\input amstex
\input amsppt.sty

\input path.sty

\magnification1200
\hsize13cm
\vsize19cm

\TagsOnLeft
\NoBlackBoxes

\def\al{\alpha}
\def\La{\Lambda}
\def\la{\lambda}
\def\Si{\Sigma}
\def\de{\delta}
\def\MUR{\operatorname{MUR}}
\def\LAT{\operatorname{LAT}}

\vskip1cm
\head \bf LITTLEWOOD-RICHARDSON WITHOUT ALGORITHMICALLY DEFINED BIJECTIONS \endhead
\vskip0,5cm
\centerline{\eightpoint BY}
\vskip1cm
\centerline{\smc Peter HOFFMANN}
\vskip1cm

\document
John Stembridge \cite {St} has recently solved the important problem of
finding a ``Little\-wood-Ri\-chard\-son rule" for $Q$-functions. His proof is
very natural combinatorially, but lengthy, if all the background is
included. It uses extensive material from Worley's thesis \cite {W} and
Sagan's similar theory of shifted tableaux \cite {Sa}. To include this
result in a forthcoming book (coauthored by John Humphreys), and in
order to keep the volume of material under control, a second proof was
sought for it and for an analogous product theorem in the $\Bbb Z/2
\times \Bbb N $-graded algebra of projective representations of $A_n$
and $S_n $ \cite {H, HH}. 

Given below is a similar, less tableau-theoretic, proof of the
usual Littlewood-Richardson rule for Schur functions. The best of the
earlier proofs have considerable combinatorial explanatory power. The
proof below explains only why the product of two Schur functions is
what it is. 

More precisely, one might say that an algorithmically defined
bijection $(a.d.b.)$ $A_\al \to B_\al$, between sets indexed by a
parameter $\al$ which runs over an infinite set, is a definition in
which the image of an element in $A_\al$ is determined using successive
applications of a least one algorithm, and where, to produce that
image, the number of needed applications of the algorithm is unbounded
as we vary the element over $A_\al$ for all $\al$. Earlier proofs
apparently use $a.d.b.$'s; the proof below does not. This remark is
included to point readers, who attempt to compare this with earlier
proofs, in the correct direction. 

Our notation will be largely from [Ma], with the ring $\La$ of
symmetric functions having the inner product $\langle\ ,\ \rangle$ 
for which the
Schur functions $s_\la$ give an orthonormal basis, as $\la$ ranges
over all partitions. For $f$ in $\La$, the operator $f ^\perp$ on
$\La$ is defined by $\langle f ^\perp (g),h \rangle = \langle
g,fh \rangle$. The Schur function can
be given by
$$s_{n_1,\dots,n_\ell }=B_{n_1}\dots B_{n_\ell } \tag 1$$
where $B_n$ is the operator $\Si _i(-1) ^ih_{n+i}e ^\perp
_i$ (see [Z; p.~69]). A proof of (1), which is more basic than quoting
the Jacobi-Trudi identity, can be had by defining elements $s ^B_\la$ by the
right hand side of (1). Obviously
$$s ^B_\la -h_\la \in \text {Span} \{h_\mu : \mu < \la\}.\tag 2$$
By an induction on length, one gets
$$h ^\perp_{a+i} (s ^B_{a,b,\dots})= \cases
s^B_{b, \dots},  & \text {if} \ i=0; \\
0, & \text {if} \ i >0.\endcases
$$
Thus, for $\mu \le \la$ (partitioning the same integer),
$$\langle h_ \mu,s ^B_\la \rangle =\cdots h ^\perp_{\mu_2} h ^\perp_{\mu_1} (s
^B_\la)=\de_{\mu,\la}.$$
Using (2), this yields, for all $\nu$ and $\la$,  
$$\langle s ^B_\nu, s ^B_ \la\rangle=\de_{\nu, \la}. \tag 3$$ 
Thus $s ^B_\la = s_\la$ by (2) and (3). 

Equation (1) may be used to extend the definition of
$s_{n_1,\dots,n_\ell }$ to any sequence $\{n_i\}$ of integers. These
elements coincide with Murnaghan's ``non-standard symbols" [Mu; p.~761],
and with the $t=0$ specialization of the extension of Hall-Littlewood
polynomials [Ma; p.~109]. Each is either zero or $\pm s_\la$ for a
partition $\la$, by the identity
$$s_{\dots,c,d,\dots}=-s_{\dots, d-1, c+1,\dots}. \tag 4$$
A natural proof of (4) using the $B$-operators is to show by
straightforward calculation that
$$B_c B_d = -B_{d-1}B_{c+1}. \tag 5$$
The elements $s_{\dots,c,c+1,\dots}$ and $s_{n_1, \dots, n_\ell }$ when
$n_\ell <0$, are both zero. 

We shall use $\tilde \la$ for the conjugate of a partition $\la$, and
also $\tilde f$ for the image of $f$ under the involution of $\La$
which interchanges $h_n$ and $e_n$. Thus for $\la=(n_1, \dots , n_\ell )$
we have
$$s_{\tilde \la} = \tilde s _\la = \tilde B _{n_1} \dots \tilde B_{n_\ell },$$
where $\tilde B_n$ is the operator $\sum _i(-1) ^i e_{n+i}h ^\perp _i.$
Equation (5) holds also for the operators $\tilde B_n$.


\definition{Definition}  For a partition $\la$ and a sequence $(t_1,
\dots, t_k)$ of non-negative integers, let $\MUR(\la;t_1,\dots, t_k)$
be the set of all matrices of $0$'s and $1$'s whose row sums are the
parts of $\la$, whose column sums are $t_1, \dots, t_k,$ and such
that
$$\sum _{} ^{}\text{ rightmost $j$ entries of row $i$} 
\ge \sum _{} ^{} \text{ rightmost $j$
entries of row $(i+1)$}$$
for $0 < j \le k$ and $1 \le i <$ length$(\la)$.
\enddefinition



\proclaim{Proposition 1}  For any partition $\la$ and sequences $m_1,
\dots, m_k,$ we have 
$$\tilde s ^\perp _\la (s_{m_1, \dots, m_k})=\sum _{t_1,\dots,t_k}
\# \MUR (\la; t_1,\dots,t_k) s_{m_1-t_1,\dots,m_k-t_k}.$$
\endproclaim

\demo{Proof} The equations 
$$\align 
e ^ \perp_i (h_n)&= \cases
h_{n-i},   & \ \text {if} \ i=0,1; \\
0,  & \ \text {if} \ i>1; \endcases\\
e ^\perp_i (xy) &= \sum _{j} e ^\perp_j (x) e ^\perp_{i-j} (y);
\endalign$$
and 
$$\sum _{i} (-1) ^ie_ih_{k-i} = \de_{0,k}$$
immediately give
$$[h ^\perp_i(y)] ^\perp(u)=\sum _{\ell } (-1) ^\ell  e_\ell y ^\perp(h_{i-\ell }u).$$ 
This follows by induction on $i$, applying $\langle \ ,u\rangle$, 
for arbitrary $u$,
to both sides. It leads immediately to
$$[\tilde B_n(y)] ^\perp (z) = \sum _{p,q} (-1) ^p e_q
y ^\perp [h_p e ^\perp_{n+p+q}(z)]. \tag 6$$
Now proceed by induction on $\ell $ to prove
$$\tilde s ^\perp_{n_1,\dots,n_\ell } (h_m x) = \sum
_{\de_i=0,1} h_{m-\Si \de_i} \tilde s
^\perp_{n_1-\de_1,\dots,n_\ell -\de_\ell } (x). \tag 7$$
This is trivial for $\ell =1$. The inductive step is
$$\align 
\tilde s ^\perp_{n_1, \dots, n_\ell } (h_m x) & = 
[\tilde B_{n_1} (\tilde s_{n_2, \dots, n_\ell })] ^\perp (h_m x) \\ 
& =\sum _{p,q} (-1) ^p e_q \tilde s ^\perp_{n_2, \dots, n_\ell } 
[h_p e^\perp_{n_1+p+q} (h_m x)] \\ 
& =\sum_{p,q} (-1) ^p e_q \tilde s ^\perp_{n_2, \dots, n_\ell } [h_ph_m e ^\perp_{n_1+p+q}(x) + h_p h_{m-1} e ^
\perp_{n_1+p+q-1}(x)] \\
& = \sum _{p,q} (-1) ^p e_q \underset {i \ge 2}\to{\sum _ {\de_i=0,1} }
 \{h_{m-(\de_2+\dots+\de_q)}\tilde s
^\perp_{n_2-\de_2,\dots,n_\ell -\de_\ell } (h_p e ^\perp_{n_1+p+q}(x)) \\
& \hskip 2,7cm +h_{m-1-(\de_2+\dots+\de_q)} \tilde 
s ^\perp_{n_2-\de_2,\dots,n_\ell -\de_\ell }
(h_p e ^\perp_{n_1+p+q}(x))\} \\
& = \sum _{\de_2,\dots,\de_\ell } \{h_{m-(\de_2+\dots+\de_q)} [\tilde
B_{n_1} (\tilde s_{n_2+\de_2,\dots,n_\ell -\de_\ell })] ^\perp(x) \\
& \hskip 2,7cm +h_{m-1(1+\de_2+\dots+\de_q)} [\tilde B_{n_1-1}(\tilde
s_{n_2-\de_2,\dots,n_\ell -\de_\ell })]^\perp (x)\},
\endalign$$
as required. 

Finally, we can prove the proposition by induction on $k$. For $k=1$,
take $x=1$ in (7). For the inductive step, with $\la = (n_1, \dots,
n_\ell )$,
$$\align 
\tilde s ^\perp_ \la (s_{m_1, \dots,m_k})&=\tilde s ^\perp_\la
\bigg( \sum _{i}
(-1) ^i h_{m_1+i}e ^\perp_i (s_{m _2,\dots,m_k})\bigg)\\
&=\sum _{i} (-1) ^1\sum _{\de_1,\dots,\de_\ell }
h_{m_1+i-(\de_1+\dots+\de_\ell )} \tilde s
^\perp_{n_1-\de_1,\dots,n_\ell-\de_\ell } e ^\perp_i (s_{m_2,\dots,m_k})\\
&=\sum \Sb {\de_1,\dots,\de_\ell } \\ 
{n_p-\de_p\ge n_{p+1} -\de_ {p+1} }\\ \endSb
B_{m_1-(\de_1+\dots+\de_\ell )}\sum s_{m_2-t_2,\dots,m_k-t_k},
\endalign$$
where the second summation is over all $E'$ in $\MUR 
(n_1-\de,\dots, n_\ell -\de_\ell ;t_2,\dots,t_k)$ for all $t_2,\dots,t_k$.
The restriction $n_p-\de_p \ge n_{p+1}-\de_{p+1}$ for all $p$ follows
by the remark after (5). Now letting $t_1=\sum \de_i$ and
$$E=\pmatrix     
\de_1 \\
\vdots&
 E'\\ 
\de_\ell \endpmatrix,$$
the sum becomes $\sum _ {t_1,\dots,t_k} \sum _ {E \in \MUR (\la;
t_1,\dots,t_k)} s_{m_1-t_1,\dots,m_k-t_k}$, as required.
\enddemo

\definition{Definition} Let $\LAT (\la; t_1,\dots,t_k)$ be
the set of sentences $\Cal S=(w_1,\dots,w_k)$, where $w_i$ is a weakly
decreasing sequence of positive integers of length $t_i$, such that the
concatenated word $w_1w_2\dots w_k$ satisfies the lattice permutation
condition of the Littlewood-Richardson rule [Ma; p. 68] and has content
(or weight) $\la$. Define
$$\LAT_k(\la)=\bigcup _{t_1,\dots,t_k} \LAT (\la;
t_1,\dots,t_k).$$
\enddefinition

\proclaim{Proposition 2} $\# \MUR(\la; t_1,\dots,t_k)=\# 
\LAT (\tilde \la;  t_1,\dots,t_k)$$.
$\endproclaim
\vskip8pt 

\demo{Proof} Let $\text {TAB}(\la;t_1,\dots,t_k)$ be the set of
(column strict) tableaux of shape $\la$ and content $(t_1,\dots,t_k)$.
We have
$$\align 
\#\MUR(\la;t_1,\dots,t_k) 
&  \overset{\hbox to0pt{\hss i)\hss}}\to= 
\# \text {TAB} (\tilde \la; t_k,\dots,t_1) 
\overset{\hbox to0pt{\hss ii)\hss}}\to= \ \# \text
{TAB} (\tilde \la;t_1,\dots,t_k) \\
&  \overset{\hbox to0pt{\hss iii)\hss}}\to= 
 \# \MUR (\la;t_k,\dots,t_1) \overset{\hbox to0pt{\hss iv)\hss}}\to= \# 
\LAT (\tilde \la;t_1,\dots,t_k), 
\endalign$$
as required. In i) and iii), a bijection is defined by having an $i$
in the $jth$ column of a tableau if and only if there is a $1$
in place $(j,k-i+1)$ of the corresponding matrix. In iv), a bijection
is defined by replacing the $1's$ in the $ith$ row of a matrix
by $1,2,\dots,n_i$ (= $i$th of part $\la$) from right to left, and then
reading the columns downwards (omitting zeros) to obtain the words of
sentence (the first word from rightmost column, etc.). A bijection for
ii) exists, since the number of tableaux of given shape does not vary
if we permute the content sequence (that is, the Schur function is
symmetric).
\enddemo

\remark{Remark} The bijections for i), iii), and iv) are entirely
elementary. No $a.d.b$'s are used there, nor in the proof $[\text
{Ma};(5.12)]$ that $s_\la$ has coefficients $\# \ \text {TAB}(\la;)$.
\endremark

\definition{Definition}  Given $\Cal S=(w_1,\dots,w_k) \in 
\LAT_k(\la)$ and a partition $(m_1,\dots,m_k)$, we say that, with
respect to $(m_1,\dots,m_k)$:

\roster
\item "i)" $\Cal S$ has a bottom line violation iff length$(w_k)>m_k$;
\item "ii)" $\Cal S$ has a $(d+1,i)$-shape violation ($\text {for} \ d\ge 0 \ \text
{and} \ 1 \le i <k$) iff
$$m_{i+1}-\text {length} (w_{i+1})=m_i -\text {length} (w_1)+d+1;$$
\item "iii)" $\Cal S$ has a $(d-1,i)$-column violation ($\text {for} \ d \ge 1 \ \text
{and} \ 1 \le i <k$) iff both
$$m_1 - \text {length}(w_i)=m_{i+1} - \text {length}(w_{i+1})+d-1$$
\endroster
and when words $w_i,w_{i+1}$ are inserted from the right on lines
$i,i+1$ of the Young diagram of $(m_1, \dots,m_k)$, there exists a
vertical pair $\left[\smallmatrix x \\ y \endsmallmatrix\right]$ 
with $x \ge y$.
\enddefinition

\remark{Remark}  When the words of $\Cal S$ are all inserted on their
respective lines as in iii), one obtains a skew tableau iff there
are no violations. In this case, we say that $\Cal S$ is
$(m_1,\dots,m_k) \text {\it {-innocent}}$. Since $s_{\mu/\la}=s
^\perp_\la(s_\mu)$, the following is therefore exactly the usual
statement of the Littlewood-Richardson rule.

\proclaim{Theorem}  $\text {For all partitions} \ \la \ \text {and} \
(m_1, \dots,m_k)$,
$$\tilde s ^\perp_\la(s_{m_1,\dots,m_k})=
\underset {\Cal S \text { is }(m_1,\dots,m_k)-\text
{innocent}}\to{\sum _ {\Cal S \in 
\LAT_k(\tilde \la)}}  s_{m_1-\text {length}(w_1),\dots,m_k-\text
{length}(w_k)}.$$
\endproclaim
\endremark

\demo{Proof}  The two previous propositions prove the analogous
equation where $\Cal S$ ranges over {\it all\/} of $\LAT_k(\tilde
\la)$. We must now prove that, in this sum, the terms coming from $\Cal S$
which have a violation add up to zero. For such $\Cal S$ choose the largest
$i$ for which there is a violation. If it is a bottom line violation
or a $(1,i)$-shape violation, the term itself is zero, by the remark after (5). 
The remaining such terms will cancel in pairs as follows,
using (4). Fix $d \ge 1, 1\le i <k$ and all the words of $\Cal S$ except
$w_i$ and $w_{i+1}$. Let $\Cal A$ be the content remaining to be used
for $w_i w_{i+1}$. Let $\Cal B$ be the submultiset of $\Cal A$ of the
form $\{j_1, j_1+1,\dots,j_p,j_p+1\}$ for which $\{j_1,\dots,j_p\}$
must occur in $w_i$, and $\{j_1+1,\dots,j_p+1\}$ in $w_{i+1}$ in
order that the lattice permutation condition should hold. We need
bijections from the set $V(\Cal A, \Cal B, q, r, d)$ of $\Cal S$ having a
$(d-1,i)$-column violation to the sets $M(\Cal A, \Cal B, q, r, d)$
having a $(d+1,i)$-shape violation; that is, bijections between sets of
two row integer arrays

\vskip10pt
\vbox{
\centerline{$V(\Cal A, \Cal B,q,r,d) $
\hskip4cm
$ M(\Cal A, \Cal B, q, r, d)$}
$$
\Einheit.3cm
\Pfad(0,0),11111111112111112\endPfad
\Pfad(0,0),21111121111111111\endPfad
\Pfad(1,0),2\endPfad
\Pfad(4,0),2\endPfad
\Pfad(5,0),2\endPfad
\Pfad(5,1),11111\endPfad
\Pfad(6,0),22\endPfad
\Pfad(9,0),22\endPfad
\Pfad(10,1),2\endPfad
\Pfad(11,1),2\endPfad
\Pfad(14,1),2\endPfad
\Label\ro{.}(1,0)
\Label\ro{.}(2,0)
\Label\ro{.}(3,0)
\Label\ro{.}(6,0)
\Label\ro{.}(7,0)
\Label\ro{.}(8,0)
\Label\ro{.}(6,1)
\Label\ro{.}(7,1)
\Label\ro{.}(8,1)
\Label\ro{.}(11,1)
\Label\ro{.}(12,1)
\Label\ro{.}(13,1)
\Label\o{\kern-10pt\overset d-1\to{\overbrace{\hbox{\hskip1.4cm}}}}(3,2)
\Label\o{\kern-10pt\overset r+1\to{\overbrace{\hbox{\hskip1.4cm}}}}(8,2)
\Label\o{\kern-10pt\overset q\to{\overbrace{\hbox{\hskip1.4cm}}}}(13,2)
\hbox{\hskip6cm}
\Pfad(0,1),111116111112111112\endPfad
\Pfad(0,1),2111111111111111\endPfad
\Pfad(1,1),2\endPfad
\Pfad(4,1),2\endPfad
\Pfad(5,0),22\endPfad
\Pfad(5,1),11111\endPfad
\Pfad(6,0),22\endPfad
\Pfad(9,0),22\endPfad
\Pfad(10,1),2\endPfad
\Pfad(11,1),2\endPfad
\Pfad(14,1),2\endPfad
\Label\ro{.}(1,1)
\Label\ro{.}(2,1)
\Label\ro{.}(3,1)
\Label\ro{.}(6,0)
\Label\ro{.}(7,0)
\Label\ro{.}(8,0)
\Label\ro{.}(6,1)
\Label\ro{.}(7,1)
\Label\ro{.}(8,1)
\Label\ro{.}(11,1)
\Label\ro{.}(12,1)
\Label\ro{.}(13,1)
\Label\o{\kern-10pt\overset d+1\to{\overbrace{\hbox{\hskip1.4cm}}}}(3,2)
\Label\o{\kern-10pt\overset r\to{\overbrace{\hbox{\hskip1.4cm}}}}(8,2)
\Label\o{\kern-10pt\overset q\to{\overbrace{\hbox{\hskip1.4cm}}}}(13,2)
\hskip4.5cm
$$}
\vskip10pt

\noindent
with weakly increasing rows, with at least one 
$\left[\smallmatrix x \\ y\\ \endsmallmatrix\right]$ 
with $x \ge y$ in $V$, and with content $\Cal A$, where $\Cal B$ is inserted as
mentioned above. Note that fixing $r$ and $q$ fixes the length of
$w_i$ and $w_{i+1}$, but differently for $V$ and $M$. When $p=0$, so
that $\Cal B$ is empty, a bijection, shown to me by Ian Goulden, is
given below, together with its inverse:

\vskip10pt
$$
\Einheit.35cm
\Pfad(0,0),11111111112111112\endPfad
\Pfad(0,0),21111121111111111\endPfad
\Pfad(5,1),11111\endPfad
\Pfad(8,0),22\endPfad
\Pfad(9,0),22\endPfad
\Label\ro{\circ}(0,0)
\Label\ro{\circ}(1,0)
\Label\ro{\circ}(2,0)
\Label\ro{\circ}(3,0)
\Label\ro{\circ}(4,0)
\Label\ro{\circ}(5,0)
\Label\ro{\circ}(6,0)
\Label\ro{\circ}(7,0)
\Label\ro{y}(8,0)
\Label\ro{x}(8,1)
\Label\ro{\circ}(9,1)
\Label\ro{\circ}(10,1)
\Label\ro{\circ}(11,1)
\Label\ro{\circ}(12,1)
\Label\ro{\circ}(13,1)
\Label\ro{\circ}(14,1)
\Label\ru{\uparrow}(8,0)
\Label\ru{\text{rightmost}}(8,-1)
\Label\ru{\text{for which $x\ge y$}}(8,-2)
\hbox{\hskip6cm}
\hbox{$\longleftrightarrow$}
\hbox{\hskip.5cm}
\Pfad(0,1),111116111112111112\endPfad
\Pfad(0,1),2111111111111111\endPfad
\Pfad(5,0),2\endPfad
\Pfad(5,1),11111\endPfad
\Pfad(8,0),22\endPfad
\Pfad(9,0),22\endPfad
\Label\ro{\circ}(0,1)
\Label\ro{\circ}(1,1)
\Label\ro{\circ}(2,1)
\Label\ro{\circ}(3,1)
\Label\ro{\circ}(4,1)
\Label\ro{\circ}(5,1)
\Label\ro{\circ}(6,1)
\Label\ro{\circ}(7,1)
\Label\ro{v}(8,0)
\Label\ro{u}(8,1)
\Label\ro{\circ}(9,1)
\Label\ro{\circ}(10,1)
\Label\ro{\circ}(11,1)
\Label\ro{\circ}(12,1)
\Label\ro{\circ}(13,1)
\Label\ro{\circ}(14,1)
\Label\ru{\uparrow}(8,0)
\Label\ru{\text{rightmost}}(10,-1)
\Label\ru{\text{for which $u\ge v$}}(10,-2)
\Label\ru{\uparrow}(5,0)
\Label\ru{\text{here if no}}(3,-1)
\Label\ru{\text{such $(u,v)$}}(3,-2)
\hskip5.3cm
$$
\vskip10pt

The general case can be reduced to the case $p=0$ as follows. If
$p>0$, adding and removing $\{j_p,j_p+1\}$ clearly gives mutually
inverse maps between $M(\Cal A, \Cal B, q, r, d)$ and $M(\Cal A', \Cal
B', q, r-1,d)$ where
$$\Cal A' = \Cal A \ \backslash \{j_p,j_p+1\} \hskip 1cm \text
{and} \hskip 1cm \Cal B' = \Cal B \ \backslash \{j_p,j_p+1\}.$$
The same holds for the sets $V(\Cal A, \Cal B, q, r, d)$ and $V(\Cal
A', \Cal B', q,r-1,d)$, noting that the existence of a pair $\left[
\smallmatrix
x \\ y \endsmallmatrix\right] $
with $x \ge y$ persists after the addition or removal. This
completes the proof.
\enddemo


\Refs\widestnumber\key{HH}

\ref\key H\by HOFFMAN (P.)  \paper A Bernstein-Type Formula for
Projective Representations of $S_n$ and $A_n$ \jour Adv. Math. 
\vol 74 \yr 1989\pages 135--143
\endref

\ref\key HH\by HOFFMAN (Peter) and HUMPHREYS (John)  \paper Hopf
Algebras and Projective Representations of $G\wr S\sb n$ and $G\wr A\sb n$
\jour Can.~ J.~ Math. \vol 38 \yr 1986 \pages 1380--1458 \endref

\ref\key Ma\by MACDONALD (I.G.) \book
Symmetric Functions and Hall Polynomials \publ Oxford Univ. Press
\publaddr Oxford \vol \yr 1979 \pages \endref

\ref\key Mu\by MURNAGHAN (F.D.) \paper The Analysis of the
Kronecker Product of Irreducible Representations of the Symmetric Group
\jour Amer.~J. of Math. 
\vol 60 \yr 1938 \pages 761--784 \endref

\ref\key Sa\by SAGAN (B.E.) \paper Shifted Tableaux, Schur
$Q$-Functions, and a Conjecture of R. Stanley \jour J.~ Comb. Theory,
Ser. A \vol 45 \yr 1987 \pages 62--103 \endref 

\ref\key St\by STEMBRIDGE (J.R.)  \paper Shifted Tableaux and the
Projective Representations of Symmetric Groups \jour Adv. Math. 
\vol  74 \yr 1989\pages 87--134 \endref

\ref\key W\by WORLEY (D.R.) \book A Theory of Shifted Tableaux, Ph.D. thesis
M.I.T \vol  \yr 1984 \pages \endref 

\ref\key Z\by ZELEVINSKI (Andrey V.) \book Representations of Finite
Classical Groups  \publ Lecture Notes in Math., vol.~869,
Springer-Verlag \publaddr Berlin \yr 1981 \endref 
\endRefs



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