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%_ ************************************************************************** %_ * The TeX source for AMS journal articles is the publishers TeX code * %_ * which may contain special commands defined for the AMS production * %_ * environment. Therefore, it may not be possible to process these files * %_ * through TeX without errors. To display a typeset version of a journal * %_ * article easily, we suggest that you retrieve the article in DVI, * %_ * PostScript, or PDF format. * %_ ************************************************************************** % Author Package file for use with AMS-LaTeX 1.2 \controldates{3-OCT-2000,3-OCT-2000,3-OCT-2000,3-OCT-2000} \documentclass{era-l} \issueinfo{6}{11}{}{2000} \dateposted{October 5, 2000} \pagespan{90}{94} \PII{S 1079-6762(00)00085-8} \copyrightinfo{2000}{American Mathematical Society} %\newcommand{\copyrightyear}{2000} %\copyrightinfo{2000}{American Mathematical Society} %\copyrightinfo[2000]{American Mathematical Society} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{xca}[theorem]{Exercise} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \numberwithin{equation}{section} % personal environment \unitlength0.1mm \usepackage{euscript} \newcommand{\Z}{\mbox{\rm\bf Z}} \newcommand{\Ext} {\mbox{\rm Ext} } \newcommand{\Sl}{\CMcal{S}} \newcommand{\lraa}[1]{\begin{picture}(80,50) \put( 0, 0){$\lra$} \put( 10, 27){\makebox[6mm]{$\scm #1$}} \end{picture} } \newcommand{\Aut} {\mbox{\rm Aut}} \newcommand{\mb}[1]{{\mbox{#1}}} \newcommand{\N}{\mbox{\rm\bf N}} \newcommand{\lra}{\begin{picture}(80,17) \put(10,10){\vector(1,0){60}} \end{picture}} \newcommand{\ti}{\times} \newcommand{\scm}{\scriptstyle} \newcommand{\scr}{\scriptsize} \newcommand{\sscr}{\tiny} \newcommand{\spi}[1]{\la #1\rl} \newcommand{\eps}{\varepsilon} \newcommand{\equ}{\Longleftrightarrow} \newcommand{\la}{\langle} \newcommand{\rl}{\rangle} \newcommand{\Hom} {\mbox{\rm Hom} } \newcommand{\ts}{\otimes} \newcommand{\sZ}{\mbox{\rm\scr\bf Z}} \newcommand{\ssZ}{\mbox{\rm\sscr\bf Z}} \newcommand{\F}{\mbox{\rm\bf F}} \newcommand{\llaiso}{\begin{picture}(80,30) \put( 0, 0){$\lla$} \put( 30, 15){$\scm\sim$} \end{picture} } \newcommand{\ub}{\underbrace} \newcommand{\ohne}{\backslash} \newcommand{\fracd}[2]{{\dis\frac{#1}{#2}}} \newcommand{\iso}{\simeq} \newcommand{\ds}{\oplus} \newcommand{\Q}{\mbox{\rm\bf Q}} \newcommand{\lraiso}{\begin{picture}(80,30) \put( 0, 0){$\lra$} \put( 25, 15){$\scm\sim$} \end{picture} } \newcommand{\hra}{\begin{picture}(100,17) \put(25,5){\vector(1,0){50}} \put(25,15){\oval(20,20)[l]} \end{picture}} \newcommand{\con}{\equiv} \newcommand{\mateckzz}[4]{ \left[ \begin{array}{cc} #1 & #2 \\ #3 & #4 \\ \end{array} \right] } \newcommand{\tm}{\subseteq} \newcommand{\lla}{\begin{picture}(80,17) \put(70,10){\vector(-1,0){60}} \end{picture}} \newcommand{\dis}{\displaystyle} \newcommand{\lraisoa}[1]{\begin{picture}(80,40) \put( 0, 0){$\lra$} \put( 10, 25){\makebox[6mm]{$\scm #1$}} \put( 25, -8){$\scm\sim$} \end{picture} } \newcommand{\vs}{\vspace*{3mm}} % end of personal environment \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}