂)∘𝕂[A] by setting, for every matrix S = (s_{μν}) ∈ GL_{n}(𝕂):

S ∘ (a_{i} | ν) = ∑_{μ=1}^{n} (a_{i} | μ) s_{μν}, i = 1, …, m, ν = 1, …, n,

and extending as an algebra endomorphism. We define a right action 𝕂[X] ∘ GL_{n}(𝕂) by setting:

(μ | x_{j}) ∘ S = ∑_{ν=1}^{n} s_{μν} (ν | x_{j}), j, μ = 1, …, n,

and extending as an algebra endomorphism. Clearly, the actions of GL_{n}(𝕂) on 𝕂[A] and 𝕂[X], regarded as actions on the tensor product 𝕂[A] ⊗ 𝕂[X], commute and, therefore, they may be combined to make 𝕂[A] ⊗ 𝕂[X] into a GL_{n}(𝕂) − GL_{n}(𝕂) - bimodule by the rule:

S ∘ (α ⊗ ξ) ∘ S′ = (S ∘ α) ⊗ (ξ ∘ S′),

for all S, S′ ∈ GL_{n}(𝕂), α ∈ 𝕂[A], ξ ∈ 𝕂[X].

16.2 Invariants and covariants

Given an integer ω, an element γ ∈ 𝕂[A] is called an invariant of weight ω if

S ∘ γ = |S|^{ω} γ, for all S ∈ GL_{n}(𝕂),

|S| the determinant of S. An element φ ∈ 𝕂[A] ⊗ 𝕂[X] is called a covariant of weight ω if

S ∘ φ ∘ S^{−1} = |S|^{ω} φ, for all S ∈ GL_{n}(𝕂).

16.3 Isobaric elements and semiinvariants. Weights

An element γ ∈ 𝕂[A] is said to be isobaric of weight π = (π_{1}, …, π_{n}) ∈ ℕ^{n} if the identity

S ∘ γ = d_{1}^{π_{1}} ⋯ d_{n}^{π_{n}} · γ

holds for all diagonal matrices S = Diag(d_{1}, …, d_{n}), d_{1}, …, d_{n} ∈ 𝕂^{∗}.

The Borel subgroup B_{n} of GL_{n}(𝕂) is the subgroup which consists of all “upper triangular” (non-singular) n × n matrices.

An element γ ∈ 𝕂[A] is called a semiinvariant of weight π if

S ∘ γ = d_{1}^{π_{1}} ⋯ d_{n}^{π_{n}} · γ,

for all S ∈ B_{n}, (d_{1}, …, d_{n}) ∈ (𝕂^{∗})^{n} the n-tuple of diagonal entries of S. Remark 16.1. Clearly, semiinvariants are isobaric elements. In particular, a semiinvariant of weight π is a homogeneous polynomial of degree d = π_{1} + π_{2} + ⋯ + π_{n} of 𝕂[A].

Note that the remarks in Subsection 15.4 apply to the study of semiinvariants, since they are homogenous elements of 𝕂[A]; the next result is just a reformulation of Theorem 15.1 and Corollary 15.3.

We recall that, given a shape λ = (λ_{1}, …, λ_{p}) ⊢ d, λ_{1} ≤ n, we denote by D_{λ} the Deruyts tableau of shape λ filled, in order, with the first λ_{1} symbols of the alphabet.

Theorem 16.1. Let γ be a homogeneous element of degree d of 𝕂[A].

The following statements are equivalent.

1. γ is a semiinvariant of weight π = (π_{1}, π_{2}, …, π_{n}).

2. The expansion of γ as a linear combination of standard bitableaux is of the form

γ = ∑_{sh(V)=λ} c_{V} (V | D_{λ}), λ ⊢ d, λ_{1} ≤ m,

with π = λ̃.

3. The expansion of γ as a linear combination of standard left symmetrized bitableaux is of the form

γ = ∑_{sh(V)=λ} d_{V} ( V | D_{λ}), λ ⊢ d, λ_{1} ≤ m,

with π = λ̃.

Claim 16.1. The preceding result implies that the weight π of a semiinvariant is an n-tuple of natural integers π = (π_{1}, …, π_{n}), with π_{1} ≥ π_{2} ≥ ⋯ ≥ π_{n}.

16.4 The map σ : 𝕂[A] → 𝕂[A] ⊗ 𝕂[X]

Given a_{i} ∈ A and x_{j} ∈ X, we set:

⟨a_{i} | x_{j}⟩ = ∑_{μ=1}^{n}(a_{i} | μ) ⊗ (μ | x_{j}), i = 1, …, m j = 1, …, n.

We explicitly note that the elements ⟨a_{i} | x_{j}⟩ are covariants of weight zero in 𝕂[A] ⊗ 𝕂[X]. Define a map

σ : 𝕂[A] → 𝕂[A] ⊗ 𝕂[X]

by setting

σ : (a_{i} | j) → ⟨a_{i} | x_{j}⟩, i = 1, …, m, j = 1, …, n

and extending as an algebra morphism. Remark 16.2. Since 𝕂 is an infinite field, we regard an element φ ∈ 𝕂[A] ⊗ 𝕂[X] as a polynomial function in the n^{2} variables (μ | x_{j}) from the space M_{n}(𝕂) to 𝕂[A] (evaluate any “variable” (μ | x_{j}) on the entry h_{μj} of a matrix H ∈ M_{n}(𝕂), μ, j = 1, …, n). The following facts follow from the definitions.

1. Let γ ∈ 𝕂[A]. Then (σγ)(S) = S ∘ γ, for all S ∈ GL_{n}(𝕂).

2. Let φ ∈ 𝕂[A] ⊗ 𝕂[X]. Then

(φ ∘ S)(H) = φ(SH) ∈ 𝕂[A],

for all S ∈ GL_{n}(𝕂), H ∈ M_{n}(𝕂). Theorem 16.2. The map σ is a 𝕂-algebra monomorphism. Furthermore, Im[σ] is the set of all φ ∈ 𝕂[A] ⊗ 𝕂[X] which satisfy

S ∘ φ = φ ∘ S, for all S ∈ GL_{n}(𝕂).

In other words, Im[σ] is the set of all covariants of weight zero in 𝕂[A] ⊗ 𝕂[X].

Proof. Notice that, since σ is an algebra morphism, it maps any element γ ∈ 𝕂[A] to a covariant of weight zero. Since

σ(a_{i} | j)(S) = ⟨a_{i} | x_{j}⟩(S) = S ∘ (a_{i} | j), for all S ∈ GL_{n}(𝕂),

it follows that

(σγ)(S) = S ∘ γ, for all γ ∈ 𝕂[A], for all S ∈ GL_{n}(𝕂).

In particular, the “evaluation of the covariant (σγ)(I) at the identity matrix I ∈ GL_{n}(𝕂)” equals γ, in symbols:

(σγ)(I) = γ, for all γ ∈ 𝕂[A],

Hence, the map σ is injective.

Let φ be a covariant of weight zero. By the previous remark, we know that

(S ∘ φ)(H) = (φ ∘ S)(H) = φ(SH),

for all S ∈ GL_{n}(𝕂), H ∈ M_{n}(𝕂).

By setting H = I, we have

φ(S) = (S ∘ φ)(I) = S ∘ φ(I), for all S ∈ GL_{n}(𝕂),

with φ(I) ∈ 𝕂[A]. Since

φ(S) = S ∘ φ(I) = [σ(φ(I))](S), for all S ∈ GL_{n}(𝕂),

and GL_{n}(𝕂) is a Zariski open set in M_{n}(𝕂), this implies the identity

φ = σ(φ(I)) in 𝕂[A] ⊗ 𝕂[X],

and, therefore, the map σ is surjective on the subalgebra of covariants of weight zero.

If φ is a covariant of weight zero, then, by Theorem 16.2, there exists a unique element γ ∈ 𝕂[A] such that σ(γ) = φ. The element γ is called the source of the covariant φ, and γ = φ(I).

16.5 Left and right spans of covariants of weight zero

Consider an element φ ∈ 𝕂[A] ⊗ 𝕂[X], and write

φ = α_{1} ⊗ β_{1} + ⋯ + α_{r} ⊗ β_{r}, α_{s} ∈ 𝕂[A], β_{s} ∈ 𝕂[X];

let r(φ) be the minimal length of all such expansions for φ. If r = r(φ), we say that the above expansion is minimal. As a matter of fact an expansion is minimal if and only if {α_{1}, …, α_{r}} and {β_{1}, …, β_{r}} are linearly independent sets. The vector spaces

L(φ) = ⟨α_{1}, …, α_{r}⟩_{𝕂}

and

R(φ) = ⟨β_{1}, …, β_{r}⟩_{𝕂}

are called the left span and the right span of φ, respectively.

The following result follows directly from the definitions.

Proposition 16.1. Let φ ∈ 𝕂[A] ⊗ 𝕂[X] be a covariant of weight zero. Then L(φ) and R(φ) are left and right 𝕂[GL_{n}(𝕂)]- submodules of 𝕂[A] and 𝕂[X], respectively.

The following result easily follows from a standard argument (see, e.g., Green [45, p. 261]).

Proposition 16.2. If φ = σ(γ), γ ∈ 𝕂[A], then

L(φ) = 𝕂[GL_{n}(𝕂)] ∘ γ,

the cyclic 𝕂[GL_{n}(𝕂)]- submodule of 𝕂[A] generated by γ.

16.6 Primary covariants and irreducible GL_{n}(𝕂)-representa-tions

A non-zero function φ ∈ 𝕂[A] ⊗ 𝕂[X] is called a primary covariant if φ is a covariant of weight zero and φ = σ(γ), where the source γ is a semiinvariant.

Theorem 16.3. If φ ∈ 𝕂[A]⊗𝕂[X] is a primary covariant, then L(φ) is an irreducible left 𝕂[GL_{n}(𝕂)]- submodule of 𝕂[A].

Proof. By the preceding proposition,

L(φ) = 𝕂[GL_{n}(𝕂)] ∘ γ,

and the source γ is a semiinvariant of a given weight π. Therefore, by Theorem 16.1, γ is of the form

γ = ∑_{sh(V)=λ} c_{V} (V | D_{λ}),

where λ is the conjugate shape of π.

By Corollary 15.4, the cyclic module 𝕂[GL_{n}(𝕂)] ∘ γ is the irreducible module spanned, as a vector space, by the basis

{ ∑_{sh(V)=λ} c_{V} (V | T ); T standard, sh(T) = λ}.

Given a weight π = (π_{1} ≥ π_{2} ≥ ⋯ ≥ π_{n}), π_{h} ∈ ℕ, consider the semiinvariant

a(π) = (D_{λ} | D_{λ}) ∈ 𝕂[A],

where the left-most D_{λ} denotes the Deruyts tableau of shape λ filled, in order, with the first letters of A, the right-most D_{λ} denotes the Deruyts tableau of shape λ filled, in order, with the first places of \underline{n}, and λ is the conjugate shape of π.

Example 16.1. Let π = (3, 3, 1). Then

a(π) = (D_{(3,2,2)} | D_{(3,2,2)}) = ( \begin{matrix}a_{1}a_{2}a_{3} & 123 \\ a_{1}a_{2} & 12 \\ a_{1}a_{2} & 12 \end{matrix} ) .

Proposition 16.3. We have:

• Given any semiinvariant γ = ∑_{sh(V)=λ} c_{V} (V | D_{λ}) ∈ 𝕂[A] of weight π = λ̃, we have the following identity:

γ = ( ∑_{sh(V)=λ} c_{V} Y (V | D_{λ} )) (a(π)),

where the A-letter polarization operator ∑_{sh(V)=λ} c_{V} Y (V | D_{λ} ) is a 𝕂[GL_{n}(𝕂)]- equivariant operator.

• Every non-zero semiinvariant γ ∈ 𝕂[A] of weight π generates an irreducible left 𝕂[GL_{n}(𝕂)]- submodule 𝕂[GL_{n}(𝕂)] ∘ γ which is isomorphic to 𝕂[GL_{n}(𝕂)] ∘ a(π).

The proof follows immediately from Theorem 16.1 and Theorem 16.3.

16.7 The Deruyts–Capelli expansion

Theorem 16.4. Let γ be an element of 𝕂[A], homogeneous of degree d ∈ ℤ^{+}. Consider the canonical expansion

γ = ∑_{\begin{matrix}λ ⊢ d \\ λ_{1} ≤ min(m, n) \end{matrix}} ∑_{sh(T)=λ} ∑_{sh(S)=λ} c_{ST}^{γ} (S | T )

with respect to the Clebsch–Gordan–Capelli basis of standard right symmetrized bitableaux, and set

γ_{λ,T} = ∑_{sh(S)=λ} c_{ST}^{γ} (S | T ),

γ_{λ,T}^{D_{λ}} = ∑_{sh(S)=λ} c_{ST}^{γ} (S | D_{λ}),

for every λ ⊢ d and T ∈ Stab_{λ}(P).

Then:

1. γ_{λ,T} belongs to the cyclic 𝕂[GL_{n}(𝕂)]-module generated by γ ∈ 𝕂[A]; 2. γ_{λ,T} belongs to the irreducible cyclic 𝕂[GL_{n}(𝕂)]-module generated by the semiin-

variant γ_{λ,T}^{D_{λ}};

3. the semiinvariant γ_{λ,T}^{D_{λ}} belongs to the cyclic 𝕂[GL_{n}(𝕂)]-module generated by γ_{λ,T};

4. the cyclic modules 𝕂[GL_{n}(𝕂)] ∘ γ_{λ,T} and 𝕂[GL_{n}(𝕂)] ∘ γ_{λ,T}^{D_{λ}} are the same.

Proof. By Proposition 12.1, we have:

1. γ_{λ,T} is the image of γ under the operator (T, T )Y induced by the (place) action of 𝕂[GL_{n}(𝕂)].

2. γ_{λ,T} is the image of γ_{λ,T}^{D_{λ}} under the operator (D_{λ}, T )Y induced by the (place) action of 𝕂[GL_{n}(𝕂)], up to a non-zero scalar factor.

3. γ_{λ,T}^{D_{λ}} is the image of γ_{λ,T} under the operator (T, D_{λ} )Y induced by the (place) action of 𝕂[GL_{n}(𝕂)], up to a non-zero scalar factor.

The next reformulation of Theorem 16.4 describes the Deruyts–Capelli expansion of a polynomial γ ∈ 𝕂[A]. As a matter of fact, the first statement is properly the Deruyts Expansion (cfr. Green, [45, Theorem 12.1]), and the second statement is the core of the crucial Capelli’s refinement of the result of Deruyts (Capelli’s polar expansion formula [25]).

Corollary 16.1 (The Deruyts–Capelli Expansion). Let γ be an element of 𝕂[A], γ homogeneous of degree d. Then:

1. for every λ ⊢ d and T ∈ Stab_{λ}(P), there exist semiinvariants

γ_{λ,T}^{D_{λ}} ,

and elements

Λ_{λ,T} ∈ 𝕂[GL_{n}(𝕂)],

such that:

γ = ∑_{λ⊢d} ∑_{sh(T)=λ} Λ_{λ,T} ∘ γ_{λ,T}^{D_{λ}};

2. the semiinvariants

γ_{λ,T}^{D_{λ}}

belong to the cyclic 𝕂[GL_{n}(𝕂)]-module generated by γ ∈ 𝕂[A].

A more explicit formulation and a more direct proof of the preceding result can be found in [21].

16.8 On the complete reducibility of the span of covariants of weight zero. A “pre-Schur” description of polynomial GL_{n}(𝕂)- irreducible representations

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

Since L(φ) = 𝕂[GL_{n}(𝕂)]∘γ, σ(γ) = φ, the problem can be solved in the algebra 𝕂[A].

Let φ ∈ 𝕂[A] ⊗ 𝕂[X] be a covariant of weight zero, and let γ ∈ 𝕂[A] be its source. Without loss of generality, we assume that γ is a “homogeneous polynomial” of degree d.

Let

γ = ∑_{λ⊢d} ∑_{sh(T)=λ} γ_{λ,T} , γ_{λ,T} ≠ 0,
γ_{λ,T} = ∑_{sh(S)=λ} c_{ST}^{γ} · (S | T ),

be the canonical expression of γ as a linear combination of standard right symmetrized bitableaux.

Corollary 16.2 (Decomposition theorem). Let φ ∈ 𝕂[A] ⊗ 𝕂[X] be a covariant of weight zero, and let γ ∈ 𝕂[A] be its source, that is φ = σ(γ). Then:

L(φ) = 𝕂[GL_{n}(𝕂)] ∘ γ

= ⨁_{λ⊢d} ( ∑_{sh(T)=λ} 𝕂[GL_{n}(𝕂)] ∘ γ_{λT}^{D_{λ}})

= ⨁_{λ⊢d} ( ∑_{sh(T)=λ} L(σ(γ_{λT}))),

where L(σ(γ_{λT})) = 𝕂[GL_{n}(𝕂)] ∘ γ_{λT}^{D_{λ}}.

16.9 Weyl’s First Fundamental Theorem

Let V be a vector space of finite dimension m. Since char(𝕂) = 0, we may identify the algebra 𝕂[A] with the algebra 𝕂[V^{⊕n}] of polynomial functions on the vector space V^{⊕n}, by reading the variable (a_{i} | μ) as the i-th coordinate function on the μ-th copy of V in the direct sum V^{⊕n}, for every i = 1, 2, …, m, μ = 1, 2, …, n.

In plain words, we identify the algebra 𝕂[A] with the algebra 𝕂[V^{⊕n}] = Sym[(V^{∗})^{⊕n}].

We recall the contravariant action of the general linear group GL_{m}(𝕂) on

𝕂[A] = 𝕂[V^{⊕n}] = Sym[(V^{∗})^{⊕n}]

is defined in the following way:

(g ∘ γ)(v_{1}, …, v_{n}) = γ(g^{−1}(v_{1}), …, g^{−1}(v_{n})),

for all g ∈ GL_{m}(𝕂), γ ∈ 𝕂[A], (v_{1}, …, v_{n}) ∈ V^{⊕n}.

In matrix notation, the diagonal action of a matrix T = (t_{hk}) ∈ GL_{m}(𝕂) on 𝕂[A] is given by

T ∘ (a_{i} | μ) = ∑_{k=1}^{m} t̄_{ik} (a_{k} | μ), i = 1, …, m, μ = 1, …, n,

where

(t̄_{hk}) = T^{−1}.

The algebra 𝕂[A] is both a left GL_{n}(𝕂)-module (see Subsection 16.1) and a left GL_{m}(𝕂)-module, and the two actions clearly commute.

Furthermore, the action of the general linear group GL_{m}(𝕂) on every homogeneous component of 𝕂[A] is implemented by the algebra of A-letter polarizations, and the action of the general linear group GL_{n}(𝕂) is implemented by the algebra of P-place polarizations, P = \underline{n} = \{1, 2, …, n\}.

Let m ≤ n.

Informally speaking, the Deruyts–Capelli expansion formula says that any homogeneous polynomial function γ in n vector variables in dimension m may be expressed as a linear combination of polarized GL_{n}(𝕂)-semiinvariants and, by Theorem 16.1, these GL_{n}(𝕂)-semiinvariants involve only the first m^{2} letterplace variables

(a_{1} | μ), …, (a_{m} | μ), μ = 1, …, m;

furthermore, these GL_{n}(𝕂)-semiinvariants may be obtained, in turn, by applying place polarization operators to the original polynomial function γ.

Since the place polarization process is a GL_{m}(𝕂)-invariantive process (i.e., the actions of GL_{m}(𝕂) and GL_{n}(𝕂) commute), the study of GL_{m}(𝕂)-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:

• Let G be a subgroup of the general linear group GL_{m}(𝕂).

A polynomial function γ ∈ 𝕂[A] = 𝕂[V^{⊕n}] is said to be a (formal) relative G-invariant if and only if the following condition holds:

(g ∘ γ) = λ(g)γ,

for all g ∈ G, λ(g) ∈ 𝕂.

• The bracket

[j_{1}, …, j_{m}], j_{h} ∈ P = \underline{n} = \{1, 2, …, n\}

is defined as follows:

[j_{1}, …, j_{m}] = det[(a_{i} | j_{h})]_{i=1,...,m, h=1,...,m,} = (−1)^{\binom{m}{2}}(a_{1}a_{2} ⋯ a_{m} | j_{1}j_{2} ⋯ j_{m}).

Clearly, any bracket [j_{1}, …, j_{m}] is a relative G-invariant, G a subgroup of the general linear group GL_{m}(𝕂).

Let γ ∈ 𝕂[A] = 𝕂[V^{⊕n}] be a polynomial function, γ homogeneous of degree d. Thanks to the Deruyts–Capelli expansion formula (Corollary 16.1), γ is expanded in the following form:

γ = ∑_{\begin{matrix}λ ⊢ d \\ λ_{1} ≤ m \end{matrix}} ∑_{sh(T)=λ} Λ_{λ,T} ∘ γ_{λ,T}^{D_{λ}} , Λ_{λ,T} ∈ 𝕂[GL_{n}(𝕂)], T ∈ Stab(P),

where any

γ_{λ,T}^{D_{λ}} = ∑_{sh(S)=λ} c_{ST}^{γ} (S | D_{λ})

belongs to the cyclic 𝕂[GL_{n}(𝕂)]-module generated by γ ∈ 𝕂[A].

Since the actions of GL_{m}(𝕂) and GL_{n}(𝕂) commute, if γ is a relative G-invariant (G a subgroup of GL_{m}(𝕂)), the GL_{n}(𝕂)-semiinvariants γ_{λ,T}^{D_{λ}} are relative G-invariants.

Note that any γ_{λ,T}^{D_{λ}} can be written in the form

γ_{λ,T}^{D_{λ}} = [1, 2, …, m]^{q_{λ}}φ_{T} ,

where

q_{λ} = #{i; λ_{i} = m},

and

φ_{T} = ± ∑_{S} c_{ST}^{γ} (S^{∗} | D_{λ^{∗}}),

where λ^{∗} = (λ_{q_{λ}+1}, λ_{q_{λ}+2}, …) and the tableaux S^{∗} of shape λ^{∗} are obtained from the tableaux S of shape λ by deleting the first q_{λ} top rows.

Since λ_{1}^{∗} < m, any φ_{T} is a polynomial in the m × m − 1 letterplace variables

(a_{1} | 1),  (a_{2} | 1), …, (a_{m} | 1)
(a_{1} | 2),  (a_{2} | 2), …, (a_{m} | 2)
⋮  ⋮  ⋮
(a_{1} | m − 1), (a_{2} | m − 1), …, (a_{m} | m − 1)

or, equivalently, φ_{T} is a polynomial function

φ_{T} (v_{1}, v_{2}, …, v_{m−1}) ∈ 𝕂[V^{⊕(m−1)}]

in the vector variables v_{1}, v_{2}, …, v_{m−1}. Hence, we have the following result.

Corollary 16.3 (Weyl’s First Fundamental Theorem). Let G be a subgroup of GL_{m}(𝕂), and let γ be a d-homogeneous relative G-invariant polynomial function in n vector variables, n ≥ m. Then,

γ = ∑_{\begin{matrix}λ ⊢ d \\ λ_{1} ≤ m \end{matrix}} ∑_{T ∈Stab_{λ}(P)} Λ_{λ,T} ∘ ([1, 2, … m]^{q_{λ}}φ_{T}),

where

Λ_{λ,T} ∈ 𝕂[GL_{n}(𝕂)],

and the φ_{T} ’s are both relative G-invariant polynomial functions in the vector variables v_{1}, v_{2}, …, v_{m−1} and GL_{n}(𝕂)-semiinvariants.

For example, since the GL_{m}(𝕂)-invariant polynomial functions which involve at most m − 1 vector variables are just the constant functions (see, e.g., Theorem 10.2 and Remark 10.1) and polarizations of brackets yield linear combinations of brackets, the following well-known result immediately follows.

Corollary 16.4 (The first fundamental theorem for vector GL_{m}(𝕂)-in-variants). Let γ be a relative GL_{m}(𝕂)-invariant polynomial function on n vector variables, n ≥ m. Then γ can be written as a homogeneous polynomial in the brackets

[j_{1}, …, j_{m}], j_{h} ∈ P = \underline{n} = \{1, 2, …, n\}.

16.10 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., [40], [92]; in this subsection, by way of application of the method of virtual variables, we provide an elementary and mechanical proof of them. Let A = \{a_{1}, a_{2}, …, a_{m}\} and P = \underline{n} = \{1, 2, …, n\} be negatively signed alphabets and let 𝕂[A] = Super[A| \underline{n}] be the commutative letterplace algebra generated by the positively signed letterplace alphabet [A | \underline{n}]. Let H_{m} be the Capelli operator in m variables:

H_{m} = det [ \begin{matrix}D_{a_{1}a_{1}} + (m − 1)I & D_{a_{1}a_{2}} & … & D_{a_{1}a_{m}} \\ D_{a_{2}a_{1}} & D_{a_{2}a_{2}} + (m − 2)I & … & D_{a_{2}a_{m}} \\ ⋮ & ⋮ & ... & ⋮ \\ D_{a_{m}a_{1}} & D_{a_{m}a_{2}} & … & D_{a_{m}a_{m}} \end{matrix} ] ,

where the expansion of the determinant is by column from left to right. Let Ω_{n} be the Cayley operator in dimension n:

Ω_{n} = det [ \begin{matrix}\frac{∂}{∂(a_{1} | 1)} & \frac{∂}{∂(a_{1} | 2)} & … & \frac{∂}{∂(a_{1} | n)} \\ \frac{∂}{∂(a_{2} | 1)} & \frac{∂}{∂(a_{2} | 2)} & … & \frac{∂}{∂(a_{2} | n)} \\ ⋮ & ⋮ & ⋮ & ⋮ \\ \frac{∂}{∂(a_{n} | 1)} & \frac{∂}{∂(a_{n} | 2)} & … & \frac{∂}{∂(a_{n} | n)} \end{matrix} ] ,

where the symbol \frac{∂}{∂(a_{i} | μ)} denotes the (formal) partial derivative with respect to the variable (a_{i} | μ). The following result is due to Capelli (see, e.g., [25], [92]). Theorem 16.5. Let γ ∈ 𝕂[A]. We have the following identities:

H_{m}(γ) = \begin{cases}0 & if m > n, \\ [a_{1}, a_{2}, …, a_{n}] Ω_{n}(γ) & if m = n, \end{cases}

where the bracket [a_{1}, a_{2}, …, a_{n}] denotes the determinant

det[(a_{i} | μ)]_{i, μ=1,2,...,m=n}.

Proof. We regard the algebra 𝕂[A] = Super[A| \underline{n}] as a subalgebra of the letterplace superalgebra Super[A ∪ \{α\}| \underline{n}], α a (virtual) positive letter. From Subsection 6.6, we recall that the action on 𝕂[A] of the Capelli operator H_{m} is the same as the action of the operator

D_{a_{1},α}D_{a_{2},α} … D_{a_{m},α}D_{α,a_{m}}D_{α,a_{m−1}} … D_{α,a_{1}}.

Consider a momomial m = ∏_{i,μ}(a_{i} | μ)^{d_{iμ}}, with d_{iμ} ∈ ℕ, in 𝕂[A]. Clearly, all the monomials in the polynomial

D_{α,a_{m}}D_{α,a_{m−1}} … D_{α,a_{1}}(m) ∈ Super[A ∪ \{α\}| \underline{n}]  (∗)

contain exactly m occurrences of the positive letter α. Note that the letterplace variables (α | μ) are of ℤ_{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 (α | μ); thus,

D_{α,a_{m}}D_{α,a_{m−1}} … D_{α,a_{1}}(m) = 0,

and the first assertion is proved, by linearity.

Let us now consider the case m = n. Set (a_{i} | μ)^{d_{iμ}−1} = 0 whenever d_{iμ} = 0, and recall that (α | μ)^{2} = 0 for μ = 1, 2, …, n. We have the identity

D_{α,a_{n}}D_{α,a_{n−1}} … D_{α,a_{1}}(m)

= ∑_{σ} ( ∏_{i,μ} d_{i,σ(i)}(a_{i} | μ)^{d_{i,μ}−δ_{i,σ(i)}}(α | σ(n))(α | σ(n − 1)) … (α | σ(1)),

where the summation is over all the permutations σ of the set {1, 2, …, n} and δ is the Kronecker symbol.

The last expression can be rewritten in the form

∑_{σ} [\frac{∂}{∂(a_{1} | σ(1))}a( ∏_{μ}(a_{1} | μ)^{d_{1μ}}) ⋯ \frac{∂}{∂(a_{n} | σ(n))}( ∏_{μ}(a_{n} | μ)^{d_{nμ}})]

(α | σ(n)) … (α | σ(1)).

Since the variables (α | σ(μ)) anticommute, we get the identities

D_{α,a_{n}}D_{α,a_{n−1}} … D_{α,a_{1}}(m)

= (α | n)(α | n − 1) … (α | 1) ( ∑_{σ}(−1)^{|σ|} \frac{∂}{∂(a_{1} | σ(1))} ⋯ \frac{∂}{∂(a_{n} | σ(n))}) (m)

= (α | n)(α | n − 1) … (α | 1) Ω_{n}(m).

Since

D_{a_{1},α}D_{a_{2},α} ⋯ D_{a_{n},α}(α | n)(α | n − 1) … (α | 1) = [a_{1}, a_{2}, …, a_{n}],

the second assertion is proved, by linearity.

17 ℤ_{2}-Graded Tensor Representations: the Berele– Regev Theory

From now on let L = {1, 2, …, n} = L_{0} be an alphabet of n positive letters, and P = P_{0} ∪ P_{1} a finite alphabet of places, with |P_{0}| = r, |P_{1}| = s.

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

We consider the following structures:

• Super_{n}[\underline{L}|P], the subspace of Super_{n}[L|P] freely generated by the letter-mul-tilinear monomials or, equivalently, by the standard symmetrized bitableaux (S | T ), where S is a multilinear tableau on L; • \underline{B}_{n}, the operator algebra linearly generated by the orthonormal letter generators

Y_{n}(S′, S ), with S and S′ multilinear tableaux over L, restricted to the subspace Super_{n}[\underline{L}|P].

• _{n}B′, the operator algebra generated by the proper place polarizations, or, equivalently, by the place orthonormal generators (T, T′ )_{n}Y, T, T′ ∈ Stab_{λ}(P), restricted to the subspace Super_{n}[\underline{L}|P].

Note that Super_{n}[\underline{L}|P] is invariant under the action of \underline{B}_{n},_{n}B′, so we have the bimodule

\underline{B}_{n} · Super_{n}[\underline{L}|P] ·_{n} B′.

By specialization of the general theory, we have that: the space Super_{n}[\underline{L}|P] is a semisimple \underline{B}_{n}-module, and it is a semisimple_{n}B′-module; the operator algebras \underline{B}_{n} and nB′ are semisimple, and one is the centralizer of the other in the endomorphism algebra of Super_{n}[\underline{L}|P]. The basic elements and operators, that is, the orthonormal letter generators Y_{n}(S′, S ), the symmetrized bitableaux (S | T ), and the place orthonormal generators (T, T′ )_{n}Y, are parametrized by tableaux such that

S′, S ∈ Stab_{λ}(L), S′, S multilinear, T, T′ ∈ Stab_{λ}(P),

where the partitions λ ⊢ n satisfy the hook condition

λ_{r+1} < s + 1.

17.1 The letter multilinear subspace as a 𝕂[S_{n}]-module

Note that a group action of the symmetric group S_{n} on Super_{n}[\underline{L}|P] is consistently defined by setting

σ · (i_{1} | y_{j_{1}})(i_{2} | y_{j_{2}}) … (i_{n} | y_{j_{n}}) = (σ(i_{1}) | y_{j_{1}})(σ(i_{2}) | y_{j_{2}}) … (σ(i_{n}) | y_{j_{n}}).

This action defines a representation

ρ : 𝕂[S_{n}] → End_{𝕂}[Super_{n}[\underline{L}|P]].

Notice that the representation of any permutation σ can be regarded as a multilinear letter Capelli column:

[ σ(1) ; ⋮ ; σ(n) | 1 ; ⋮ ; n ] = D_{σ(1)α_{1}} ⋯ D_{σ(n)α_{n}}D_{α_{n}n} ⋯ D_{α_{1}1} = ρ(σ).

Indeed, we have the following result.

Proposition 17.1. The operator algebra induced by the action of the symmetric group coincides with the operator algebra generated by the multilinear orthonormal letter generators:

\underline{B}_{n} = ρ(𝕂[S_{n}]).

Proof. [20] To begin with, we show that each operator associated to a permutation belongs to the algebra \underline{B}_{n}. Indeed, for any permutation σ ∈ S_{n}, and for any multilinear standard symmetrized bitableau (T | U ), we have

σ · (T | U ) = (σ · T | U )

= ∑_{S′} c_{T S′}^{σ}(S′ | U )

= ( ∑_{S′,S} c_{SS′}^{σ} Y_{n}(S′, S))(T | U ),

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 ρ(𝕂[S_{n}]). Indeed, by Proposition 10.1 (Factorization Theorem), the action of γ_{n}(S′, S) on Super_{n}[\underline{L}|P] is the same, up to a scalar factor, as the action of an operator of the form

D_{S′C}D_{CT} D_{T D}D_{DS},

where T denotes any multilinear standard tableau of the same shape as S′ and S, filled with positive virtual letters α_{1}, …, α_{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

∑_{σ} D_{σ(1)α_{1}}D_{σ(2)α_{2}} … D_{σ(n)α_{n}} ∑_{τ}(−1)^{τ}D_{α_{1}τ(1)}D_{α_{2}τ(2)} … D_{α_{n}τ(n)}

= ∑_{σ,τ} ±D_{σ(1)α_{1}}D_{σ(2)α_{2}} … D_{σ(n)α_{n}}D_{α_{1}τ(1)}D_{α_{2}τ(2)} … D_{α_{n}τ(n)}.

where σ, τ range over suitable sets of permutations. Finally, notice that each summand acts on Super_{n}[\underline{L}|P] in the same way as a permutation.

Remark 17.1. 1. The action of γ_{n}(S′, S ) is, up to sign, the same as the action of

π_{S′S} ∑_{\begin{matrix}τ ∈ R(S) \\ ξ ∈ C(S) \end{matrix}} (−1)^{|ξ|}τξ,

where R(S) and C(S) are the row-stabilizer and the column-stabilizer of the tableau S, respectively, and π_{S′S} ∈ S_{n} is the permutation such that π_{S′S}(S) = S′.

The element

e_{S′S}^{∗} = π_{S′S} ∑_{\begin{matrix}τ ∈ R(S) \\ ξ ∈ C(S) \end{matrix}} (−1)^{|ξ|}τξ ∈ 𝕂[S_{n}]

is called a generalized (dual) Young symmetrizer. Clearly, e_{S′S}^{∗} = π_{S′S}e_{SS}^{∗}, for any pair of multilinear tableaux (S′, S).

2. Let S′ ∈ T ab(L) be any multilinear tableau. Then,we have the following result:

(S′ | T ) = ±γ_{n}(S′, S )⟨S | T ⟩,

where ⟨S | T ⟩ is the tableau monomial in Super_{n}[\underline{L}|P] associated to the pair (S, T) and S ∈ T ab(L) is any multilinear tableau such that sh(S) = sh(T). As a matter of fact, (in short notation) we have

(S′ | T ) = S′C CD \underline{DT} = ±S′C CD DS \underline{ST} = ±γ_{n}(S′, S )⟨S | T ⟩.

17.2 The ℤ_{2}-graded tensor representation theory

Let

W = ⟨P⟩_{𝕂} = ⟨P_{0}⟩_{𝕂} ⊕ ⟨P_{1}⟩_{𝕂} = W_{0} ⊕ W_{1},

then

Super_{n}[\underline{L}|P] ≅ T^{n}[W_{0} ⊕ W_{1}]

via the linear isomorphism

F : (1 | y_{i_{1}}) … (n | y_{i_{n}}) ↦ y_{i_{1}} ⊗ … ⊗ y_{i_{n}}.

Therefore, any construction and any result pertaining to the multilinear letter subspace Super_{n}[\underline{L}|P] may be carried over to the space T^{n}[W_{0} ⊕ W_{1}] of the n-tensors over the ℤ_{2}-graded vector space W = W_{0} ⊕ W_{1}.

The natural action of the symmetric group S_{n} on Super_{n}[\underline{L}|P], gives rise to a representation

𝕂[S_{n}] → End_{𝕂}[T^{n}[W_{0} ⊕ W_{1}]],

which is called a Berele–Regev ℤ_{2}-graded representation of S_{n}.

Example 17.1. Let y_{1}, y_{2}, y_{3} ∈ P, |y_{1}| = 0, |y_{2}| = |y_{3}| = 1. Let n = 3. We have:

(12) · y_{1} ⊗ y_{2} ⊗ y_{3} = F ((12) · (1 | y_{1})(2 | y_{2})(3 | y_{3}))
= F ((2 | y_{1})(1 | y_{2})(3 | y_{3}))
= F ((1 | y_{2})(2 | y_{1})(3 | y_{3})) = y_{2} ⊗ y_{1} ⊗ y_{3}.
(23) · y_{1} ⊗ y_{2} ⊗ y_{3} = F ((23) · (1 | y_{1})(2 | y_{2})(3 | y_{3}))
= F ((1 | y_{1})(3 | y_{2})(2 | y_{3}))
= F (−(1 | y_{1})(2 | y_{3})(3 | y_{2})) = −y_{1} ⊗ y_{3} ⊗ y_{2}.

On the other hand, the action of pl(P) on Super_{n}[\underline{L}|P] gives rise to a representation

End_{𝕂}[T^{n}[W_{0} ⊕ W_{1}]] ← pl(W)

of the general linear Lie superalgebra pl(W). Therefore, by specializing the results of Section 12 to the bimodule

𝕂[S_{n}] · T^{n}[W_{0} ⊕ W_{1}] · pl(W_{0} ⊕ W_{1}),

we get the following results.

Theorem 17.1.

1. The subalgebras of End_{𝕂}[T^{n}[W_{0} ⊕ W_{1}] induced by the actions of

𝕂[S_{n}] and pl(W_{0} ⊕ W_{1})

are the centralizers of each other.

2. A complete decomposition of T^{n}[W_{0} ⊕ W_{1}] with respect to the action of the symmetric group S_{n} is given by:

T^{n}[W_{0} ⊕ W_{1}] = F [Super_{n}[\underline{L}|P]]

= ⨁_{{λ∈H(P) \atop λ⊢n}} ⨁_{{T ∈Stab(P) \atop sh(T)=λ}} F [⟨(S | T ), S ∈ Stab(L), S multilinear⟩];

3. A complete decomposition of T^{n}[W_{0}⊕W_{1}] with respect to the action of the general linear Lie superalgebra pl(W_{0} ⊕ W_{1}) is given by:

T^{n}[W_{0} ⊕ W_{1}] = F [Super_{n}[\underline{L}|P]]

= ⨁_{{λ∈H(P) \atop λ⊢n}} ⨁_{S∈Stab(L), S multilinear sh(S)=λ} F [⟨(S | T ), T ∈ Stab(P)⟩] .

The irreducible 𝕂[S_{n}]-submodules and pl(W_{0} ⊕ 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} ⊕ W_{1}].

First of all, we notice that any decomposable tensor y_{i_{1}} ⊗ … ⊗ y_{i_{n}} ∈ T^{n}[W_{0} ⊕ W_{1}] is (up to a sign) the image under F of a bitableau monomial ⟨S | T ⟩, where S ∈ T ab(L) is a multilinear tableau, T ∈ T ab(P), sh(S) = sh(T) ⊢ n. Clearly, this representation is not unique.

For example, let y_{1}, y_{2}, y_{3} ∈ P, |y_{1}| = 0, |y_{2}| = |y_{3}| = 1; then

y_{1} ⊗ y_{2} ⊗ y_{3} = F (⟨ 123 y_{1}y_{2}y_{3} ⟩) = F ((1 | y_{1})(2 | y_{2})(3 | y_{3}))

= F (− ⟨ \begin{matrix}13 & y_{1}y_{3} \\ 2 & y_{2} \end{matrix} ⟩) = F (−(1 | y_{1})(3 | y_{3})(2 | y_{2})) .

Thanks to Remark 17.1, we get the following results.

Remark 17.2. The irreducible 𝕂[S_{n}]-submodule

F [⟨(S | T ), S ∈ Stab_{λ}(L), S multilinear⟩]

equals

{α · F (⟨U | T ⟩) ; α ∈ 𝕂[S_{n}]e_{UU}^{∗}} ⊆ T^{n}[W_{0} ⊕ W_{1}],

for every multilinear tableau U ∈ Stab_{λ}(L) and, thus, it is 𝕂[S_{n}]-isomorphic to the (minimal) left ideal 𝕂[S_{n}]e_{UU}^{∗} of 𝕂[S_{n}].

Example 17.2. Let y_{1}, y_{2}, y_{3} ∈ P, and let |y_{1}| = 0, |y_{2}| = |y_{3}| = 1. Consider the tableaux U, U′ ∈ T ab(L) and T ∈ Stab(P),

U = {12 \atop 34} U′ = {13 \atop 24} T = \begin{matrix}y_{1}y_{3} \\ y_{2}y_{3} \end{matrix} .

Note that

F (⟨ \begin{matrix}12 & y_{1}y_{3} \\ 34 & y_{2}y_{3} \end{matrix} ⟩) = y_{1} ⊗ y_{3} ⊗ y_{2} ⊗ y_{3}, F (⟨ \begin{matrix}13 & y_{1}y_{3} \\ 24 & y_{2}y_{3} \end{matrix} ⟩) = −y_{1} ⊗ y_{2} ⊗ y_{3} ⊗ y_{3}.

The irreducible 𝕂[S_{n}]-submodule

F [⟨(S | T ), S ∈ Stab_{(2,2)}(L), S multilinear⟩] ⊆ T^{4}[W_{0} ⊕ W_{1}]

equals

{α · (y_{1} ⊗ y_{3} ⊗ y_{2} ⊗ y_{3}); α ∈ 𝕂[S_{n}]e_{UU}^{∗}}={α · (−y_{1} ⊗ y_{2} ⊗ y_{3} ⊗ y_{3}); α ∈ 𝕂[S_{n}]e_{U′U′}^{∗}} .

Remark 17.3. The irreducible pl(W_{0} ⊕ W_{1})-submodule

F [⟨(S | T ), T ∈ Stab_{λ}(P)⟩]

equals

e_{SS}^{∗} [T^{n}[W_{0} ⊕ W_{1}]] ,

the image of the tensor space T^{n}[W_{0} ⊕ W_{1}] under the action of the (dual) Young symmetrizer e_{SS}^{∗}.

Remark 17.4. If P is trivially ℤ_{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 S_{n} ([76], [77], [92]).

18 The Symmetric Group

From now on let L = L_{1} = P = P_{1} = {1, 2, …, n}, an alphabet of n negative symbols, and consider the letterplace algebra (in commutative letterplace variables) Super_{n}[L|P].

In the sequel, given a partition λ ⊢ n, we will write S_{λ1}, S_{λ2}, …, S_{λ\underline{f}_{λ}} to mean the list of all multilinear standard tableaux of shape λ over L = P = {1, 2, …, 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_{λi}.

We recall that the “doubly indexed” Young symmetrizers (of shape λ ⊢ n) are the elements of the group algebra 𝕂[S_{n}] defined as follows:

e_{ij} = π_{ij}( ∑_{\begin{matrix}τ ∈ R(S_{j}) \\ ξ ∈ C(S_{j}) \end{matrix}} (−1)^{|τ|}τξ),

where R(S_{j}) and C(S_{j}) are the row-stabilizer and the column-stabilizer of S_{j}, respectively, and π_{ij} ∈ S_{n} is the (unique) permutation such that

π_{ij}(S_{j}) = S_{i}.

In the classical notation, e_{ij} = π_{ij}e_{j}, where e_{j} denotes the Young symmetrizer associated to the tableau S_{j} (see, e.g., [53]).

We consider the module

\underline{\underline{B}}_{n} · Super_{n}[\underline{L}|\underline{P}],

where • Super_{n}[\underline{L}|\underline{P}] is the subspace of Super_{n}[L|P] spanned by all the doubly multilinear

monomials, that is, monomials of the form

(τ(1) | 1)(τ(2) | 2) ⋯ (τ(n) | n), τ ∈ S_{n},

or, equivalently, by the doubly multilinear symmetrized bitableaux (S | T ), S and T multilinear tableaux on L = P. Notice that the set of right symmetrized bitableux

(S_{λi}, S_{λj} ), i, j = 1, 2, …, \underline{f}_{λ}, λ ⊢ n,

is a 𝕂-linear basis of Super_{n}[\underline{L}|\underline{P}].

• \underline{\underline{B}}_{n} is the algebra of operators on Super_{n}[\underline{L}|\underline{P}] linearly generated by the restrictions of the orthonormal letter generators Y_{n}(S′, S ), S and S′ multilinear tableaux on L. Y_{n}(S′, S ), We hardly need to recall that the set of restricted operators

Y_{n}(S_{λi}, S_{λj} ), i, j = 1, …, \underline{f}_{λ} λ ⊢ n

is a 𝕂-linear basis of \underline{\underline{B}}_{n}. Therefore, this algebra is a semisimple subalgebra of End_{𝕂}(Super_{n}[\underline{L}|\underline{P}]), and admits a complete decomposition

\underline{\underline{B}}_{n} = ⨁_{λ⊢n} ⨁_{j=1,2,...,\underline{f}_{λ}} ⟨Y_{n}(S_{λi}, S_{λj} ), i = 1, 2, …, \underline{f}_{λ}⟩_{𝕂} = ⨁_{λ⊢n} \underline{\underline{B}}_{λ},

where each simple component \underline{\underline{B}}_{λ} is isomorphic to M_{\underline{f}_{λ}}, the full 𝕂-algebra of square matrices of order \underline{f}_{λ}.

18.1 The doubly multilinear subspace as a 𝕂[S_{n}]-module

The action of the symmetric group S_{n} on Super_{n}[\underline{L}|\underline{P}] defined by setting

σ · ((τ(1) | 1)((τ(2) | 2) ⋯ (τ(n) | n)) = (στ(1) | 1)(στ(2) | 2) ⋯ (στ(n) | n)

induces a faithful representation

\underline{ρ} : 𝕂[S_{n}] → End_{𝕂}[Super_{n}[\underline{L}|\underline{P}]].

By specializing the argument of Proposition 17.1, we get:

Proposition 18.1. The operator algebra induced by the action of the symmetric group coincides with the operator algebra generated by the multilinear orthonormal letter generators:

\underline{ρ}[𝕂[S_{n}]] = \underline{\underline{B}}_{n},

Furthermore, \underline{ρ} induces a 𝕂-algebra isomorphism 𝕂[S_{n}] ≅ \underline{\underline{B}}_{n}.

We remark that the Young–Capelli symmetrizers are, up to a sign, the representations of the classical two-parameter Young symmetrizers:

\underline{ρ}(e_{ij}) = (−1)^{(_{2}^{n})}γ_{n}(S_{i}, S_{j} ).

18.2 Complete decompositions of the group algebra 𝕂[S_{n}] as a left regular module.

The map

F : Super_{n}[\underline{L}|\underline{P}] → 𝕂[S_{n}], (τ(1) | 1)(τ(2) | 2) ⋯ (τ(n) | n) ↦ τ

is an isomorphism from the module

𝕂[S_{n}] · Super_{n}[\underline{L}|\underline{P}]

to the module

𝕂[S_{n}] · 𝕂[S_{n}].

The symmetrized bitableaux are sent, by this isomorphism to the classical two-para-meter Young symmetrizers:

F ((S_{i} | S_{j} )) = (−1)^{(_{2}^{n})}e_{ij}.

By combining Theorem 12.1 with Proposition 18.1, we get immediately the following theorem.

Theorem 18.1 (Complete decomposition of 𝕂[S_{n}] as a left regular module).

𝕂[S_{n}] = F [Super_{n}[\underline{L}|\underline{P}]]

= F [ ⨁_{λ⊢n} ⨁_{j=1,2,...,\underline{f}_{λ}} ⟨(S_{λi} | S_{λj} ), i = 1, 2, …, \underline{f}_{λ}⟩_{𝕂}]

= ⨁_{λ⊢n} ⨁_{j=1,2,...,\underline{f}_{λ}} F [⟨(S_{λi} | S_{λj} ), i = 1, 2, …, \underline{f}_{λ}⟩_{𝕂}]

= ⨁_{λ⊢n} ⨁_{j=1,2,...,\underline{f}_{λ}}⟨π_{ij}e_{j}; i = 1, 2, …, \underline{f}_{λ}⟩_{𝕂}

= ⨁_{λ⊢n} ⨁_{j=1,2,...,\underline{f}_{λ}} 𝕂[S_{n}]e_{j},

where the outer sum is the isotypic decomposition, and

⟨π_{ij}e_{j}; i = 1, 2, …, \underline{f}_{λ}⟩_{𝕂} = 𝕂[S_{n}]e_{j}, j = 1, 2, …, f_{λ}^{M},

are minimal left ideals of 𝕂[S_{n}].

The irreducible 𝕂[S_{n}]-module

S_{λj} = ⟨(S_{λi} | S_{λj} ), i = 1, 2, …, \underline{f}_{λ}⟩_{𝕂} = F^{−1}[𝕂[S_{n}] · e_{j}]

is the Specht module (of the first kind) associated to the multilinear standard tableau S_{λj} of shape λ (see, e.g., [28], [73]).

18.3 On the coefficients h_{λ} (Lemma 8.3 and Theorem 10.1).

Proposition 18.2. The coefficient h_{λ} which appears in Lemma 8.3 and Theorem 10.1 equals the product of hook lengths of the shape λ.

Proof. We already know that the coefficient h_{λ} depends only on the shape λ. Let S_{i} be the i-th multilinear standard tableau of shape λ. The Triangularity Theorem (Theorem 10.1 ) specializes to the following identity:

γ_{n}(S_{i}, S_{i} )(S_{i} | S_{i} ) = (−1)^{(_{2}^{n})}θ_{S_{i}S_{i}}^{−+}h_{λ}(S_{i} | S_{i} ),

with θ_{S_{i}S_{i}}^{−+} = 1. Then, the following identities hold:

F (\underline{ρ}(e_{ii})(S_{i} | S_{i} )) = F ((−1)^{(_{2}^{n})}γ_{n}(S_{i}, S_{i} )(S_{i} | S_{i} ))

= F (h_{λ}(S_{i} | S_{i} ))

= (−1)^{(_{2}^{n})}h_{λ}e_{ii}.

On the other hand, we have:

F (\underline{ρ}(e_{ii})(S_{i} | S_{i} )) = e_{ii} · F ((S_{i} | S_{i} )) = (−1)^{(_{2}^{n})}e_{ii}^{2}.

Hence e_{ii}^{2} = h_{λ}e_{ii}. Since it is well-known that e_{ii}^{2} = ∏ (hook lenghts of λ) e_{ii}, (see, e.g., [40], [73], [53], [72]), the assertion follows.

18.4 The Young natural form of irreducible representations

In the following, we specialize the constructions and the results of Section 13. Given any partition λ ⊢ n and any multilinear standard place tableau S_{λj} of shape λ, the module structure 𝕂[S_{n}] · S_{λj} induces a surjective algebra morphism

\underline{ν}_{λj} : 𝕂[S_{n}] → End_{𝕂}[S_{λj}].

By choosing the basis of the (S_{λi} | S_{λj} )’s in S_{λj}, the morphism \underline{ν}_{λ,j} induces an irreducible matrix representation

\overline{\underline{ν}}_{λj} : 𝕂[S_{n}] → M_{\underline{f}_{λ}},

where, for every σ ∈ S_{n},

\overline{\underline{ν}}_{λj}(σ) = [c_{hk}^{λ}(\underline{ν}_{λj}(σ))].

For every λ ⊢ n, the module structure 𝕂[S_{n}]·Super_{n}^{M}[\underline{L}|\underline{P}] induces a surjective algebra morphism

\underline{ρ}_{λ} : 𝕂[S_{n}] → \underline{\underline{B}}_{λ};

by choosing the basis of the Y (S_{λi} | S_{λj} ) in \underline{\underline{B}}_{n}, the morphism \underline{ρ}_{λ} induces an irreducible matrix representation

\overline{\underline{ρ}}_{λ} : 𝕂[S_{n}] → M_{\underline{f}_{λ}},

where, for every σ ∈ S_{n},

\overline{\underline{ρ}}_{λ}(σ) = [d_{hk}^{λ}(\underline{ρ}_{λ}(σ))].

From Proposition 13.1, it follows that the irreducible representations \overline{\underline{ρ}}_{λ} and \overline{\underline{ν}}_{λ,j} are equal.

We now specialize Theorem 13.1 to the multilinear case, thereby obtaining a simple combinatorial interpretation of the coefficients

c_{hk}^{λ}(\underline{ν}_{λj}(σ)) = d_{hk}^{λ}(\underline{ρ}_{λ}(σ)).

We have:

C^{λ}(σ) = Θ^{λ}(I)^{−1} × Θ^{λ}(σ),

where the entries θ_{ij}^{λ}(σ) of the matrix Θ^{λ}(σ) are the symmetry transition coefficients θ_{S_{i}S_{j}}^{−+}(σ), defined by the relations

DS σ · \underline{T C} = θ_{ST}^{−+}(σ) \underline{DC}.

Notice that

θ_{P Q}^{−+}(σ) = θ_{P σQ}^{−+}.

These coefficients admit a simple combinatorial description; specifically

θ_{P Q}^{−+} = \begin{cases}(−1)^{|β|} & if there exists a (unique) α ∈ C(P), β ∈ R(Q) : αP = βQ, \\ 0 & otherwise, \end{cases}

where R(Q) and C(P) are the subgroups of S_{n} which are the row-stabilizer of Q and the column-stabilizer of P, respectively.

The matrix Θ^{λ}(I)^{−1} is the same as the transition matrix from the (normalized) generalized Young symmetrizers \frac{1}{h_{λ}} e_{ij} to the Young natural units γ_{ij} in 𝕂[S_{n}], (see e.g., [53, 72]). As a matter of fact, we have

\underline{ρ}(γ_{ij}) = \underline{ρ}( ∑_{h} ϱ_{ih}^{λ} \frac{1}{h_{λ}} e_{hj}) = ∑_{h} ϱ_{ih}^{λ} \frac{(−1)^{(_{2}^{n})}}{h_{λ}} γ(S_{h}, S_{j} ) = Y (S_{i}, S_{j} ).

Therefore, \overline{\underline{ρ}}_{λ} = \overline{\underline{ν}}_{λ,j} is indeed the Young natural form of the irreducible matrix representations of S_{n}, for every λ ⊢ n (see, e.g., [53, 41]).

Part IV

A Glimpse of the General Representation Theory of Finite Dimensional Lie Superalgebras

19 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 [54]. Shortly afterwards, Kac founded the representation theory of these Lie superalgebras (cf. [55], [56]).

The list of complex finite dimensional simple Lie superalgebras L = L_{0} ⊕ L_{1} consists of two essentially different parts, namely, classical and Cartan Lie superalgebras; classical Lie superalgebras are those for which the Lie subalgebra L_{0} is reductive.

The list of classical Lie superalgebras is in turn divided into two parts, basic and 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., [56]).

A basic classical Lie superalgebra L = L_{0} ⊕ L_{1} is said to be of type I whenever its ℤ_{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 A(m, n), the super-analogues of the special linear Lie algebras (for details, see, e.g., [54], [55], [56], [39]).

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 ([23], [24]) and Soergel [83] which are, in turn, inspired by the work of Bernstein, Gelfand and Gelfand (see [6], [7]).

20 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. [55], [56]), who also introduced the two main classes of modules, namely, the class of highest weight representations and the class of modules known nowadays as Kac modules.

The highest weight representations V(Λ) (with integral weights) are irreducible modules, but they are not, in general, of finite dimension. Given a general linear Lie superalgebra gl(m|n), let O_{m|n} be the category whose objects are all ℤ_{2}-graded gl(m|n)-modules with integral weights which are finitely generated gl(m|n)-modules and are locally finite dimensional over a distinguished Borel subalgebra; any irreducible module M ∈ O_{m|n} is isomorphic to an integral highest weight module V(Λ) (see, e.g., [55], [56], [39], [23], [24], [83]).

The Kac modules \overline{V}(Λ), Λ a 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 V(Λ) is a quotient module of the Kac module \overline{V}(Λ), Λ a dominant integral weight, and the module \overline{V}(Λ) is isomorphic to the highest weight module V(Λ) (that is, \overline{V}(Λ) is an irreducible module) if and only if its highest weight Λ is a typical weight.

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

The characters of the modules \overline{V}(Λ) were computed by Kac (cf. [55], [56]) in the case of typical highest weights. After that, there were several conjectures and partial results dealing with atypical highest weights (see, e.g., [8], [5], [80], [87], [51], [57], [67]), before the complete solution of the problem was given by Serganova (see [78], [79]), using a mixture of algebraic and geometric techniques (see, e.g., [23]).

Inspired by some ideas of borrowed from the work of Lascoux, Leclerc and Thibon [62], Brundan [23] 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 [59].

20.1 ℤ_{2}-homogeneous bases of V = V_{0} ⊕ V_{1} and consistent ℤgraduations of a general linear Lie superalgebra 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 ℤ_{2}.

Let V = V\overline{0} ⊕ V\overline{1} be a ℤ_{2}-graded vector space, dim(V\overline{0}) = m, dim(V\overline{1}) = n, and let L = \{x_{1}, …, x_{m}, x_{m+1}, …, x_{m+n}\} be a ℤ_{2}-homogeneous basis of V, |x_{i}| = \overline{0} for every i = 1, …, m, |x_{i}| = \overline{1} for every i = m + 1, …, m + n.

In the following, we will write

gl(m|n) = gl(m|n)\overline{0} ⊕ gl(m|n)\overline{1}

for the general linear Lie superalgebra pl(V) of V = V\overline{0} ⊕ V\overline{1}.

In matrix notation (see Subsection 3.3) — with respect to the choice of the ℤ_{2}- homogeneous basis L = {x_{1}, …, x_{m}, x_{m+1}, …, x_{m+n}} — the Lie superalgebra

gl(m|n) = gl(m|n)\overline{0} ⊕ gl(m|n)\overline{1}

is the algebra of all (m + n) × (m + n)-square matrices, where the set

{E_{x_{i},x_{j}}; i, j = 1, …, m} ∪ {E_{x_{i},x_{j}}; i, j = m + 1, …, m + n}

is a basis of gl(m|n)\overline{0}, and

{E_{x_{i},x_{j}}; i = 1, …, m, j = m+1, …, m+n}∪{E_{x_{i},x_{j}}; i = m+1, …, m+n, j = 1, …, m}

is a basis of gl(m|n)\overline{1}.

The choice of the basis L = {x_{1}, …, x_{m}, x_{m+1}, …, x_{m+n}} induces a consistent ℤgraduation on gl(m|n) (see, e.g., [74]). Specifically, we have the following direct sum decomposition:

gl(m|n) = ⨁_{k∈ℤ} gl(m|n)_{k}, k ∈ ℤ,

where

• gl(m|n)_{−1} is the span of the set {E_{x_{i},x_{j}}; i = 1, …, m, j = m + 1, …, m + n},

• gl(m|n)_{0} is the span of the set {E_{x_{i},x_{j}}; i, j = 1, …, m} ∪ {E_{x_{i},x_{j}}; i, j = m + 1, …, m + n},

• gl(m|n)_{1} is the span of the set {E_{x_{i},x_{j}}; i = m + 1, …, m + n, j = 1, …, m},

• and gl(m|n)_{k} = (0), for every k ≠ −1, 0, 1.

Remark 20.1 (Scheunert, [74]). One may define another consistent ℤ-graduation on 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 ℤ-graded algebra

g′ = g′_{−1} ⊕ g′_{0} ⊕ g′_{1},

where

g′_{−1} = gl(m|n)_{1}, g′_{0} = gl(m|n)_{0}, g′_{1} = gl(m|n)_{−1}

and the supertransposition map

M → M^{T}

from gl(m|n) to g′ such that

E_{x_{i},x_{j}}^{T} = (−1)^{(|x_{i}|+|x_{j}|)|x_{i}|} E_{x_{j},x_{i}}, i, j = 1, 2, …, m + n.

The map

M → −M^{T}

is an isomorphism of ℤ-graded Lie superalgebras from gl(m|n) to g′.

20.2 The supertrace. A consistent, supersymmetric, invariant bilinear form on gl(m|n)

Let M = (m_{ij})_{i,j=1,...,m+m} be a matrix in gl(m|n) = gl(m|n)\overline{0} ⊕ gl(m|n)\overline{1}. The supertrace of M is the number

str(M) = ∑_{i=1}^{m} m_{ii} − ∑_{i=m+1}^{m+n} m_{ii}.

Following Kac [54], we define a bilinear form on gl(m|n) by setting

(M, N) = str(MN), M, N ∈ gl(m|n).

This bilinear form ( , ) is

• consistent, that is

(M, N) = 0, for all M ∈ gl(m|n)\overline{0}, N ∈ gl(m|n)\overline{1};

• supersymmetric, that is

(M, N) = (−1)^{|M||N|} (N, M), for all ℤ_{2} − homogeneous M, N ∈ gl(m|n);

• invariant with respect to the adjoint action, that is

([M, P], N) = (M, [P, N]), for all M, P, N ∈ gl(m|n).

20.3 Distinguished triangular decompositions and roots of the general linear Lie superalgebra gl(m|n)

Let h ⊆ gl(m|n)\overline{0} ⊂ gl(m|n) be the Cartan subalgebra of all diagonal matrices, and let

{ε_{i}; i = 1, …, m + n}

be the canonical basis of the dual space h^{∗} (in plain words, the evaluation of ε_{i} on a diagonal matrix in h equals its i-th diagonal entry). Since the adjoint representation of h in gl(m|n) is diagonalizable, we may consider the so-called distinguished choice for a triangular decomposition of gl(m|n) (see, e.g., [54], [66]). This decomposition is characterized by the following conditions:

• gl(m|n) = n^{−}⊕h⊕n^{+}, where n^{−} and n^{+} are subalgebras such that [h, n^{+}] ⊂ n^{+}, [h, n^{−}] ⊂ n^{−},

• gl(m|n)_{−1} ⊂ n^{+} and gl(m|n)_{1} ⊂ n^{−}.

In plain words, the subalgebras n^{+} and n^{−} are the algebras of all strictly upper and lower matrices, respectively. The algebra b = h ⊕ n^{+} is the distinguished Borel subalgebra and the set of positive roots - with respect to the choice of the Borel subalgebra b - is the set of linear functionals

Δ_{+} = Δ_{\overline{0},+} ∪ Δ_{\overline{1},+} ⊂ h^{∗},

where

Δ_{\overline{0},+} = \{ε_{i} − ε_{j}; 1 ≤ i < j ≤ m\} ∪ \{ε_{i} − ε_{j}; m + 1 ≤ i < j ≤ m + n\}

is the set of positive roots of ℤ_{2}-degree \overline{0}, and

Δ_{\overline{1},+} = \{ε_{i} − ε_{j}; i = 1, …, m, j = m + 1, …, m + n\},

is the set of positive roots of ℤ_{2}-degree \overline{1}. The corresponding set of simple roots (the distinguished set, see, e.g., [66]) is given by

Π = {α_{1} = ε_{1} − ε_{2}, …, α_{m} = ε_{m} − ε_{m+1}, …, α_{m+n−1} = ε_{m+n−1} − ε_{m+n}}.

Thus, in the distinguished set there is only one simple root of ℤ_{2}-degree \overline{1}, the root ε_{m} − ε_{m+1}. As usual, we put

ρ\overline{0} = \frac{1}{2} ∑_{α∈Δ_{\overline{0},+}} α, ρ\overline{1} = \frac{1}{2} ∑_{α∈Δ_{\overline{1},+}} α, ρ = ρ\overline{0} − ρ\overline{1}.

Consider the symmetric bilinear form ( , ) on h^{∗} induced by the invariant supersymmetric bilinear form on gl(m|n); in the natural basis {ε_{i}; i = 1, …, m + n} it takes the form • (ε_{i}, ε_{j}) = δ_{ij}, for all i, j = 1, …, m,

• (ε_{i}, ε_{j}) = 0, for all i = 1, …, m, j = m + 1, …, m + n,

• (ε_{i}, ε_{j}) = −δ_{ij}, for all i, j = m + 1, …, m + n.

Note that the positive roots of ℤ_{2}-degree \overline{1} are isotropic, that is (α, α) = 0, for all α ∈ Δ_{\overline{1},+}.

Remark 20.2.

1.

(ρ, α_{i}) = \frac{1}{2}(α_{i}, α_{i}) = \begin{cases}1 & for 1 ≤ i < m, \\ 0 & for i = m, \\ −1 & for m < i < m + n. \end{cases}

2.

(ρ, ϵ_{i} − ϵ_{j}) = ∑_{h=i}^{j−1}(ρ, α_{h}) = ∑_{h=i}^{j−1} \frac{1}{2}(α_{h}, α_{h})

= \begin{cases}j − i & for 1 ≤ i < j ≤ m, \\ 2m + 1 − i − j & for 1 ≤ i ≤ m < j ≤ m + n, \\ i − j & for m < i < j ≤ m + n. \end{cases}

Remark 20.3.

1. The supersymmetric bilinear form ( ,) is non-degenerate on gl(m|n).

2. Regarded as a gl(m|n)\overline{0}-module (with respect to the adjoint action), gl(m|n)\overline{1} splits into the direct sum of the irreducible submodules:

gl(m|n)\overline{1} = gl(m|n)_{−1} ⊕ gl(m|n)_{1}, \overline{1} ∈ ℤ_{2}, −1, 1 ∈ ℤ.

3. The irreducible gl(m|n)\overline{0}-modules gl(m|n)_{−1} and gl(m|n)_{1} are the irreducible gl(m|n)\overline{0}-modules with highest weight vectors E_{x_{1},x_{m+n}} and E_{x_{m+1},x_{m}}, respectively.

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

20.4 Highest weight modules with integral weights and Verma modules

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

Let

X(m|n) = {μ ∈ h^{∗}; μ = ∑_{i=1}^{m+n} μ_{i} ε_{i}, μ_{i} ∈ ℤ}

be the set of all linear functionals on h with integral coefficients with respect to the standard basis {ε_{i}; i = 1, …, m + n}.

Given a gl(m|n)-supermodule M and an integral weight μ ∈ X(m|n), define the μweight subspace M(μ) of M with respect to h as usual:

M(μ) = {m ∈ M; h · m = μ(h)m for all h ∈ h}.

A gl(m|n)-supermodule M is said to be a module with integral weights whenever it satisfies the following condition:

M = ⨁_{μ∈X(m|n)} M(μ).

Following Brundan ([23], [24]) and Soergel [83], the category O_{m|n} is the category of all ℤ_{2}-graded gl(m|n)-modules M with integral weights which are finitely generated gl(m|n)-modules and are locally finite dimensional over b (that is, all finitely generated b-submodules of M are finite dimensional ℂ-vector spaces). The category O_{m|n} is the (integral weight) analogue of the category O for semisimple Lie algebras, introduced by Bernstein, Gelfand and Gelfand in [7].

For every Λ = ∑_{i=1}^{m+n} Λ_{i}ε_{i} ∈ X(m|n), define a ℤ_{2}-graded one-dimensional b-module < v_{Λ} > by setting

h(v_{Λ}) = Λ(h)v_{Λ}, h ∈ h
n^{+}(v_{Λ}) = 0,

where v_{Λ} is a ℤ_{2}-homogeneous vector of degree ∑_{i=m+1}^{m+n} Λ_{i} (mod 2).

Define the Verma module Ṽ(Λ) as the induced gl(m|n)-module

Ṽ(Λ) = U(gl(m|n)) ⊗_{U(b)} < v_{Λ} > ∈ O_{m|n},

where U(gl(m|n)) and U(b) are the universal enveloping algebras of gl(m|n) and b, respectively. Remark 20.4. The fact that the Verma modules are objects in the category O_{m|n} follows from the Poincaré–Birkhoff–Witt theorem (see, e.g., [54], [74]).

The gl(m|n)-module Ṽ(Λ) contains a unique maximal submodule I(Λ) and, therefore, the quotient module

V(Λ) = Ṽ(Λ)/I(Λ)

is an irreducible gl(m|n)-module.

The modules V(Λ) are usually called highest weight modules (with integral weights) (see, e.g., [39]); the highest weight of the module V(Λ) is clearly the linear functional Λ ∈ h and its unique — up to a scalar factor — highest weight vector is the class vector of the vector 1 ⊗_{U(b)} < v_{Λ} > ∈ Ṽ(Λ).

The following result is essentially due to Kac (cf. [55], [56]).

Theorem 20.1 (see, e.g., [23], [24]). The set

{V(Λ); Λ ∈ X(m|n)}

is a complete set of pairwise non-isomorphic irreducibles in O_{m|n}.

20.5 Kac modules and typical modules

The category F_{m|n} is the category of all finite dimensional gl(m|n)-modules with integral weights.

We recall that an integral weight Λ ∈ h^{∗}, Λ = ∑_{i=1}^{m+n} Λ_{i} ε_{i} ∈ X(m|n) is said to be an integral dominant weight whenever it satisfies the condition

a_{i} = \frac{2(Λ, α_{i})}{(α_{i}, α_{i})} ≥ 0,

for every simple root α_{i} ∈ Π, i ≠ m, that is,

Λ_{1} ≥ Λ_{2} ≥ ⋯ ≥ Λ_{m}, and Λ_{m+1} ≥ Λ_{m+2} ≥ ⋯ ≥ Λ_{m+n}.

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

Let V\overline{0}(Λ) be the irreducible gl(m|n)\overline{0}-module with highest integral dominant weight Λ ∈ X^{+}(m|n); V\overline{0}(Λ) is turned into a ℤ_{2}-graded module by specifying that its highest weight is a ℤ_{2}-homogeneous vector of degree ∑_{i=m+1}^{m+n} Λ_{i} (mod 2).

Consider the subalgebra p = gl(m|n)_{0} ⊕ gl(m|n)_{−1} ⊂ gl(m|n).

The gl(m|n)\overline{0}-module V\overline{0}(Λ) is extended to a p-module by setting

gl(m|n)_{−1} · V\overline{0}(Λ) = 0.

The induced gl(m|n)-module

\overline{V}(Λ) = U(gl(m|n)) ⊗_{U(p)} V\overline{0}(Λ)

is said to be a Kac module.

Remark 20.5. The Kac module \overline{V}(Λ) is a finite dimensional vector space isomorphic to the tensor product ⋀(gl(m|n)_{1}) ⨂ V\overline{0}(Λ).

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

In general, we have the following result.

Proposition 20.1 ([55], [56], [39], [23]). Let Λ ∈ X^{+}(m|n) be an integral dominant weight. Then, the following statements are equivalent.

1. V(Λ) = \overline{V}(Λ).

2. The Kac module \overline{V}(Λ) is an irreducible gl(m|n)-module.

3. (Λ + ρ, α) ≠ 0, for all positive roots α ∈ Δ_{+}.

A finite dimensional irreducible gl(m|n)-module V(Λ) with highest weight Λ is called typical whenever one of the equivalent conditions 1–3 of Proposition 20.1 is satisfied.

The following result is essentially due to Kac (cf. [55], [56]).

Theorem 20.2 (see, e.g., [23], [24]). • The set

{V(Λ); Λ ∈ X^{+}(m|n)}

is a complete set of pairwise non-isomorphic irreducibles in F_{m|n}.

• Any finite dimensional irreducible gl(m|n)-module is either typical or it can be obtained from V(Λ) for some integral dominant weight Λ ∈ X^{+}(m|n) by tensoring with a one dimensional representation.

20.6 Covariant modules, Schur modules and letterplace superalgebras

Let V = V\overline{0} ⊕ V\overline{1} be a ℤ_{2}-graded vector space, dim(V\overline{0}) = m, dim(V\overline{1}) = n, and let L = {x_{1}, …, x_{m}, x_{m+1}, …, x_{m+n}} be a distinguished ℤ_{2}-homogeneous basis of V, |x_{i}| = \overline{0} for every i = 1, …, m, |x_{i}| = \overline{1} for every i = m + 1, …, m + n.

Let λ ⊢ N be a partition, and let λ ∈ H(L), where

H(L) = {λ = (λ_{1} ≥ λ_{2} ≥ ⋯); λ_{m+1} < n + 1}

is the hook set of L (see Subsection 5.4).

The covariant modules V_{λ} are usually defined as the finite dimensional irreducible gl(m|n)-modules that appear in the Berele–Regev complete decomposition of the gl(m|n)-module V^{⊗N} (see also Sergeev [80]).

We know that the covariant modules V_{λ} admit a quite explicit description as submodules of a letterplace algebra, and Theorem 12.1 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.

Proposition 20.2. Let λ be a partition, and let λ ∈ H(L). The covariant module V_{λ} is gl(m|n)-isomorphic to the module

S_{λ} = ⟨(S | D_{λ}); S ∈ T ab_{λ}(L)⟩_{𝕂} ⊆ Super[L|P],

where P = P\overline{1} = \{1, 2, …, t\}, t ≥ λ_{1} and D_{λ} is a Deruyts tableau ∈ T ab_{λ}(P).

Remark 20.6. 1. The covariant module S_{λ} has a standard basis (see Subsection 8.3):

{(S | D_{λ}); S ∈ Stab_{λ}(L)}.

Since the place tableau D_{λ} is of Deruyts type, the bitableaux (S | D_{λ}) are skewsymmetric in the rows of D_{λ}. From Subsection 8.3, it follows that the process of expanding the generators (S | D_{λ}), S ∈ T ab_{λ}(L), of S_{λ} into linear combinations of the standard basis elements is ruled by the following special form of the Straightening Law (see Theorem 8.1):

∑_{(v)} ( \begin{matrix}uv_{(1)} & x \\ v_{(2)}w & y \end{matrix} ) = 0,

where x = 12 ⋯ h, y = 12 ⋯ k, h ≥ k, and v is a word of length greater than h. The above identities are known as the “exchange rules” (see, e.g., [46]).

2. The covariant module S_{λ} has a Clebsch–Gordan–Capelli basis (see Subsection 9.4);

namely,

{( S | D_{λ}); S ∈ Stab_{λ}(L)}.

Let S ∈ T ab_{λ}(L). We recall that the content c(S) of the tableau S is the vector

c(S) = (c(S, x_{1}), …, c(S, x_{m}), c(S, x_{m+1}), …, 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, …, m + n.

Let λ ⊢ N be a partition, and let λ ∈ H(L). An element

φ = ∑_{i=1}^{m+n} φ_{i} ε_{i}, φ_{i} ∈ ℤ

of the dual space h^{∗} is called a standard weight of the covariant module S_{λ} if there exists a standard tableau S ∈ Stab_{λ}(L) such that

(φ_{1}, φ_{2}, …, φ_{m+n}) = c(S).

From Remark 20.5, it follows that the module S_{λ} is h-diagonalizable with weight decomposition

S_{λ} = ⨁_{φ} S_{λ}(φ),

where the direct sum is over the set of all standard weights φ of the covariant module S_{λ}.

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

The standard tableau

F_{λ} = (w_{1}, w_{2}, …, w_{p}) ∈ T ab_{λ}(L)

is the tableau defined by the following conditions:

• w_{i} = x_{i}^{λ_{i}}, for all i ∈ ℤ^{+}, i ≤ m,

• w_{i} = x_{m+1}x_{m+2} ⋯ x_{m+λ_{i}}, for all i ∈ ℤ^{+}, m < i ≤ m + n.

Note that F_{λ} is a tableau of Co-Deruyts type if n = 0, and F_{λ} is a tableau of Deruyts type if m = 0.

The bitableau (F_{λ} | D_{λ}) plays a crucial role in the theory of the covariant module S_{λ}. We notice that the bitableau (F_{λ} | D_{λ}) is (up to a scalar factor) the unique weight vector of S_{λ} of weight

Λ_{λ} = ∑_{i=1}^{m} λ_{i}ε_{i} + ∑_{j=1}^{n} ν_{m+j}ε_{m+j},

where ν_{m+j} = max{0, λ̃_{j}−m}, and λ̃ = (λ̃_{1} ≥ λ̃_{2} ≥ ⋯) denotes the conjugate partition of the partition λ.

We have the following results.

Lemma 20.1. Consider the set of ordered (m + n)-tuples in ℕ^{m+n}

{c(S); S ∈ Stab_{λ}(L)}

as a linearly ordered set with respect to the lexicographic order. Then

c(F_{λ}) = max{c(S); S ∈ Stab_{λ}(L)}.

Proof. The assertion immediately follows from Remark 20.5.

Proposition 20.3. 1. The bitableau (F_{λ} | D_{λ}) is annihilated by all strictly upper polarizations D_{x_{i}x_{j}}, i < j, i, j = 1, 2, …, m + n.

2. The elements of the basis {( S | D_{λ}); S ∈ Stab_{λ}(L)} of S_{λ} may be obtained from (F_{λ} | D_{λ}) by iterated actions of strictly lower polarization D_{x_{i}x_{j}}, i > j, i, j = 1, 2, …, m + n.

3. The bitableau (F_{λ} | D_{λ}) is (up to a scalar factor) the unique element of S_{λ} which is annihilated by all strictly upper polarizations D_{x_{i}x_{j}}, i < j, i, j = 1, 2, …, m+n.

Proof. 1) Since D_{x_{i}x_{j}}(F_{λ} | D_{λ}), i < j, is a linear combination of bitableaux (T | D_{λ}) with c(T) > c(F_{λ}) in the lexicographic order, the assertion follows from Remark 20.5 and Lemma 20.1.

2) We know that the action of the Capelli operator [S̃ | F̃_{λ}]_{−} on the bitableau (F_{λ} | D_{λ}) yields the following result:

[S̃ | F̃_{λ}]_{−} ((F_{λ} | D_{λ})) = c_{F_{λ}}( S | D_{λ}),

where c_{F_{λ}} is a non-zero integer (see Subsection 8.2).

When we devirtualize the Capelli operator [S̃ | F̃_{λ}]_{−}, we may write it as a linear combination of products

D_{1} D_{2} D_{3},

where D_{1} is a product of strictly lower polarizations D_{x_{i}x_{j}}, i > j, i, j = 1, 2, …, m + n,

D_{2} is a product of diagonal polarizations D_{x_{i}x_{i}}, i = 1, 2, …, m+n, and D_{3} is a product of strictly upper polarizations D_{x_{h}x_{k}}, h < k, h, k = 1, 2, …, m + n, by the “easy” part of the Poincaré–Birkhoff–Witt theorem (see, e.g., [54], [74], [58]).

Then, by assertion 1), the action of the operator [S̃ | F̃_{λ}]_{−} on (F_{λ} | D_{λ}) is the same as the action of an operator which may be written as a linear combination of products of strictly lower polarization D_{x_{i}x_{j}}, i > j, i, j = 1, 2, …, m + n.

3) If an element of F ∈ S_{λ} satisfies the conditions of assertion 1), then the cyclic module generated by F (with respect to the actions of the strictly lower polarization D_{x_{i}x_{j}}, i > j, i, j = 1, 2, …, m + n) equals the gl(m|n)-irreducible module S_{λ}. Thus, the assertion immediately follows from Lemma 20.1.

Proposition 20.3 implies the following result.

Corollary 20.1. The vector (F_{λ} | D_{λ}) is the highest weight vector of the covariant module S_{λ}, and, thus, S_{λ} is isomorphic to the highest weight module V(Λ_{λ}) with highest weight Λ_{λ} (with respect to the distinguished set of positive roots defined in Subsection 20.3).

Covariant modules are not, in general, typical modules.

Proposition 20.4. The covariant module S_{λ} is a typical module if and only if λ_{m} ≥ n.

Proof. From Remark 20.2, it follows that:

•

(Λ_{λ} + ρ, ϵ_{i} − ϵ_{j}) = (Λ_{λ}, ϵ_{i} − ϵ_{j}) + (ρ, ϵ_{i} − ϵ_{j})

= \begin{cases}λ_{i} − λ_{j} + j − i > 0 & for 1 ≤ i < j ≤ m, \\ λ_{i} + ν_{j} + 2m + 1 − i − j & for 1 ≤ i ≤ m < j ≤ m + n, \\ −ν_{i} + ν_{j} + i − j < 0 & for m < i < j ≤ m + n. \end{cases}

•

λ_{i} + ν_{j} + 2m + 1 − i − j ≠ 0, for 1 ≤ i ≤ m < j ≤ m + n.

In particular, for i = m we have the set of conditions

λ_{m} + ν_{m+1} ≠ 0,
λ_{m} + ν_{m+2} ≠ 1,
λ_{m} + ν_{m+3} ≠ 2,
⋮
λ_{m} + ν_{m+n} ≠ n − 1,

which is equivalent to the single condition

λ_{m} ≥ n.

In turn, this condition implies

λ_{i} + ν_{j} + 2m + 1 − i − j ≥ n + 0 + 2m + 1 − m − (m + n) = 1, for 1 ≤ i ≤ m < j ≤ m + n.

Remark 20.7. 1. The highest weights Λ_{λ} were computed by Van der Jeugt et al. in [87] and by Cheng and Wang in [26].

2. The covariant modules are “tame” modules in the sense of Kac and Wakimoto [57].

Th recent result [66] by Moens and Van der Jeugt has a significant consequence in the theory of supersymmetric Schur polynomials s_{λ}(x/y). These polynomials appeared in the work of Berele and Regev [5] and turn out to be the characters of the covariant modules, regarded as irreducible gl(m|n)-modules. Since the covariant modules are tame, one can apply the character formula of Kac and Wakimoto [57], and a nice determinantal formula holds for the polynomials s_{λ}(x/y) (cf. [66, Formula (1.17)]).

20.7 The basic plethystic superalgebras S(S^{k}(V)) and ⋀(S^{k}(V))

In a series of rather recent papers, Cheng and Wang ([26], [27]) and Sergeev ([81], [82]) independently rediscovered our complete decomposition Theorem 12.1 from [11] for the letterplace algebra (regarded as the supersymmetric algebra of the tensor product of a pair of finite dimensional ℤ_{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 Howe duality (see, e.g [49], [50] and Theorem 12.3).

Furthermore, they describe the complete decompositions of the supersymmetric and superexterior algebras S(S^{2}(V)) and ⋀(S^{2}(V)) of the supersymmetric square of the natural representations of gl(m|n), and ask (see [26, Introduction]) “whether the results concerning the decomposition of S(S^{2}(V)) (respectively ⋀(S^{2}(V))) and the highest weight vectors in these modules may also be obtained with extra insights from the combinatorial approach in [11] 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[L|P] can be explicitly described by combining Corollary 20.1 and Theorem 12.1.

With reference to the modules S(S^{2}(V)) and ⋀(S^{2}(V)), we mention that they have been systematically studied by Brini, Huang and Teolis [16] and Grosshans [47], in the more general setting of “plethystic superalgebras”.

For the convenience of the reader, we recall some basic definitions from [16].

Let W = W\overline{0} ⊕ W\overline{1} be a finite dimensional ℤ_{2}-graded vector space.

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

We recall that S^{k}(W) is a ℤ_{2}-graded vector space, where

S^{k}(W)\overline{0} = ⨁_{m}(Sym^{k−2m}(W\overline{0}) ⊗ Λ^{2m}(W\overline{1})),

and

S^{k}(W)\overline{1} = ⨁_{m}(Sym^{k−2m−1}(W\overline{0}) ⊗ Λ^{2m+1}(W\overline{1})).

Let V = V\overline{0} ⊕ V\overline{1} be a finite dimensional ℤ_{2}-graded vector space, dim(V\overline{0}) = m, dim(V\overline{1}) = n.

The basic k-th plethystic (super)symmetric superalgebra of V = V\overline{0} ⊕ V\overline{1} is the superalgebra

S(S^{k}(V)) = Sym(S^{k}(V)\overline{0}) ⊗ Λ(S^{k}(V)\overline{1}),

and the basic k-th plethystic (super)exterior superalgebra of V = V\overline{0} ⊕ V\overline{1} is the superalgebra

⋀(S^{k}(V)) = Λ(S^{k}(V)\overline{0}) ⊗ Sym(S^{k}(V)\overline{1}).

Let S^{h}(S^{k}(V)) be the ℤ-homogeneous component of degree h of S(S^{k}(V)),

S^{h}(S^{k}(V)) = ⨁_{p}(Sym^{p}(S^{k}(V)\overline{0}) ⊗ Λ^{h−p}(S^{k}(V)\overline{1}),

and let ⋀^{h}(S^{k}(V)) be the ℤ-homogeneous component of degree h of ⋀(S^{k}(V),

⋀^{h}(S^{k}(V)) = ⨁_{p}(Λ^{p}(S^{k}(V)\overline{0}) ⊗ Sym^{h−p}(S^{k}(V)\overline{1}).

The ℤ_{2}-graded vector spaces S^{h}(S^{k}(V)) and ⋀^{h}(S^{k}(V)) are in a natural way pl(V)- modules.

In [16] and [47], the modules S^{h}(S^{k}(V)) and ⋀^{h}(S^{k}(V)) are shown to be epimorphic images of suitable letterplace algebras under the so-called umbral operator U (see also [46]). The operator U is a pl(V))-equivariant operator.

Since the letterplace algebras are semisimple modules (Theorem 12.1), it follows that the modules S^{h}(S^{k}(V)) and ⋀^{h}(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 S_{λ} in a complete decomposition is described in a rather implicit way; these multiplicities are shown to be equal to the 𝕂-linear dimensions of suitable representations of the symmetric group S_{h} ([16, Theorem 12], and [47, Theorems 13 and 17]).

In the case k = 2 (i.e., the case of supersymmetric matrices), the situation is much more satisfactory: the pl(V)-modules S^{h}(S^{2}(V)) and ⋀^{h}(S^{2}(V)) are described in detail in Sections 4 and 6 of [16]. In particular, Straightening Laws for both the modules of type S^{h}(S^{2}(V)) and for the modules of type ⋀^{h}(S^{2}(V)) are provided; these Straightening Laws are closely related to the work of De Concini and Procesi [32] and Rota and Stein (cf. [70], [71]). Furthermore, two classes of Clebsch–Gordan–Capelli bases are exhibited for both classes of modules in [16, Sections 3 and 4].

We have the following structure theorems.

Theorem 20.3 (see [16], [47], [26], [82]). 1. The pl(V)-module S^{h}(S^{2}(V)) is a multiplicity-free module. We have the following complete decomposition result:

S^{h}(S^{2}(V)) ≅ ⨁_{λ} S_{λ},

where the direct sum ranges over all the partitions λ = (λ_{1}, λ_{2}, …) ⊢ 2h, λ_{i} even for every i, λ_{m+1} ≤ n.

2. The pl(V)-module ⋀^{h}(S^{2}(V)) is a multiplicity-free module. We have the following complete decomposition result:

⋀^{h}(S^{2}(V)) ≅ ⨁_{λ} S_{λ},

where the direct sum ranges over all the partitions λ = (λ_{1}, λ_{2}, …) ⊢ 2h which are obtained by nesting (q + 1, q)-hooks, λ_{m+1} ≤ n.

References

[1] A. Akin, D. Buchsbaum, J. Weyman, Schur functors and Schur complexes, Adv. Math. 44 (1982), 207–278.

[2] A. Balentekin, I. Bars, Dimension and character formula for Lie supergroups, J. Phys. A. 22 (1981), 1149–1162.

[3] M. Barnabei, A. Brini, G.-C. Rota, On the exterior calculus of invariant theory, J. Alg. 96 (1985), 120–160.

[4] A. Berele, A. Regev, Hook Young diagrams, combinatorics and representations of Lie superalgebras, Bull. Amer. Math. Soc. 8 (1983), 337–339.

[5] A. Berele, A. Regev, Hook Young Diagrams with applications to combinatorics and to representations of Lie superalgebras, Adv. Math. 64 (1987), 118–175.

[6] I. N. Bernstein, I. M. Gelfand, S. I. Gelfand, Structure of representations generated by vectors of highest weight, Funk. Anal. Appl. 5 (1971), 1–9.

[7] I. N. Bernstein, I. M. Gelfand, S. I. Gelfand, A category of g-modules, Funk. Anal. Appl. 10 (1976), 87–92.

[8] J. Bernstein, D. Leites, Character formulae for irreducible representations of Lie superalgebras of series gl and sl, C. R. Acad. Bulg. Sci. 33 (1980), 1049–1051.

[9] H. Boerner, Representations of groups, Elsevier North–Holland, Amsterdam, 1970.

[10] F. Bonetti, D. Senato, A. Venezia, The Robinson–Schensted correspondence for the fourfold algebra. Boll. Un. Mat. Ital. B (7) 2 (1988), no. 3, 541–554.

[11] A. Brini, A. Palareti, A. Teolis, Gordan–Capelli series in superalgebras, Proc. Natl. Acad. Sci. USA 85 (1988), 1330–1333.

[12] A. Brini, A. Teolis, Young–Capelli symmetrizers in superalgebras, Proc. Natl. Acad. Sci. USA 86 (1989), 775–778.

[13] A. Brini, A. Teolis, Capelli bitableaux and ℤ-forms of general linear Lie superalgebras, Proc. Natl. Acad. Sci. USA 87 (1990), 56–60.

[14] A. Brini, A. Teolis, Capelli’s theory, Koszul maps, and superalgebras, Proc. Natl. Acad. Sci. USA 90 (1993), 10245–10249.

[15] A. Brini, A. Teolis, Remark on the branching theorem and supersymmetric algebras, J. Algebra 142 (1991), 255–271.

[16] A. Brini, R.Q. Huang, A. Teolis, The umbral symbolic method for supersymmetric tensors, Adv. Math. 96 (1992), 123–193.

[17] A. Brini, A. Teolis, Capelli’s method of variabili ausiliarie, superalgebras and geometric calculus, in Invariant Methods in Discrete and Computational Geometry (N.White, Ed.), Kluwer 1995, pp. 59–75.

[18] A. Brini, F. Regonati, A. Teolis, Multilinear algebra over supersymmetric rings, Adv. Math. 145 (1999), 98–158.

[19] A. Brini, F. Regonati, A. Teolis, Grassmann geometric calculus, invariant theory and superalgebras, in Algebraic combinatorics and computer science (H. Crapo and D. Senato, Eds.), Springer Italia, Milan, 2001, 151–196.

[20] A. Brini, F. Regonati, A. Teolis, The method of virtual variables and Representations of Lie Superalgebras, in Clifford algebras — applications to mathematics, physics, and engineering (Abłamowicz, R., ed.), Progress in Mathematical Physics, vol. 34, Birkhäuser, Boston, 2004, 245–263.

[21] A. Brini, F. Regonati, A. Teolis, Combinatorics and representations of general linear Lie superalgebras over letterplace superalgebras, in Computer Algebra and Clifford Algebra with Applications (H. Li, P. Olver, G. Sommer, Eds.), Lecture Notes in Computer Sciences, vol. 3519, Springer-Verlag, 2005, 239–257.

[22] A. Brini, F. Regonati, A. Teolis, Combinatorics, transvectants and superalgebras. An elementary constructive approach to Hilbert’s finiteness theorem, Adv. Appl. Math. 37 (2006), 287–308, Special issue dedicated to A. Regev (A. Berele and Y. Roichman, eds.).

[23] J. Brundan, Kazhdan–Lusztig polynomials and character formulae for the Lie superalgebra gl(m|n), J. Amer. Math. Soc. 16 (2002), 185–231.

[24] J. Brundan, Tilting modules for Lie superalgebras, Communications in Algebra 32 (2004), 2251–2268.

[25] A. Capelli, Lezioni sulla teoria delle forme algebriche, Pellerano, Napoli, 1902.

[26] S.-J. Cheng, W. Wang, Howe duality for Lie superalgebras, Compositio Math. 128 (2001), 55–94.

[27] S.-J. Cheng, W. Wang, Remarks on the Schur–Howe–Sergeev duality, Lett. Math. Phys. 52 (2000), 143–153.

[28] M. Clausen, Letter place algebras and a characteristic-free Approach to the Representation Theory of the General Linear and Symmetric Groups, I, Adv. Math. 33 (1979), 161–191.

[29] L. Corwin, Y. Ne’eman, S. Steinberg, Graded Lie algebras in mathematics and physics, Rev. Mod. Phys. 47 (1975), 573–604.

[30] C.W. Curtis, Pioneers of representation theory: Frobenius, Burnside, Schur, and Brauer, History of Mathematics, vol. 15, Amer. Math. Soc., 1999.

[31] C. De Concini, D. Eisenbud, C. Procesi, Young diagrams and determinantal varieties, Invent. Math. 56 (1980), 129–165.

[32] C. De Concini, C. Procesi, A characteristic-free approach to invariant theory, Adv. Math. 21 (1976), 330–354.

[33] J. Deruyts, Essai d’une théorie générale des formes algébriques, Mém. Soc. Roy. Sci. Liège 17 (1892), 1–156.

[34] J. Désarménien, J. P. S. Kung, G.-C. Rota, Invariant theory, Young bitableaux and combinatorics, Adv. Math. 27 (1978), 63–92.

[35] J. Désarménien, An algorithm for the Rota straightening formula, Discrete Math. 30 (1980), 51–68.

[36] J. Dieudonné , J. Carrel, Invariant theory, old and new, Academic Press, 1971.

[37] P.H. Dondi, P.D. Jarvis, Diagram and superfield techniques in the classical superalgebras, J. Phys. A. 14 (1981), 547–563.

[38] P. Doubilet, G.-C. Rota, J.A Stein, On the foundations of combinatorial theory IX. Combinatorial methods in invariant theory, Studies in Appl.Math. 53 (1974), 185–216.

[39] L. Frappat, P. Sorba, A. Sciarrino, Dictionary on Lie superalgebras, Academic Press, 2000.

[40] W. Fulton, J. Harris, Representation theory. A first course, Graduate Texts in Mathematics, vol. 129, Springer-Verlag, New York, 1991.

[41] A. Garsia, T.J. Mc Larnan, Relations between Young’s natural and the Kazhdan–Lusztig representation of S_{n}, Adv. Math. 69 (1988) 32–92.

[42] R. Goodman, N.R. Wallach, Representations and invariants of the classical groups, Encyclopedia of Mathematics and its Applications, vol. 68, Cambridge University Press, Cambridge, 1998.

[43] J.H. Grace, A. Young, The algebra of invariants, Cambridge University Press, Cambridge, 1903.

[44] J.A. Green, Polynomial representations of GL_{n}, Lecture Notes in Mathematics, vol. 830, Springer Verlag, Berlin, 1980.

[45] J.A. Green, Classical invariants and the general linear group, Progress in Mathematics, vol. 95, Birkhäuser, 1991, 247–272.

[46] F.D. Grosshans, G.-C. Rota, J.A. Stein, Invariant theory and superalgebras, Amer. Math. Soc., Providence, RI, 1987.

[47] F.D. Grosshans, The Symbolic method and representation theory, Adv. Math. 98 (1993), 113–142.

[48] F.D. Grosshans, Algebraic homogeneous spaces and invariant theory, Lecture Notes in Mathematics, vol. 1673, Springer, 1997.

[49] R. Howe, Remarks on classical invariant theory, Trans. Amer. Math. Soc. 313(1989), 539–570.

[50] R. Howe, Perspective on invariant theory: Schur duality, multiplicity-free actions and beyond, The Schur Lectures, Israel Math. Conf. Proc., 8 (1992), 1–182

[51] J.W.B. Hughes, R.C. King, J. Van der Jeugt, On the composition factors of Kac modules for the Lie superalgebra sl(m/n), J. Math. Phys. 33 (1992), 470–491.

[52] F. Iachello, Symmetry in physics, Eur. Phys. J. A 20 (2004),1–3.

[53] G. James, A. Kerber, The Representation theory of the symmetric group, Encyclopedia of Mathematics and Its Applications, vol. 16, Addison–Wesley, Reading, MA, 1981

[54] V. Kac, Lie Superalgebras, Adv. Math. 26 (1977), 8–96.

[55] V. Kac, Characters of typical representations of classical Lie superalgebras, Communications in Algebra 5 (1977), 889–897.

[56] V. Kac, Representations of classical Lie superalgebras, Lecture Notes in Mathematics, Springer, Berlin, 1978, vol. 676, pp. 597–626.

[57] V. Kac, M. Wakimoto, Integrable highest weight modules over affine superalgebras and number theory, Progress in Mathematics 123 (1994), 415–456.

[58] I. Kaplansky, Superalgebras, Pacific J. Math. 86 (1980), 93–98.

[59] D. Kazhdan, G. Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), 175–184.

[60] B. Kostant, Graded manifolds, graded Lie theory and prequantization, Lecture Notes in Mathematics, Springer, Berlin, 1975, vol. 570, pp. 177–306.

[61] R. La Scala, V. Nardozza, D. Senato, Super RSK-algorithms and super plactic monoid, Int. J. Algebra Comp. 16 (2006), 377–396.

[62] A. Lascoux, B. Leclerc, J.Y. Thibon, Hecke algebras at roots of unity and crystal bases of quantum affine algebras, Comm. Math. Phys. 181 (1996), 205–263.

[63] I.G. MacDonald, Symmetric functions and Hall polynomials, Oxford Mathematical Monographs, Clarendon Press, Oxford, 1995.

[64] M. Marcus, Finite dimensional multilinear algebra, vol. 2, Dekker, New York, 1975.

[65] D.G. Mead, Determinantal ideals, identities and the Wronskian, Pacific J. Math. 42 (1972), 165–175.

[66] E.M. Moens, J. Van der Jeugt, A determinantal formula for supersymmetric Schur polynomials, J. Algebraic Combin. 17 (2003), 203–307.

[67] I. Penkov, V. Serganova, Generic irreducible representations of finite dimensional Lie superalgebras, Internat. J. Math. 5 (1994), 389–419.

[68] C. Procesi, Lie Groups. An approach through invariants and representations, Springer Universitext, Springer Verlag, 2005.

[69] F. Regonati, On the combinatorics of Young–Capelli symmetrizers, preprint.

[70] G.-C. Rota, J.A. Stein, Standard basis in supersymplectic algebras, Proc. Natl. Acad. Sci. U.S.A. 86 (1989), 2521–2524.

[71] G.-C. Rota, J.A. Stein, Supersymmetric Hilbert space, Proc. Natl. Acad. Sci. U.S.A. 87 (1990), 653–657.

[72] D.E. Rutherford, Substitutional analysis, Edinburgh University Publications: Science and Mathematics, Edinburgh-London, 1948.

[73] B.E. Sagan, The symmetric group. Representations, combinatorial algorithms, and symmetric functions, Springer Graduate Texts in Mathematics, vol. 203, Springer Verlag, 2001.

[74] M. Scheunert, The theory of Lie superalgebras: an introduction, Lecture Notes in Math., vol. 716, Springer Verlag, New York, 1979.

[75] M. Scheunert, R. B. Zhang, The general linear supergroup and its Hopf superalgebra of regular functions. J. Algebra 254 (2002), 44–83.

[76] I. Schur, Über eine Klasse von Matrizen, die sich einer gegebenen Matrix zuordnen lassen, in I. Schur Gesammelte Abhandlungen 1, pp.1 –71, Springer–Verlag, 1973.

[77] I. Schur, Über die rationalen Darstellungen der allgemeinen linearen Gruppen, in I. Schur Gesammelte Abhandlungen 3, pp.65 –85, Springer–Verlag, 1973.

[78] V. Serganova, Kazhdan–Lusztig polynomials for the Lie superalgebra gl(m|n), Adv. Sov. Math. 16 (1993), 151–165.

[79] V. Serganova, Kazhdan–Lusztig polynomials and character formula for the Lie superalgebra gl(m|n), Selecta Math. 2 (1996), 607–651.

[80] A. Sergeev, Tensor algebra of the identity representation as a module over the Lie superalgebras GL(n, m) and Q(n) (in Russian), Math. USSR Sbornik 51 (1985), 419– 427.

[81] A. Sergeev, An analog of the classical invariant theory for Lie superalgebras I, Michigan Math. J. 49 (2001), 113–146.

[82] A. Sergeev, An analog of the classical invariant theory for Lie superalgebras II, Michigan Math. J. 49 (2001), 147–168.

[83] W. Soergel, Character formulas for tilting modules over Kac–Moody algebras, Represent. Theory 2 (1998), 432–448.

[84] M. Sweedler, Hopf algebras, Benjamin, New York, 1969.

[85] J. Towber, Two new functors from modules to algebras, J. Algebra 47 (1977), 80–109.

[86] H.W. Turnbull, The theory of determinants, matrices and invariants, 3rd ed., Dover, New York, 1960.

[87] J. Van der Jeugt, J.W.B. Hughes, R.C. King, J. Thierry–Mieg, Character formulas for irreducible modules on the Lie superalgebra sl(m/n), J. Math. Phys. 31 (1990), 2278– 2304.

[88] V.S. Varadarajan, Supersymmetry for mathematicians: an introduction, American Mathematical Society, Providence, R.I., 2004.

[89] A.H. Wallace, Invariant matrices and the Gordan–Capelli series, Proc. London Math. Soc. 2 (1952), 98–127.

[90] R. Weitzenböck, Komplex–Symbolik, Teubner, Leipzig, 1908.

[91] R. Weitzenböck, Invariantentheorie, Noordhoff, Groningen, 1923.

[92] H. Weyl, The classical groups, Princeton Univ. Press, Princeton, NY, 1946.

[93] A. Young, The collected papers of Alfred Young, Univ. of Toronto Press, Toronto, 1977.

Andrea Brini Dipartimento di Matematica, Università di Bologna 40126 Bologna, Italy E-mail: brini@dm.unibo.it