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\controldates{3-OCT-2000,3-OCT-2000,3-OCT-2000,3-OCT-2000}
 
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\dateposted{October 5, 2000}
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\begin{document}

\title{A one-box-shift morphism between Specht modules}

%    Information for first author
\author{Matthias K\"unzer}
%    Address of record for the research reported here
\address{Fakult\"at f\"ur Mathematik, Universit\"at Bielefeld, Postfach 
100131, 33501 Bielefeld}
\email{kuenzer@mathematik.uni-bielefeld.de}

%    General info
\subjclass[2000]{Primary 20C30}

\date{July 14, 2000}

\commby{David J. Benson}

\keywords{Symmetric group, Specht module}

\begin{abstract}
We give a formula for a morphism between Specht modules over 
$(\mbox{\rm\bf Z}/m){\CMcal S}_n$, where $n\geq 1$, and where the 
partition indexing the
target Specht module arises from that indexing the source Specht module 
by a downwards shift of one box, $m$ being the box shift length. Our 
morphism can be reinterpreted integrally as an
extension of order $m$ of the corresponding Specht lattices.
\end{abstract}

\maketitle

\setcounter{section}{-1}

\section{Notation}

We write composition of maps on the right, $\lraa{\alpha}\lraa{\beta} = 
\lraa{\alpha\beta}$. Intervals are to be read as subsets of $\Z$. Let 
$n\geq 1$, let
$\Sl_n = \Aut_\mb{\scr Sets} [1,n]$ denote the symmetric group on $n$ 
letters and let $\eps_\sigma$ denote the sign of a permutation 
$\sigma\in\Sl_n$. Let
\[
\begin{array}{rcl}
\N & \lraa{\lambda} & \N_0 \\
i  & \lra           & \lambda_i
\end{array}
\]
be a {\it partition} of $n$, i.e.\ assume $\sum_i \lambda_i = n$ and 
$\lambda_i \geq \lambda_{i+1}$ for $i\in \N$. Let
\[
[\lambda] := \{ i\ti j \in\N\ti\N \; |\; j \leq\lambda_i \}
\]
denote the {\it diagram} of $\lambda$. We say that $i\ti j\in 
[\lambda]$ lies in row $i$ and in column $j$. A {\it $\lambda$-tableau} 
is a bijection
\[
\begin{array}{rcl}
[\lambda] & \lraisoa{[a]} & [1,n] \\
i\ti j    & \lra          & a_{i,j}.
\end{array}
\]
The element $\sigma\in\Sl_n$ acts on the set $T^\lambda$ of 
$\lambda$-tableaux via composition $[a] \lraa{\sigma} [a]\sigma$. Let 
$F^\lambda$ be the free $\Z$-module on $T^\lambda$ with the induced
operation of $\Sl_n$. Let
\[
\begin{array}{rclcrcl}
[\lambda]  & \lraa{\rho} & \N & \hspace*{2cm} & [\lambda] & 
\lraa{\kappa} & \N \\
i\ti j     & \lra        & i  &               & i\ti j    & \lra        
  & j  
\end{array}
\]
denote the projections. We denote by $\{ a\} := [a]^{-1}\rho$ the {\it 
$\lambda$-tabloid} associated to the $\lambda$-tableaux $[a]$. The free 
$\Z$-module on the
set of tabloids, equipped with the inherited $\Sl_n$-operation, is 
denoted by $M^\lambda$. Let
\[
C_{[a]} := \{ \sigma\in\Sl_n \; |\; [a]^{-1}\kappa = 
([a]\sigma)^{-1}\kappa \}
\]
be the {\it column stabilizer} of $[a]$. Let the {\it Specht lattice} 
$S^\lambda$ be the $\Z\Sl_n$-sublattice of $M^\lambda$ generated over 
$\Z$ by the {\it $\lambda$-polytabloids}
\[
\spi{a} := \sum_{\sigma\in C_{[a]}} \{ a\}\sigma\eps_\sigma.
\]
Let $\lambda'$ denote the {\it transposed partition} of $\lambda$, 
i.e.\ $j\leq \lambda_i \equ i\ti j\in [\lambda] \equ i\leq\lambda'_j$.

\section{Carter-Payne}

Let $d\in [1,n]$ be the number of shifted boxes. Let $1\leq s < t\leq n$, 
$s$ being the row of $[\lambda]$ from which the boxes are shifted, and $t$ 
being the row
into which the boxes are shifted. Suppose
\[
\mu_i :=
\left\{
\begin{array}{ll}
\lambda_i - d & \mb{ for } i = s, \\
\lambda_i + d & \mb{ for } i = t, \\
\lambda_i     & \mb{ else}
\end{array}
\right.
\]
defines a partition of $n$. Let the {\it box shift length} be denoted by
\[
m := (\lambda_s - s) - (\lambda_t - t) - d.
\]
Let $m[p] := p^{v_p(m)}$ be the $p$-part of $m$. Using \cite{CL74}, 
{\sc Carter} and {\sc Payne} proved the following

\begin{theorem}[\cite{CP80}]
Let $K$ be an infinite field of characteristic $p$. Suppose $d < m[p]$. 
Then
\[
\Hom_{K\Sl_n}(K\ts_{\sZ}S^\lambda,K\ts_{\sZ}S^{\mu}) \neq 0.
\]
\end{theorem}

\section{Integral reinterpretation}

Assume $d = 1$, i.e.\ $[\mu]$ arises from $[\lambda]$ by a 
one-box-shift. The condition $d < m[p]$ translates into $p|m$.

As we will see below, this particular case of the result of {\sc 
Carter} and {\sc Payne} already holds over $K = \F_p$. So we obtain a 
nonzero element in
\[
\Hom_{\sZ\Sl_n}(S^\lambda/p S^\lambda, S^\mu/p S^\mu) \llaiso 
\Hom_{\sZ\Sl_n}(S^\lambda, S^\mu/p S^\mu).
\]
We consider a part of the long exact 
$\Ext^\ast_{\sZ\Sl_n}(S^\lambda,-)$-sequence on
\[
0\lra S^\mu\lraa{p} S^\mu \lra S^\mu/pS^\mu\lra 0,
\]
viz.\
%\[
%\begin{array}{rrccccl}
%0 & \lra & \ub{\Hom_{\sZ\Sl_n}(S^\lambda,S^\mu)}_{=\; 0} & \lraa{p} & 
%%\ub{\Hom_{\sZ\Sl_n}(S^\lambda,S^\mu)}_{=\; 0} & \lra & 
%%\Hom_{\sZ\Sl_n}(S^\lambda,S^\mu/pS^\mu)  \\
  %& \lra & \Ext^1_{\sZ\Sl_n}(S^\lambda,S^\mu)            & \lraa{p} & 
%%\Ext^1_{\sZ\Sl_n}(S^\lambda,S^\mu). \\
%\end{array}
%\]
\begin{eqnarray}
& 0 & \lra  \ub{\Hom_{\sZ\Sl_n}(S^\lambda,S^\mu)}_{=\; 0} \lraa{p} 
\ub{\Hom_{\sZ\Sl_n}(S^\lambda,S^\mu)}_{=\; 0} \lra  
\Hom_{\sZ\Sl_n}(S^\lambda,S^\mu/pS^\mu)  \nonumber\\
  & \; & \lra  \Ext^1_{\sZ\Sl_n}(S^\lambda,S^\mu)  \lraa{p} 
\Ext^1_{\sZ\Sl_n}(S^\lambda,S^\mu).\nonumber
\end{eqnarray}
Mapping our morphism into $\Ext^1$, we obtain a nonzero element of 
$\Ext^1_{\sZ\Sl_n}(S^\lambda,S^\mu)$ which is annihilated by $p$. 
Conversely, the $p$-torsion elements of $\Ext^1$
are given by morphisms modulo $p$.

Since $n!$ annihilates $\Ext^1_{\sZ\Sl_n}(S^\lambda,S^\mu)$, 
replacement of $p$ by $n!$ shows that any element in $\Ext^1$ is given 
by a modular morphism modulo $n!$,
\[
\Hom_{\sZ\Sl_n}(S^\lambda,S^\mu/n!\, S^\mu)\lraiso 
\Ext^1_{\sZ\Sl_n}(S^\lambda,S^\mu).
\]
Therefore, in order to get hold of the whole $\Ext^1$, we need to 
calculate modulo prime powers in general.

\section{One-box-shift formula}

We keep the assumption $d = 1$. Let $s' := \lambda_s$ and let $t' := 
\lambda_t + 1$. A {\it path} of length $l\in [1,s'-t']$ is a map
\[
\begin{array}{rcl}
[0,l] & \lraa{\gamma} & [\lambda]\cup [\mu] \\
k     & \lra          & \alpha_k \ti \beta_k
\end{array}
\]
such that $k < k'$ implies $\beta_k < \beta_{k'}$, and such that 
$\alpha_0\ti\beta_0 = t \ti t'$ and $\beta_l = s'$. For a 
$\lambda$-tableau $[a]$, we define the
$\mu$-tableau $[a^\gamma]$ by
\[
\begin{array}{rcll}
a^\gamma_{i,j}               & := & a_{i,j}                       & 
\mbox{ for } i\ti j\in [\mu]\ohne (\gamma([1,l]) \cup \N\ti \{ s'\}), \\
a^\gamma_{\alpha_k,\beta_k}  & := & a_{\alpha_{k+1},\beta_{k+1}}  & 
\mbox{ for } k\in [0,l-1], \\
a^\gamma_{i,s'}              & := & a_{i,s'}                      & 
\mbox{ for } i < \alpha_l, \\
a^\gamma_{i,s'}              & := & a_{i+1,s'}                    & 
\mbox{ for } i \geq \alpha_l.
\end{array}
\]
For $i\in [t'+1,s'-1]$, we denote
\[
X_i := (s' - \lambda'_{s'}) - (i - \lambda'_i).
\]
Let
\[
x_\gamma := (-1)^{\alpha_l+1}\fracd{\prod_{i\in [t'+1,s'-1],\; \mu'_i > 
\mu'_{i+1}} X_i}{\prod_{k\in [1,l-1]} X_{\beta_k}}.
\]
Let $\Gamma$ be the set of paths of some length $l\in [1, s'-t']$.

\begin{theorem}[\cite{K99}, 4.3.31, cf.\ 0.7.1] The abelian group 
$\Hom_{\sZ\Sl_n}(S^\lambda,S^\mu/mS^\mu)$ contains an element $f$ of 
order $m = (\lambda_s - s) - (\lambda_t - t) - 1$ which
is given by the commutative diagram of $\Z\Sl_n$-linear maps
\begin{center}
\begin{picture}(250,350)
\put(   0, 270){$[a]$}
\put(  50, 280){\vector(1,0){130}}
\put( 200, 270){$\sum_{\gamma\in\Gamma}x_\gamma\ts\spi{a^\gamma}$}
\put( -10, 200){$F^\lambda$}
\put(  50, 210){\vector(1,0){130}}
\put( 200, 200){$\Q\ts_{\sZ} S^\mu$}
\put(-100, 200){$[a]$}
\put( -80, 180){\vector(0,-1){130}}
\put(-100,   0){$\spi{a}$}
\put(  10, 180){\vector(0,-1){130}}
\put( 210, 140){\vector(0,1){40}}
\put( 200, 100){$S^\mu$}
\put( 210,  80){\vector(0,-1){40}}
\put(  50, 190){\vector(2,-1){130}}
\put( 400, 100){$\spi{b}$}
\put( 420,  80){\vector(0,-1){40}}
\put( 400,   0){$\spi{b} + mS^\mu$.}
\put( 420, 140){\vector(0,1){40}}
\put( 400, 200){$1\ts\spi{b}$}
\put( -10,   0){$S^\lambda$}
\put(  50,  10){\vector(1,0){130}}
\put( 110,  20){$\scm f$}
\put( 200,   0){$S^\mu/m S^\mu$}
\end{picture}
\end{center}
\end{theorem}

Reducing modulo a prime dividing $m$, this recovers the case $d = 1$ of 
the result of {\sc Carter} and {\sc Payne}. By the long exact sequence 
as above, but with $p$ replaced by $m$, we obtain a
nonzero element in $\Ext^1_{\sZ\Sl_n}(S^\lambda,S^\mu)$ of order $m$. 
\footnote{I do not know the structure of 
$\Ext^1_{\ssZ\Sl_n}(S^\lambda,S^\mu)$ as an abelian group. At least 
{\it in case
$n\leq 7$,} direct computation yields that the projection of our 
element to its $2'$-part generates this $2'$-part. We have, however, 
for example
$\Ext^1_{\ssZ\Sl_6}(S^{(4,1^2)},S^{(3,1^3)})_{(2)}\iso\Z/2\ds\Z/2$.}

The proof of this theorem proceeds by showing that a sufficient set of 
Garnir relations in $F^\lambda$ is annihilated by $F^\lambda\lra 
S^\mu/m S^\mu$.

\section{Example}

Let $n = 9$, $\lambda = (4,3,2)$, $\mu = (3,3,2,1)$, $t' = 1$ and $s' = 
4$, whence $m = 6$, $X_2 = 4$, $X_3 = 2$. We obtain a morphism of order 
$6$ that maps
\[
\begin{array}{rcl}
S^{(4,3,2)} & \lraa{f} & S^{(3,3,2,1)}/6\; S^{(3,3,2,1)} \\
\la\begin{array}{llll}
1 & 4 & 7 & 9 \\
2 & 5 & 8 &   \\
3 & 6 &   &  
\end{array}\rl
& \lra
& \;\;\; 4^0 2^0
\setlength{\arraycolsep}{1pt}
\left(
\la\begin{array}{ccc}
1        & \fbox{7} & \fbox{9} \\
2        & 5        & 8 \\
3        & 6        &   \\
\fbox{4} &          &   \\
\end{array}\rl
+
\la\begin{array}{ccc}
1        & 4        & \fbox{9} \\
2        & \fbox{7} & 8 \\
3        & 6        &   \\
\fbox{5} &          &   \\
\end{array}\rl
+
\la\begin{array}{ccc}
1        & 4        & \fbox{9} \\
2        & 5        & 8 \\
3        & \fbox{7} &   \\
\fbox{6} &          &  
\end{array}\rl \right. \vs \\
& & \hspace*{10mm} \left. +
\setlength{\arraycolsep}{1pt}
\la\begin{array}{ccc}
1        & \fbox{8} & 7 \\
2        & 5        & \fbox{9} \\
3        & 6        &   \\
\fbox{4} &          &   
\end{array}\rl
+
\la\begin{array}{ccc}
1        & 4        & 7 \\
2        & \fbox{8} & \fbox{9} \\
3        & 6        &   \\
\fbox{5} &          &  
\end{array}\rl
+
\la\begin{array}{ccc}
1        & 4        & 7 \\
2        & 5        & \fbox{9} \\
3        & \fbox{8} &   \\
\fbox{6} &          &  
\end{array}\rl
\right)\vs \\
& &  + 4^1 2^0
\setlength{\arraycolsep}{1pt}
\left(
\la\begin{array}{ccc}
1        &  \: 4 & \: \fbox{9} \\
2        &  \: 5 & \: 8 \\
3        &  \: 6 &   \\
\fbox{7} &       &  
\end{array}\rl
+
\la\begin{array}{ccc}
1        &  \: 4 & \: 7 \\
2        &  \: 5 & \: \fbox{9} \\
3        &  \: 6 &   \\
\fbox{8} &       &  
\end{array}\rl
\right)\vs\\
& &  + 4^0 2^1
\setlength{\arraycolsep}{1pt}
\left(
\la\begin{array}{ccc}
1        & \fbox{9} & \: 7 \\
2        & 5        & \: 8 \\
3        & 6        &      \\
\fbox{4} &          &     
\end{array}\rl
+
\la\begin{array}{ccc}
1        & 4        & \: 7 \\
2        & \fbox{9} & \: 8 \\
3        & 6        &      \\
\fbox{5} &          &     
\end{array}\rl
+
\la\begin{array}{ccc}
1        & 4        & \: 7 \\
2        & 5        & \: 8 \\
3        & \fbox{9} &      \\
\fbox{6} &          &      
\end{array}\rl
\right)\vs\\
& &  + 4^1 2^1
\setlength{\arraycolsep}{1pt}
\left(
\la\begin{array}{ccc}
1        & \: 4 & \;\; 7 \\
2        & \: 5 & \;\; 8 \\
3        & \: 6 &        \\
\fbox{9} &      &        
\end{array}\rl
\right).\\
\end{array}
\]
The $[0,l-1]$-part of the respective path is highlighted.

\section{Motivation}

We consider the rational Wedderburn isomorphism
\[
\begin{array}{rcl}
\Q\Sl_n & \lraiso & \prod_\lambda (\Q)_{n_\lambda\ti n_\lambda} \\
\sigma  & \lra    & (\rho^\lambda_\sigma)_\lambda
\end{array}
\]
where $\lambda$ runs over the partitions of $n$ and where 
$\rho^\lambda_\sigma$ denotes the matrix describing the operation of 
$\sigma\in\Sl_n$ on $S^\lambda$ {\it with respect to a chosen
tuple of integral bases.} The restriction
\[
\Z\Sl_n\hra \prod_\lambda (\Z)_{n_\lambda\ti n_\lambda}
\]
of this isomorphism, viewed as an embedding of abelian groups, has 
index \footnote{{\bf Question.} {\it Given a central primitive 
idempotent
$e^\lambda$ of $\Gamma := \prod_\lambda (\Z)_{n_\lambda\ti n_\lambda}$, 
what is the index of $e^\lambda\Z\Sl_n$ in $e^\lambda\Gamma$ ?} Cf.\ 
(\cite{K99}, Section 1.1.3).}
\[
\prod_\lambda \left(\frac{n!}{n_\lambda}\right)^{n_\lambda^2/2}.
\]
In particular, for $n\geq 2$ it is no longer an isomorphism.

Suppose, for partitions $\lambda$ and $\mu$ of $n$ and for some modulus 
$m\geq 2$, we are given a $\Z\Sl_n$-linear map
\[
S^\lambda\lraa{g} S^\mu/m S^\mu.
\]
Let $G$ be the matrix, with respect to the chosen integral bases of 
$S^\lambda$ and $S^\mu$, of a lifting of $g$ to a $\Z$-linear map 
$S^\lambda\lra S^\mu$. The $\Z\Sl_n$-linearity of
$g$ reads
\[
G \rho^\mu_\sigma - \rho^\lambda_\sigma G  \in m(\Z)_{n_\lambda\ti 
n_\mu} \hspace*{1cm} \mb{for all $\sigma\in\Sl_n$.}
\]
Thus such a morphism yields a {\it necessary} condition for a tuple of 
matrices to lie in the image of the Wedderburn embedding.

For example, the evaluations of our one-box-shift morphism at hook 
partitions, i.e.\ at $\lambda = (k,1^{n-k})$ and $\mu = (k-1,1^{n-k+
1})$, $k\in [2,n]$, furnish
a long exact sequence. In the (simple) case of $n = p$ prime, and 
localized at $(p)$, the set of necessary conditions imposed by these 
morphisms already turns out to be sufficient for a tuple of
matrices over $\Z_{(p)}$ to lie in the image of the localized 
Wedderburn embedding (\cite{K99}, Section 4.2.1). Therefore, it is advisable 
to chose a tuple of locally integral bases adapted to this long
exact sequence. For instance, we obtain
\[
\begin{array}{rcl}
\Z_{(3)}\Sl_3 & \lraiso & \left\{ a\ti\mateckzz{b}{c}{d}{e}\ti f 
\;\Big|\; a\con_3 b,\; d\con_3 0,\; e\con_3 f\right\} \\
& \tm & \Z_{(3)}\ti \mateckzz{\Z_{(3)}}{\Z_{(3)}}{\Z_{(3)}}{\Z_{(3)}} 
\ti \Z_{(3)}, 
\end{array}
\]
the embedding {\it not} being written in the combinatorial standard 
polytabloid bases.

For an approach to the general case, see (\cite{K99}, Chapters 3 and 5). 
Further examples may be found in (\cite{K99}, Chapter 2).

\section{Acknowledgments}

This result being part of my thesis, I'd like to repeat my thanks to my 
advisor {\sc Steffen K\"onig.}

\bibliographystyle{amsplain}
\begin{thebibliography}{10}

\bibitem{CL74} R.\ W.\ Carter and G.\ Lusztig, \textit{On the modular 
representations of the general linear and symmetric groups}, Math.\ Z.\ 
\textbf{136} (1974), 139--242. 
\MR{50:7364}
\bibitem{CP80} R.\ W.\ Carter and M.\ T.\ J.\ Payne, \textit{On 
homomorphisms between Weyl modules and Specht modules}, Math.\ Proc.\ 
Camb.\ Phil.\ Soc.\ \textbf{87} (1980), 419--425.
\MR{81h:20048}
\bibitem{J78} G.\ D.\ James, \textit{The representation theory of the 
symmetric groups}, SLN 682, 1978.
\MR{80g:20019}
\bibitem{K99} M.\ K\"unzer, \textit{Ties for the $\Z\Sl_n$}, thesis, 
http://www.mathematik.uni-bielefeld.de/$\scm\sim$kuenzer, Bielefeld, 
1999.

\end{thebibliography}

\end{document}