\documentclass[12pt]{article} \input epsf.tex \usepackage{amssymb,latexsym,times} %\usepackage{amssymb,latexsym} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %\textheight 230mm \textwidth 150mm \hoffset -16mm %\voffset -16mm %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \catcode`\@=11 \renewcommand\subsection{\@startsection{subsection}{2}{\z@}% {-3.25ex\@plus -1ex \@minus -.2ex}% %{-1.5ex \@plus .2ex}% {-0.01 mm} {\normalfont\large\bfseries}} \newcommand{\doublebro}{[\hspace*{-.5mm}[} \newcommand{\doublebrf}{]\hspace*{-.5mm}]} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \catcode`\@=11 \renewcommand\subsubsection{\@startsection{subsubsection}{2}{\z@}% {-3.25ex\@plus -1ex \@minus -.2ex}% %{-1.5ex \@plus .2ex}% {-0.01 mm} {\normalfont\bfseries}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newtheorem{example}{Example}%[subsection] \newtheorem{theorem}[example]{Theorem}%[section] \newtheorem{corollary}[example]{Corollary}%[section] \newtheorem{definition}[example]{Definition}%[section] \newtheorem{proposition}[example]{Proposition}%[section] \newtheorem{lemma}[example]{Lemma}%[section] \newtheorem{remark}[example]{Remark}%[section] \newtheorem{conjecture}[example]{Conjecture} \def\resp{{\em resp.$\ $}} \def\etc{{\em etc.}} \def\proof{\medskip\noindent {\it Proof --- \ }} \def\finex{\hfill $\Diamond$} \def\Rem{\medskip\noindent {\it Remark --- \ }} \def\cqfd{\hfill $\Box$ \bigskip} \def\adots{\mathinner{\mkern2mu\raise1pt\hbox{.} \mkern3mu\raise4pt\hbox{.}\mkern1mu\raise7pt\hbox{.}}} \def\<{\langle\,} \def\>{\,\rangle} \def\fsf{k\!\not< \! \bb{A}\!\not>} \def\cf{{\it cf.$\ $}} \def\mat{{\rm Mat}} \def\hom{{\rm Hom}} \def\ext{{\rm Ext}} \def\End{{\rm End}} \def\shuf{\sqcup\!\sqcup} \def\ie{{\em i.e. }} \def\eg{{\em e.g. }} \def\tab{{\rm Tab\, }} \def\sym{{\rm Sym}} \def\inv{{\rm inv}} \def\SG{\mathfrak S} \def\HH{\mathfrak H} \def\SB{\mathfrak s} \def\csym{\hbox{\sl Sym}} \def\l{\lambda} \def\a{\alpha} \def\de{\delta} \def\b{\beta} \def\ga{\gamma} \def\i{\iota} \def\N{{\mathbb N}} \def\Z{{\mathbb Z}} \def\C{{\mathbb C}} \def\R{{\mathbb R}} \def\Q{{\mathbb Q}} \def\FF{{\mathbb F}} \def\T{{\cal T}} \def\F{{\cal F}} \def\P{{\cal P}} \def\B{{\mathbf B}} \def\PB{{\bf P}} \def\IB{{\bf I}} \def\FB{{\bf F}} \def\EAHB{\widehat{\bf H}} \def\BB{{\cal B}} \def\U{{\cal U}} \def\V{{\cal V}} \def\LL{{\mathfrak L}} \def\I{{\cal I}} \def\J{{\cal J}} \def\SS{{\cal S}} \def\Tn{{\cal T}^{(n)}} \def\sgn{{\rm sgn\, }} \def\tr{{\rm Tr}} \def\mod{\mbox{\ mod\ }} \def\ch{{\rm ch\, }} \def\wt{{\rm wt}} \def\g{\mathfrak g} \def\h{\mathfrak h} \def\gl{\mathfrak{gl}} \def\Sl{\mathfrak{sl}} \def\Rg{\mathfrak R} \def\n{\mathfrak n} \def\bb{\mathfrak b} \def\slchap{\widehat{\mathfrak{sl}}} \def\glchap{\widehat{\mathfrak{gl}}} \def\ASG{\widetilde{\mathfrak S}} \def\EASG{\widehat{\mathfrak S}} \def\H{\widehat H} \def\AH{\widetilde H} \def\A{{\cal A}} \def\AA{{\bf A}} \def\G{{\cal G}} \def\LL{{\cal L}} \def\L{\Lambda} \def\desc{{\rm desc}} \def\K{{\cal K}} \def\O{{\cal O}} \def\bar{\overline} \def\UT{\underline{\underline{\theta}}\,} \def\Fr{{\rm Fr}} \def\sp{{\rm spin}} \def\<{\langle} \def\>{\rangle} \def\op{{\rm op}} \def\Im{{\rm im\,}} \def\pr{{\rm pr\,}} \def\PP{{\ssym P}} \def\E{{\cal E}} \def\FI{{\bf F}_\infty} \def\HI{{\bf H}_\infty} \def\deg{{\rm deg}} \def\BB{{\cal B}} \def\CC{{\cal C}} \def\m{\mu} \def\mm{{\mathbf m}} \def\nn{{\bf n}} \def\p{{\bf p}} \def\q{{\bf q}} %def\a{{\bf a}} %def\b{{\bf b}} \def\s{{\bf s}} \def\t{{\bf t}} \def\M{{\cal M}} \def\RR{{\cal R}} \def\lt{{\widetilde{\lambda}}} \def\le{\leqslant} \def\ge{\geqslant} \def\veps{\varepsilon} \def\rad{{\rm rad}} \def\Si{\Sigma} \def\si{\sigma} \def\th{s} \def\vp{\varphi} \def\inte{\rm int} \def\ul{\underline{\l}} \def\um{\underline{\m}} \def\un{\underline{\nu}} \def\ua{\underline{\a}} \def\ub{\underline{\b}} \def\inf{\prec} \def\sup{\succ} \def\infe{\preceq} \def\supe{\succeq} \def\De{\Delta} \def\irr{{\rm Irr\,}} \def\con{{\rm Con}} \def\sy{{\rm Sy}} \def\ssy{{\rm SSy}} \def\pa{{\rm Pa}} \def\es{\emptyset} \def\ind{{\rm Ind}} \def\z{\zeta} \def\ra{\rightarrow} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % TABLEAUX MACROS %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newdimen\Squaresize \Squaresize=14pt \newdimen\Thickness \Thickness=0.5pt \def\Square#1{\hbox{\vrule width \Thickness \vbox to \Squaresize{\hrule height \Thickness\vss \hbox to \Squaresize{\hss#1\hss} \vss\hrule height\Thickness} \unskip\vrule width \Thickness} \kern-\Thickness} \def\Vsquare#1{\vbox{\Square{$#1$}}\kern-\Thickness} \def\blk{\omit\hskip\Squaresize} \def\young#1{ \vbox{\smallskip\offinterlineskip \halign{&\Vsquare{##}\cr #1}}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \title{\bf A Littlewood-Richardson rule for evaluation representations of $U_q(\slchap_n)$} \author{Bernard {\sc Leclerc}} \date{} \begin{document} \maketitle %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % ABSTRACT %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \vskip 1cm \begin{abstract}\noindent We give a combinatorial description of the composition factors of the induction product of two evaluation modules of the affine Iwahori-Hecke algebra of type $GL_m$. Using quantum affine Schur-Weyl duality, this yields a combinatorial description of the composition factors of the tensor product of two evaluation modules of the quantum affine algebra $U_q(\slchap_n)$. \end{abstract} %First page headline in AmS-LaTeX for S\'eminaire Lotharingien de Combinatoire %--first part \thispagestyle{myheadings} \font\rms=cmr8 \font\its=cmti8 \font\bfs=cmbx8 \markright{\its S\'eminaire Lotharingien de Combinatoire \bfs 50 \rms (2004), Article~B50e\hfill} \def\thepage{} % % \vskip 0.6cm %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % SECTION 1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Introduction} \label{SECT1} \subsection{}\label{1.1} Let $H_m$ denote the Iwahori-Hecke algebra of type $A_{m-1}$ over $\C(t)$. This is a semisimple associative algebra isomorphic to the group algebra $\C(t)[\SG_m]$ of the symmetric group. Hence its simple modules $S(\l)$ are parametrized by the partitions $\l$ of~$m$. Consider a decomposition $m=m_1+m_2$, and two partitions $\l^{(1)}$ and $\l^{(2)}$ of $m_1$ and $m_2$, respectively. Then we have a $H_{m_1}$-module $S(\l^{(1)})$ and a $H_{m_2}$-module $S(\l^{(2)})$, and we can form the induced module \[ S(\l^{(1)})\odot S(\l^{(2)}) := \ind_{H_{m_1}\otimes H_{m_2}}^{H_m}\left( S(\l^{(1)})\otimes S(\l^{(2)})\right). \] Here, $H_{m_1}\otimes H_{m_2}$ is identified to a subalgebra of $H_m$ in the standard way. Using again the isomorphism $H_m\cong\C(t)[\SG_m]$, we see that the multiplicity of a simple $H_m$-module $S(\m)$ in $S(\l^{(1)})\odot S(\l^{(2)})$ is equal to the classical Littlewood-Richardson coefficient $c_{\l^{(1)}\l^{(2)}}^\mu$ (see \eg \cite{Mcd}). \subsection{} Let now $\H_m$ be the affine Iwahori-Hecke algebra over $\C(t)$ (see \ref{evaluation} below). For each invertible $z\in\C(t)$ we have a surjective {\em evaluation homomorphism} $\tau_z : \H_m \ra H_m$. Pulling back the simple $H_m$-module $S(\l)$ via $\tau_z$ we obtain a simple $\H_m$-module $S(\l;z)$ called an {\em evaluation module}. In analogy with \ref{1.1}, given two invertible elements $z_1$ and $z_2$ of $\C(t)$, we can then form the induced $\H_m$-module \[ S(\l^{(1)};z_1)\odot S(\l^{(2)};z_2) := \ind_{\H_{m_1}\otimes \H_{m_2}}^{\H_m} \left( S(\l^{(1)};z_1)\otimes S(\l^{(2)};z_2)\right). \] It turns out that if we fix $\l^{(1)},\l^{(2)}$ and vary the spectral parameters $z_1, z_2$, this module is generically irreducible, that is, it is simple except for a finite number of values of the ratio $z_1/z_2$. In \cite[Theorem 36]{LNT} a combinatorial description of these special values was given. In this note we shall make this result more precise by describing all the composition factors of $S(\l^{(1)};z_1)\odot S(\l^{(2)};z_2)$ at these critical values $z_1/z_2$. We shall also prove that, in contrast with the classical Littlewood-Richardson rule, all the composition factors appear with multiplicity one. The composition factors occuring in a product will be described using the combinatorics of Lusztig's {\em symbols}, that is, of certain two-row arrays introduced by Lusztig for parametrizing the irreducible complex representations of the classical reductive groups over finite fields \cite{Lu79,Lu84}. \subsection{} We will derive our combinatorial formula from some explicit calculations of canonical bases in level 2 representations of the quantum algebra $U_v(\Sl_{n+1})$ performed in \cite{LM}. More precisely, by dualizing \cite[Theorem 3]{LM}, we get a formula for the expansion of the product of two quantum flag minors on the dual canonical basis of $U_v(\Sl_{n+1})$ (Theorem~\ref{th2}). Using then Ariki's theorem as in \cite{LNT}, we obtain immediately the above-mentioned Littlewood-Richardson rule for induction products of two evaluation modules over affine Hecke algebras (Theorem~\ref{th1}). Finally, by means of the quantum affine analogue of the Schur-Weyl duality developed by Cherednik, Chari-Pressley and Ginzburg-Reshetikhin-Vasserot, we can deduce from this rule a similar one for the tensor product of two evaluation modules over the quantum affine algebra $U_q(\slchap_N)$. %First page headline in AmS-LaTeX for S\'eminaire Lotharingien de Combinatoire %--restoring the headers and pagenumbering \pagenumbering{arabic} \addtocounter{page}{1} %\markboth{\SMALL TOUFIK MANSOUR}{\SMALL EQUIDISTRIBUTION AND SIGN-BALANCE %ON $132$-AVOIDING PERMUTATIONS} % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Composition factors of induced $\H_m$-modules} \label{SECT2} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{}\label{evaluation} Let $\H_m$ be the affine Hecke algebra of type $GL_m$ over $\C(t)$. It has invertible generators $T_1,\ldots ,T_{m-1},y_1,\ldots ,y_m$ subject to the relations \begin{eqnarray*} &&T_iT_{i+1}T_i=T_{i+1}T_iT_{i+1},\hskip 1.1cm (1\le i\le m-2),\label{EQ_T1}\\ &&T_iT_j=T_jT_i,\hskip 2.9cm (\vert i-j\vert>1),\label{EQ_T2}\\ &&(T_i-t)(T_i+1)=0, \hskip 1.6cm(1\le i\le m-1),\label{EQ_T3}\\ &&y_iy_j=y_jy_i, \hskip 3.1cm (1\le i,j\le m),\label{EQ_YY}\\ &&y_jT_i=T_iy_j, \hskip 3cm (j \not= i,i+1),\\ &&T_iy_iT_i=t\,y_{i+1}, \hskip 2.5cm (1\le i\le m-1).\label{EQ_Y} \end{eqnarray*} The subalgebra $H_m$ generated by the $T_i$'s is the Iwahori-Hecke algebra of type $A_{m-1}$. For any invertible $z\in\C(t)$ we have a unique algebra homomorphism $\tau_z : \H_m \ra H_m$ such that \[ \tau_z(T_i)=T_i,\quad \tau_z(y_1) = z,\qquad (i=1,\ldots,m-1). \] This is called the {\em evaluation at $z$}. We also have an algebra automorphism $\si_z : \H_m \ra \H_m$ such that \[ \si_z(T_i)=T_i,\quad \si_z(y_i) = zy_i,\qquad (i=1,\ldots,m-1). \] This is called the {\em shift by $z$}. \subsection{}\label{reduction} As mentioned in the introduction, given two partitions $\l^{(1)}$ and $\l^{(2)}$, the structure of the induced $\H_m$-module \[ S(\l^{(1)};z_1)\odot S(\l^{(2)};z_2) \] depends essentially on the ratio $z_1/z_2$. Indeed, by twisting this module with the shift automorphism $\si_z$ we obtain the induced module \[ S(\l^{(1)};zz_1)\odot S(\l^{(2)};zz_2). \] For example, it is known that if $z_1/z_2\not\in t^\Z$ then $S(\l^{(1)};z_1)\odot S(\l^{(2)};z_2)$ is irreducible. Therefore, we can assume without loss of generality that \begin{equation}\label{assumption} z_i=t^{a_i},\quad a_i\in\Z,\quad a_i\ge\ell(\l^{(i)}),\qquad (i=1,2), \end{equation} where as usual $\ell(\l)$ denotes the length of the partition $\l$. Since $S(\l^{(1)};z_1)\odot S(\l^{(2)};z_2)$ and $S(\l^{(2)};z_2)\odot S(\l^{(1)};z_1)$ have the same composition factors with the same multiplicities, we can also assume that $a_1\le a_2$. \subsection{} It will be convenient to write partitions in weakly {\em increasing} order. Given a partition $\l$ and an integer $a\ge\ell(\l)$ we can make $\l$ into a non-decreasing sequence $(\l_1,\ldots ,\l_a)$ of length $a$ by setting $\l_j=0$ for $j=1,\ldots ,a-\ell(\l)$. We can then associate to $(\l,a)$ the increasing sequence \begin{equation}\label{beta} \b=(\b_1,\ldots ,\b_a), \quad \b_j=j+\l_j. \end{equation} In this way, given $(\l^{(i)},a_i)\ (i=1,2)$ as in \ref{reduction}, we obtain a {\em symbol} \begin{equation}\label{symbol} S=\pmatrix{\b^{(2)} \cr \b^{(1)}} = \pmatrix{\b^{(2)}_1,\ldots ,\b^{(2)}_{a_2} \cr \b^{(1)}_1,\ldots ,\b^{(1)}_{a_1}}. \end{equation} For example, the symbol attached to the pairs $((1,1,2),3)$ and $((2,3),5)$ is \[ S=\pmatrix{1 &2 &3 &6 &8\cr 2& 3& 5}. \] Conversely, given a symbol $S$, \ie a two-row array as in Eq.~(\ref{symbol}) with \[ 1\le \b^{(i)}_1<\cdots<\b^{(i)}_{a_i}\quad (i=1,2), \] there is a unique pair $(\l^{(i)},a_i)\ (i=1,2)$ whose symbol is~$S$. \subsection{} The symbol $S$ of Eq.~(\ref{symbol}) is said to be {\em standard} if $\b^{(2)}_k \le \b^{(1)}_k$ for $k\le a_1$. In \cite[\S 2.5]{LM} we have defined the {\em pairs} of a standard symbol $S$, and the set $\CC(S)$ of all symbols $\Si$ obtained from $S$ by permuting some of its pairs. As shown in \cite[Lemma 9]{LM}, these notions are equivalent to the notion of admissible involution of Lusztig \cite{Lu02}. For the convenience of the reader we shall recall these definitions. Let $S={\b\choose\ga}$ be a standard symbol. We define an injection $\psi : \ga \longrightarrow \b$ such that $\psi(j)\le j$ for all $j\in\ga$. To do so it is enough to describe the subsets \[ \ga^l = \{j\in\ga \mid \psi(j)=j-l\},\qquad (0\le l \le n). \] We set $\ga^0 = \ga \cap \b$ and for $l\ge 1$ we put \[ \ga^l = \{j\in\ga - (\ga^0 \cup \cdots \cup \ga^{l-1}) \mid j-l\in \b-\psi(\ga^0 \cup \cdots \cup \ga^{l-1})\}. \] Observe that the standardness of $S$ implies that $\psi$ is well-defined. \begin{example} {\rm Take \[ S=\pmatrix{1 & 3 & 5 & 8 & 9 \cr 3 & 6 & 7 & 10}\,. \] Then \[ \ga^0 = \{3\},\ \ga^1 = \{6,10\},\ \ga^2=\cdots =\ga^5=\emptyset, \ \ga^6 = \{7\}. \] Hence \[ \psi(3)=3,\ \psi(6)=5,\ \psi(7)=1,\ \psi(10)=9. \] } \end{example} The pairs $(j,\psi(j))$ with $\psi(j)\not = j$ (that is, $j\not \in \b\cap\ga$) will be called the pairs of $S$. Given a standard symbol $S$ with $p$ pairs, we denote by $\CC(S)$ the set of all symbols obtained from $S$ by permuting some pairs in $S$ and reordering the rows. We consider $S$ itself as an element of $\CC(S)$, hence $\CC(S)$ has cardinality $2^p$. \subsection{}\label{young} Given a partition $\l$ and an integer $a$ we call {\em Young diagram of $(\l,a)$} the Young diagram of $\l$ in which each cell $(i,j)$ is filled with the integer $i-j+a$. For instance, if $\l=(2,3)$ and $a=5$ then the Young diagram of $(\l,a)$ is \[ \young{4 & 5\cr 5 & 6 & 7\cr} \] The rows of the Young diagram of $(\l,a)$ yield a {\em multisegment} \[ \mm(\l,a):=\sum_{1\le k\le a} [k,k+\l_k-1]. \] This is a formal sum (or multiset) of intervals in $\Z$, in which we discard the empty intervals corresponding to the $k$'s with $\l_k=0$. Thus, continuing with the same example, we have \[ \mm((2,3),5)=[4,5]+[5,7]. \] Similarly, we attach to a pair $(\l^{(i)},a_i)\ (i=1,2)$ or to its symbol $S$ the multisegment \[ \mm(S) = \mm(\l^{(1)},a_1) + \mm(\l^{(2)},a_2). \] \subsection{} To each multisegment \[ \mm:=\sum_k [\a_k,\b_k] \] is attached an irreducible $\H_m$-module $L_\mm$, where $m=\sum_k (\b_k+1-\a_k)$ (see \eg \cite[\S 2.1]{LNT}). \subsection{} Let us assume that the pair $(\l^{(i)},a_i)\ (i=1,2)$ satisfies the conditions of \ref{reduction}. Let $\Si$ denote the symbol attached to this pair. We can now state: \begin{theorem}\label{th1} The composition factors of $S(\l^{(1)};t^{a_1})\odot S(\l^{(2)};t^{a_2})$ are the modules $L_{\mm(S)}$ where $S$ runs through the set of standard symbols such that $\Si\in\CC(S)$. Each of them occurs with multiplicity one. \end{theorem} % Theorem~\ref{th1} will be deduced from Theorem~\ref{th2} below. \begin{example}\label{exa1} {\rm Let $(\l^{(1)},a_1)=((1,4),2)$ and $(\l^{(2)},a_2)=((1,2,3),4)$. The corresponding symbol is \[ \Si=\pmatrix{1 &3 &5 &7\cr 2& 6}. \] The standard symbols $S$ such that $\Si\in\CC(S)$ are \[ \pmatrix{1 &2 &5 &6\cr 3& 7},\quad \pmatrix{1 &2 &5 &7\cr 3& 6},\quad \pmatrix{1 &3 &5 &6\cr 2& 7},\quad \pmatrix{1 &3 &5 &7\cr 2& 6}. \] It follows that the composition factors of $S((1,4);t^2)\odot S((1,2,3);t^4)$ are the $L_\mm$ where $\mm$ is one the following multisegments: \[ \nn_1=[1,2]+[2,6]+[3,4]+[4,5],\quad \nn_2=[1,2]+[2,5]+[3,4]+[4,6],\quad \] \[ \nn_3=[1,1]+[2,2]+[2,6]+[3,4]+[4,5],\quad \nn_4=[1,1]+[2,2]+[2,5]+[3,4]+[4,6]. \] } \end{example} \subsection{} By restriction to the finite Hecke algebra $H_m$ the irreducible $\H_m$-modules $L_{\mm(S)}$ decompose into direct sums of Specht modules. The sum of all these Specht modules is given by the (classical) Littlewood-Richardson rule for the product $S(\l^{(1)})\odot S(\l^{(2)})$. It would be interesting to find a combinatorial description of the splitting of $S(\l^{(1)})\odot S(\l^{(2)})$ thus obtained. \begin{example}{\rm Let us continue Example~\ref{exa1}. The restrictions to $H_{11}$ of the 4 irreducible $\H_{11}$-modules are as follows: \begin{eqnarray*} L_{\nn_1}\downarrow &=& S(1,3,7) \oplus S(2,2,7) \oplus S(2,3,6) \oplus S(1,1,3,6)\\ && \oplus\ S(1,2,2,6) \oplus S(1,2,3,5) \oplus S(2,2,2,5),\\ L_{\nn_2}\downarrow &=& S(1,4,6)\oplus S(2,3,6)\oplus S(2,4,5) \oplus S(1,1,4,5) \oplus S(1,2,3,5)\\ &&\oplus\ S(1,2,4,4)\oplus S(3,3,5) \oplus S(1,3,3,4) \oplus S(2,2,3,4),\\ L_{\nn_3}\downarrow &=& S(1,1,2,7)\oplus S(1,2,2,6)\oplus S(1,1,1,2,6) \oplus S(1,1,2,2,5),\\ L_{\nn_4}\downarrow &=& S(1,1,3,6) \oplus S(1,2,3,5) \oplus S(1,1,1,3,5)\oplus S(1,1,2,3,4). \end{eqnarray*} This gives a splitting of $S(1,4) \odot S(1,2,3)$. } \end{example} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Canonical bases} \label{SECT3} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{} Fix $n \ge 2$ and let $\g=\Sl_{n+1}$. We consider the quantum enveloping algebra $U_v(\g)$ over $\Q(v)$ with Chevalley generators $e_j, f_j, t_j \ (1\le j \le n)$. The simple roots and the fundamental weights are denoted by $\a_k$ and $\L_k \ (1\le k \le n)$ respectively. The irreducible representation of $U_v(\g)$ with highest weight $\L$ is denoted by $V(\L)$. We denote by $U_v(\n)$ the subalgebra of $U_v(\g)$ generated by $e_j \ (1\le j \le n)$. \subsection{} Let $\B$ (\resp $\B^*$) denote the canonical basis (\resp the dual canonical basis) of $U_v(\n)$ (\cite{Lu90}, \cite{BZ}; see also \cite[\S 3]{LNT}). The elements of $\B$ and $\B^*$ are naturally labelled by the multisegments $\mm$ supported on $[1,n]$. We shall denote them by $b_\mm$ and $b^*_\mm$ respectively. The vectors $b^*_\mm$ for which $\mm$ is of the form \[ \mm = \mm(\l,a) \] for some partition $\l$ and some integer $a$ are called {\em quantum flag minors}. Indeed, by \cite{BZ}, they can be expressed as quantum minors of a triangular matrix whose entries are iterated brackets of the $e_i$'s (see \cite[\S 5.2]{LNT}). \subsection{} Let $(\l^{(i)},a_i)\ (i=1,2)$ be as in \ref{reduction}. We also assume that the multisegments \[ \mm_i = \mm(\l^{(i)},a_i)\quad (i=1,2) \] are supported on $[1,n]$. Let $\Si$ be the symbol attached to the pair $(\l^{(i)},a_i)\ (i=1,2)$. For a standard symbol $S$ such that $\Si\in\CC(S)$ we denote by $n(S,\Si)$ the number of pairs of $S$ which are permuted to get $\Si$. Finally, we denote by $N_j(\l,a)$ the number of cells of the Young diagram of $(\l,a)$ containing the integer~$j$. \begin{theorem}\label{th2} We have \[ b^*_{\mm_1}b^*_{\mm_2} = v^{-N_{a_1}(\l^{(2)},\,a_2)} \sum_S v^{n(S,\,\Si)}\,b^*_{\mm(S)} \] where the sum runs through all standard symbols $S$ such that $\Si\in\CC(S)$. \end{theorem} \begin{example}\label{exa2} {\rm We take $(\l^{(1)},a_1)$ and $(\l^{(2)},a_2)$ as in Example~\ref{exa1}. Hence \[ \mm_1 = [1,1]+[2,5],\qquad \mm_2=[2,2]+[3,4]+[4,6]. \] Then $N_{a_1}(\l^{(2)},\,a_2)=N_2((1,2,3),4)=1$, and we obtain, using the notation of Example~\ref{exa1}, \[ b^*_{\mm_1}\, b^*_{\mm_2} = v^{-1} (v^2\,b^*_{\nn_1} + v\,b^*_{\nn_2} + v\,b^*_{\nn_3} + b^*_{\nn_4}). \] } \end{example} \subsection{} {\it Proof of Theorem~\ref{th2}.} Following \cite[\S 7.2]{LNT}, we will replace calculations of products of elements of $\B^*$ by calculations of dual canonical bases of finite-dimensional representations of $U_v(\g)$. \subsubsection{} \label{part1} Let $U_v(\n^-)$ denote the subalgebra of $U_v(\g)$ generated by the $f_i$'s, and let $x \mapsto x^\sharp$ denote the algebra isomorphism from $U_v(\n)$ to $U_v(\n^-)$ defined by $e_i^\sharp = f_i\ (i=1,\ldots, n)$. Let $\L$ be a dominant integral weight and let $u_\L$ be a highest weight vector of the irreducible module $V(\L)$. Then the map $\pi_\L : x\mapsto x^\sharp u_\L$ projects the canonical basis $\B$ of $U_v(\n)$ to the union of the canonical basis $\B(\L)$ of $V(\L)$ with the set $\{0\}$. The dual map $\pi_\L^*$ gives an embedding of the dual canonical basis $\B^*(\L)$ of $V(\L)\simeq V(\L)^*$ into the dual canonical basis $\B^*$ of $U_v(\n)\simeq U_v(\n)^*$. \subsubsection{} \label{part2} In particular the subset of $\B^*$ obtained by embedding the bases $\B^*(\L_a)\ (1\le a\le n)$ of the fundamental representations is precisely the subset of quantum flag minors. It is well known that $V(\L_a)$ is a minuscule representation whose bases $\B^*(\L_a)$ and $\B(\L_a)$ coincide. Moreover the elements of these bases are naturally labelled by the pairs $(\l,a)$ whose Young diagram (as defined in \ref{young}) contains only cells numbered by integers between $1$ and $n$. Denoting them by $b^*_{(\l,a)}$ we have \[ \pi^*_{\L_a}(b^*_{(\l,a)}) = b^*_{\mm(\l,a)} \] Equivalently, we can also label the elements of $\B^*(\L_a)$ by one-row symbols $\b$ as in Eq.~(\ref{beta}) with $\b_i\le n+1$. \subsubsection{} \label{part3} Similarly, the basis $\B^*(\L_{a_1})\otimes \B^*(\L_{a_2})$ is naturally labelled by the set of symbols $S$ as in Eq.~(\ref{symbol}) with $\b^{(i)}_{a_i}\le n+1\ (i=1,2)$. Using the theory of crystal bases \cite{K1,K2} one can see that the basis $\B^*(\L_{a_1}+\L_{a_2})$ has a natural labelling by the subset of standard symbols \cite[\S 2.3]{LM}. Moreover, denoting by $b^*_S$ the element of $\B^*(\L_{a_1}+\L_{a_2})$ labelled by the standard symbol $S$ we have, using also the notation of \ref{young}, \[ \pi^*_{\L_{a_1}+\L_{a_2}}(b^*_S) = b^*_{\mm(S)}. \] \subsubsection{} \label{part4} Let $\i : V(\L_{a_1}+\L_{a_2}) \ra V(\L_{a_1})\otimes V(\L_{a_2})$ be the $U_v(\g)$-module embedding which maps $u_{\L_{a_1}+\L_{a_2}}$ to $u_{\L_{a_1}}\otimes u_{\L_{a_2}}$, and let $\i^* : V(\L_{a_1})\otimes V(\L_{a_2}) \ra V(\L_{a_1}+\L_{a_2})$ be its dual. Let $b^*_i\in \B^*(\L_{a_i})\ (i=1,2)$ and denote by $b^*_{\mm_i} = \pi^*_{\L_{a_i}}(b^*_i)\ (i=1,2)$ the corresponding quantum flag minors. It is shown in \cite[\S 7.2.7]{LNT} that the image of $b^*_1\otimes b^*_2$ under the composition of maps $\pi^*_{\L_{a_1}+\L_{a_2}} \circ \i^*$ coincides up to a power of $v$ with the product $b^*_{\mm_1}b^*_{\mm_2}$. Hence to calculate the $\B^*$-expansion of $b^*_{\mm_1}b^*_{\mm_2}$ it is enough to calculate the matrix of the map $\i^*$ with respect to the bases $\B^*(\L_{a_1})\otimes\B^*(\L_{a_1})$ and $\B^*(\L_{a_1}+\L_{a_2})$. \subsubsection{} \label{part5} The matrix of $\i$ with respect to the bases $\B(\L_{a_1}+\L_{a_2})$ and $\B(\L_{a_1})\otimes\B(\L_{a_1})$ was calculated in \cite[Theorem 3]{LM} in terms of Lusztig's symbols. Transposing this matrix we obtain the desired matrix of $\i^*$. Using \ref{part2} and \ref{part3}, we then get the formula of Theorem~\ref{th2}. \cqfd \subsection{} {\it Proof of Theorem~\ref{th1}.} By \cite[\S 3.7]{LNT} the multisegments $\mm$ indexing the composition factors $L_\mm$ of $S(\l^{(1)};t^{a_1})\odot S(\l^{(2)};t^{a_2})$ are those occuring in the right-hand side of the formula of Theorem~\ref{th2}. Moreover the composition multiplicities are obtained by specializing $v$ to $1$ in the coefficients of this formula. Hence they are all equal to $1$. \cqfd %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Tensor products of $U_q(\slchap_N)$-modules} \label{SECT4} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{} Let $U_q(\slchap_N)$ be the quantized affine algebra of type $A_{N-1}^{(1)}$ with parameter $q$ a square root of~$t$ (see for example \cite{CP} for the defining relations of $U_q(\slchap_N)$). The quantum affine Schur-Weyl duality between $\H_m$ and $U_q(\slchap_N)$ \cite{CP, Ch, GRV} gives a functor $\F_{m,N}$ from the category of finite-dimensional $\H_m$-modules to the category of level $0$ finite-dimensional representations of $U_q(\slchap_N)$. If $N\ge m$, $\F_{m,N}$ maps the simple modules of $\H_m$ to simple modules of $U_q(\slchap_N)$. However, the image of a non-zero simple $\H_m$-module may be the zero $U_q(\slchap_N)$-module. More precisely, the simple $\H_m$-module $L_\mm$ is mapped to a non-zero simple $U_q(\slchap_N)$-module if and only if all the segments occuring in $\mm$ have length $\le N-1$. In this case the Drinfeld polynomials of $\F_{m,N}(L_\mm)$ are easily calculated from $\mm$ (see \cite{CP}). The functor $\F_{m,N}$ transforms induction product into tensor product, that is, for $M_1$ in $\CC_{m_1}$ and $M_2$ in $\CC_{m_2}$ one has \[ \F_{m_1+m_2,N}(M_1\odot M_2) = \F_{m_1,N}(M_1)\otimes\F_{m_2,N}(M_2)\,. \] \subsection{} The image under $\F_{m,N}$ of an evaluation module for $\H_m$ is an evaluation module for $U_q(\slchap_N)$, and all evaluation modules of $U_q(\slchap_N)$ can be obtained in this way, by varying $m \in \N^*$. \subsection{} By application of the Schur functor $\F_{m,N}$ to Theorem~\ref{th1} we thus obtain a combinatorial description of all composition factors of the tensor product of two evaluation modules of $U_q(\slchap_N)$. \begin{example} {\rm We continue Example~\ref{exa1} and Example~\ref{exa2}. The image of the $\H_5$-module $L_{\mm_1}$ under $\F_{5,N}$ is the evaluation module $V(\mm_1)$ of $U_q(\slchap_N)$ with Drinfeld polynomials \begin{eqnarray*} P_1(u)& =& u - q^{-2},\\ P_2(u)& =& P_3(u)\ \ =\ \ 1, \\ P_4(u)& =& u - q^{-7},\\ P_k(u)& =& 1,\quad (5\le k\le N-1). \end{eqnarray*} This is a non-zero module if and only if $N\ge 5$. Similarly, the image of the $\H_6$-module $L_{\mm_2}$ under $\F_{6,N}$ is the evaluation module $V(\mm_2)$ of $U_q(\slchap_N)$ with Drinfeld polynomials \begin{eqnarray*} P_1(u) &=& u - q^{-4},\\ P_2(u) &=& u - q^{-7}, \\ P_3(u) &=& u - q^{-10},\\ P_k(u) &=& 1,\quad (4\le k\le N-1). \end{eqnarray*} This is a non-zero module if and only if $N\ge 4$. The images of the $\H_{11}$-modules $L_{\mm_1}, L_{\mm_2}, L_{\mm_3}$, $L_{\mm_4}$ under $\F_{11,N}$ are the modules $V(\nn_1), V(\nn_2), V(\nn_3), V(\nn_4)$ with respective Drinfeld polynomials \begin{eqnarray*} \qquad\quad\ P_1(u) &=& 1,\\ P_2(u) &=& (u-q^{-3})(u-q^{-7})(u-q^{-9}),\\ P_3(u) &=& P_4(u)\ \ =\ \ 1, \\ P_5(u) &=& u - q^{-8},\\ P_k(u) &=& 1,\quad (6\le k\le N-1); \end{eqnarray*} \begin{eqnarray*} P_1(u) &=& 1,\\ P_2(u) &=& (u-q^{-3})(u-q^{-7}),\\ P_3(u) &=& u-q^{-10}, \\ P_4(u) &=& u - q^{-7},\\ P_k(u) &=& 1,\quad (5\le k\le N-1); \end{eqnarray*} \begin{eqnarray*} P_1(u) &=& (u-q^{-2})(u-q^{-4}),\\ P_2(u) &=& (u-q^{-7})(u-q^{-9}),\\ P_3(u) &=& P_4(u)\ \ =\ \ 1, \\ P_5(u) &=& u - q^{-8},\\ P_k(u) &=& 1,\quad (6\le k\le N-1); \end{eqnarray*} \begin{eqnarray*} P_1(u) &=& (u-q^{-2})(u-q^{-4}),\\ P_2(u) &=& u-q^{-7},\\ P_3(u) &=& u-q^{-10}, \\ P_4(u) &=& u - q^{-7},\\ P_k(u) &=& 1,\quad (5\le k\le N-1). \end{eqnarray*} The modules $V(\nn_1)$ and $V(\nn_3)$ are non-zero only if $N\ge 6$. Hence $V(\mm_1)\otimes V(\mm_2)$ has only two composition factors $V(\nn_2)$ and $V(\nn_4)$ for $N=5$, and four composition factors $V(\nn_1), V(\nn_2)$, $V(\nn_3), V(\nn_4)$ for $N\ge 6$. } \end{example} \subsection{} We note that our result implies the following \begin{theorem}\label{th3} All composition factors of the tensor product of two evaluation modules of $U_q(\slchap_N)$ occur with multiplicity one. \end{theorem} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % BIBLIOGRAPHY %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %\newpage \bigskip \begin{thebibliography}{ABC} \scriptsize % %\bibitem[\bf Ar1]{Ar1}{\sc S. Ariki}, {\it On the decomposition numbers %of the Hecke algebra of $G(n,1,m)$}, J. Math. Kyoto Univ. % {\bf 36} (1996), 789--808. % \bibitem[\bf BZ]{BZ}{\sc A. Berenstein, A. Zelevinsky}, {\it String bases for quantum groups of type $A_r$}, Advances in Soviet Math. 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