Multisegments

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 09 April 2012

Multisegments

$$\begin{array}{lc}{A}_{n-1}& \begin{array}{c} 2 3 n-1 n \end{array}\\ A_{\infty }& \begin{array}{c} -2 -1 0 1 2 \end{array}\\ A_{n-1}& \begin{array}{c} 0 2 3 n-1 n \end{array}\end{array}$$

Let $$I=\{\begin{array}{ll}\{1,2,...,l-1\},& in\; type\; A_{l-1},\\ \mathbb{Z},& in\; type\; A_{\infty },\\ \mathbb{Z}/l\mathbb{Z},& in\; type\; A_{l-1}^{\left(1\right)}.\end{array}\phantom{\}}$$ The elements of $I$ index the (isomorphism classes) of simple representations of the quiver.

Consider a sheet of graph paper with diagonals indexed by $\mathbb{Z}.$ The content $c\left(b\right)$ of a box $b$ on this sheet of graph paper is $$c\left(b\right)=\text{the diagonal number of the box} b.$$ Let $$[i;d)=[i,i+d-1]=(d;i+d-1]=\begin{array}{c} i i+d-1 \end{array}$$ denote a sequence of boxes in a row which has length $d$ with the leftmost box of content $i$ and the rightmost box of content $i+d-1.$ The set of segments is $$R^{+}=\{\begin{array}{ll}\left\{[i;d) \right| i\in I, 1\le d\le l-i\},& in\; type\; A_{l-1},\\ \left\{[i;d) \right| i\in I, d\in {\mathbb{Z}}_{\ge 0}\},& in\; type\; A_{\infty },\\ \left\{[i;d) \right| i\in I, d\in \mathbb{Z}/l\mathbb{Z}\},& in\; type\; A_{l-1}^{\left(1\right)}.\end{array}\phantom{\}}$$ The elements of $R^{+}$ index the (isomorphism classes) of indecomposable (nilpotent) representations of the quiver.

A multisegment is a (unordered) collection of segments, i.e. an elements of $$\tilde{B}\left(\infty \right)=\sum _{\alpha \in R^{+}}{\mathbb{Z}}_{\ge 0}\alpha .$$ For example $$\begin{array}{ll}\begin{array}{c} 3 4 5 6 7 3 4 5 6 7 5 6 7 1 2 3 4 5 3 4 5 \end{array}=2[3;5)+[5;3)+[1;5)+[3;3)& (MS\; 1)\end{array}$$ (the numbers in the boxes in the picture are the contents of the boxes).

A multisegment is aperiodic if it does not contain $$[0;d)+[1;d)+\cdots +[l-1;d),for\; any\; d\in {\mathbb{Z}}_{>0}.$$ Pictorially, a multisegment is aperiodic if it does not contain a box of height $l.$ Let $$B\left(\infty \right)=\left\{aperiodic\; multisegments\right\}.$$ In types $A_{l-1}$ and $A_{\infty },$ $B\left(\infty \right)=\tilde{B}\left(\infty \right).$ The elements of $B\left(\infty \right)$ index the isomorphism classes of nilpotent representations of the quiver.

The partial order

Consider an (infinite) sheet of graph paper which has its diagonals labeled consecutively by $$ ...,-2,-1,0,1,2,....$$ The content $c\left(b\right)$ of a box $b$ on this sheet of graph paper is $$c\left(b\right)=the\; diagonal\; number\; of\; the\; box\; b.$$ A segment is a row of boxes on a sheet of graph paper with diagonals indexed by $\mathbb{Z}.$

Consider a graph paper with diagonals indexed by $\mathbb{Z}.$ A segment is a sequence of boxes $$[i,j]=\begin{array}{c} i j \end{array}$$ in a row with the leftmost box of content $i$ and the rightmost box of content $j.$ A multisegment is a (unordered) collection of segments. For example $$\begin{array}{c} 3 4 5 6 7 3 4 5 6 7 5 6 7 1 2 3 4 5 3 4 5 \end{array}has\lambda \left([3,7]\right)=2, \lambda \left([5,7]\right)=1, \lambda \left([1,5]\right)=1, \lambda \left([3,5]\right)=1,$$ and $\lambda \left([i,j]\right)=0$ for all other segments $[i,j]$ (the numbers in the boxes in the picture are the contents of the boxes). Alternatively a multisegment $\lambda$ can be viewed as a function $$\lambda :\left\{segments\right\}\to {\mathbb{Z}}_{\ge 0}where\lambda \left([i,j]\right)=(\# \; of\; rows\; [i,j] \; in\; \lambda ).$$ The set of segments is ordered by inclusion. Define $$\begin{array}{ll}\lambda \left(\supseteq [i,j]\right)=\sum _{[r,s]\supseteq [i,j]}\lambda \left([r,s]\right).& (MS\; 2)\end{array}$$ Then $$\lambda \left([i,j]\right)=\lambda \left(\supseteq [i-1,j+1]\right)-\lambda \left(\supseteq [i-1,j]\right)-\lambda \left(\supseteq [i,j+1]\right)+\lambda \left(\supseteq [i,j]\right)$$

and so the multisegment $\lambda$ can be specified by the numbers $\lambda \left(\supseteq [i,j]\right).$ Note that $$\lambda \left(\supseteq \left[i\right]\right)=(\# \; of\; boxes\; in\; \lambda \; in\; diagonal\; i).$$ Define a partial order on multisegments by $$\lambda \ge \mu if\lambda \left(\supseteq [i,j]\right)\ge \mu \left(\supseteq [i,j]\right)for\; all\; segments\; [i,j].$$

If $[b,c]\subseteq [a,d]$ are segments define a degeneration $$R_{[b,c],[a,d]}:\left\{multisegments\right\}\to \left\{multisegments\right\}$$ by $$\begin{array}{rcl}{R}_{[b,c],[a,d]}\lambda \left([a,d]\right)& =& \lambda \left([a,d]\right)-1,\\ R_{[b,c],[a,d]}\lambda \left([b,d]\right)& =& \lambda \left([b,d]\right)+1,\\ R_{[b,c],[a,d]}\lambda \left([a,c]\right)& =& \lambda \left([a,c]\right)+1,\\ R_{[b,c],[a,d]}\lambda \left([b,c]\right)& =& \lambda \left([b,c]\right)-1,\\ R_{[b,c],[a,d]}\lambda \left([i,j]\right)& =& \lambda \left([i,j]\right),if\; [i,j]\ne [a,d],[a,c],[b,d],[b,c].\end{array}$$ The degeneration $R_{[b,c],[a,d]}\lambda$ is elementary if $$\lambda \left([i,j]\right)=0 \; for\; all\; [b,c]\subseteq [i,j]\subseteq [a,d] \; except\; [i,j]=[b,c],[a,c],[b,d] \; or\; [a,d].$$ Pictorially a degeneration takes $$\mathrm{PICTURE}\to \mathrm{PICTURE}$$ and $$\mathrm{PICTURE}\to \mathrm{PICTURE}for\; c=b-1,$$ or, equivalently, $$\mathrm{PICTURE}\to \mathrm{PICTURE}.$$

Let $A_{\infty }$ be the quiver $(I,{\Omega }^{+})$ with $$I=\mathbb{Z},{\Omega }^{+}=\{i\to i+1 | i\in \mathbb{Z}\}.$$ Fix an $I-$graded vector space $$V=\underset{i\in I}{⨁}{V}_i,$$ and let $$\begin{array}{rcl}{E}_V& =& \underset{i\to i+1}{⨁}\mathrm{Hom}(V_i,V_{i+1}),\\ G{L}_V& =& \prod _{i\in \mathbb{Z}}GL\left(V_i\right),which\; acts\; on\; E_V, \; and\\ {𝒩}_V& =& \{x\in E_V | x \; is\; a\; nilpotent\; element\; of\; \mathrm{Hom}(V,V)\}.\end{array}$$ The map $$\begin{array}{ccc}{𝒩}_V& \to & \left\{multisegments\right\}\\ & x& \mapsto & {\lambda }_x\end{array}given\; by$$ $$\lambda (\supseteq \left[i\right])=\dim \left(V_i\right)and\lambda \left(\supseteq [i,j]\right)=\mathrm{rank}(\lambda :V_i\to \cdots \to V_j)$$ provides a bijection $$\left\{multisegments\; \lambda \right| \lambda \left(\supseteq \left[i\right]\right)=\dim \left(V_i\right)\}\leftrightarrow \left\{G{L}_V \; orbits\; in\; {𝒩}_V\right\}.$$

Let $\lambda$ and $\mu$ be multisegments and let ${𝕆}_{\lambda }$ and ${𝕆}_{\mu }$ be the corresponding orbits in ${𝒩}_V/G{L}_V.$ Then the following are equivalent:

1. $\lambda \ge \mu ,$
2. $\stackrel{\_}{{𝕆}_{\lambda }}\supseteq {𝕆}_{\mu },$
3. $\lambda =R_{i_1}\cdots R_{i_r}\mu$ for some sequence of elementary degenerations $R_{i_1},...,R_{i_r}.$

		Proof.
	(1)⇒(2): $$\mathrm{PICTURE}+\epsilon \mathrm{PICTURE}\cong \mathrm{PICTURE},$$ and so $${𝕆}_{\mathrm{PICTURE}}\subseteq \stackrel{\_}{{𝕆}_{\mathrm{PICTURE}}}.$$ (2)⇒(3): If ${𝕆}_{\mu }\subseteq \stackrel{\_}{{𝕆}_{\lambda }}$ then $$\mu \left(\supseteq [i,j]\right)=\mathrm{rank}(\mu :V_i\to \cdots \to V_j)\le \mathrm{rank}(\lambda :V_i\to \cdots \to V_j)=\lambda \left(\supseteq [i,j]\right).$$ (3)⇒(1): Assume $\lambda \left(\supseteq [i,j]\right)\ge \mu \left(\supseteq [i,j]\right)$ for all segments $[i,j].$ Find (THIS STILL NEEDS DOING) a sequence $R_{i_1}\cdots R_{i_r}$ of elementary degenerations which takes $\mu$ to $\lambda ,$ i.e. $$R_{i_1}\cdots R_{i_r}\mu =\lambda .$$ $\square$

Hecke algebra representations

Let $\widetilde{H_k}$ be the affine Hecke algebra at an $l^{\mathrm{th}}$ root of unity so that $q^l=1$ (all $l=\infty$ if desired). For each $b\in \tilde{B}\left(\infty \right)$ let $$b=\sum _j[s_j,n_j)and\; define\; the\; \text{standard module}M\left(b\right)={\mathrm{Ind}}_{{\tilde{H}}_{\nu }}^{{\tilde{H}}_k}\left({ℂ}_s\right),$$ where $\nu =(n_1,...,n_r)$ and $k=n_1+\cdots +n_r.$ The simple ${\tilde{H}}_k-$modules are indexed by $b\in B\left(\infty \right)$ and are determined by the equations $$\left[M\left(b\right)\right]=\left[L\left(b\right)\right]+\sum _{\genfrac{}{}{0ex}{}{b\prime >b}{b\prime \in B\left(\infty \right)}}{d}_{b\prime b}\left[L(b\prime )\right],b\in B\left(\infty \right),d_{b\prime b}\in {\mathbb{Z}}_{\ge 0},$$ in the Grothendieck group of ${\tilde{H}}_k\left(q\right)-$modules.

The Fock space representation of $U_v{\widehat{\mathrm{𝔰𝔩}}}_l$

The crystal graph

Let $$\lambda =\left[\begin{array}{cccc}{(\lambda +\rho )}_1& {(\lambda +\rho )}_2& \cdots & {(\lambda +\rho )}_n\\ {(\mu +\rho )}_1& {(\mu +\rho )}_2& \cdots & {(\mu +\rho )}_n\end{array}\right]=\left(\begin{array}{cccc}{(\lambda +\rho )}_1& {(\lambda +\rho )}_2& \cdots & {(\lambda +\rho )}_n\\ d_1& d_2& \cdots & d_n\end{array}\right]$$ be a multisegment and assume that it is ordered so that

1. ${(\lambda +\rho )}_i\ge {(\lambda +\rho )}_{i+1},$
2. ${(\mu +\rho )}_i\le {(\mu +\rho )}_{i+1}$ if ${(\lambda +\rho )}_i={(\lambda +\rho )}_{i+1}.$

These conditions are equivalent to saying that

1. The $\mathrm{𝔤𝔩}\left(n\right)-$weight $\lambda$ is integrally dominant,
2. $\mu =w\circ \nu$ where $\nu$ is integrally dominant and $w$ is longest in its coset $W_{\lambda +\rho }w{W}_{\mu +\rho }.$

Place

1. $-1$ above each ${(\lambda +\rho )}_j=i,$
2. $+1$ above each ${(\lambda +\rho )}_j=i-1,$
3. $0$ above each ${(\lambda +\rho )}_j\ne i,i-1.$

Then, ignoring 0s, read the sequence of +1s, -1s left to right and successively cancel adjacent $(-1,+1)$ pairs to get a sequence of the form $$\begin{array}{rl}cogood& good\\ \downarrow & \downarrow \\ {\displaystyle \underset{conormal\; modes}{\underbrace{+1 +1 \cdots +1}}}& {\displaystyle \underset{normal\; modes}{\underbrace{-1 -1 \cdots -1}}}\end{array}$$ The -1s in this sequence are the normal nodes and the +1s are the conormal nodes. The good node is the leftmost normal node and the cogood node is the rightmost conormal node.

Define $$\mathrm{wt}\left(\lambda \right)=\sum _{i\in I}-\left(number\; of\; boxes\; of\; content\; i \; in\; \lambda \right){\alpha }_i,and$$ $${\epsilon }_i\left(\lambda \right)=\left(number\; of\; normal\; nodes\right),{\phi }_i\left(\lambda \right)=\left(number\; of\; conormal\; nodes\right),$$ $$\begin{array}{rcl}{\tilde{e}}_i\lambda & =& (same\; as\; \lambda \; but\; with\; the\; good\; node\; {(\lambda +\rho )}_j=i \; changed\; to\; i-1),\\ {\tilde{f}}_i\lambda & =& (same\; as\; \lambda \; but\; with\; the\; cogood\; node\; {(\lambda +\rho )}_j=i-1 \; changed\; to\; i),\end{array}$$ for each $i\in I.$

If this algorithm is being executed where $I=\mathbb{Z}/l\mathbb{Z}$ then take

1. ${(\lambda +\rho )}_j=l,$ when $i=0$ and ${(\lambda +\rho )}_j\equiv 0 \mathrm{mod}l,$ and
2. ${(\lambda +\rho )}_j=0,$ when $i=1$ and ${(\lambda +\rho )}_j\equiv 0 \mathrm{mod}l.$

1. In type $A_{l-1}^{\left(1\right)},$ $B\left(\infty \right)$ is the connected component of $\varnothing$ in the crystal graph $\tilde{B}\left(\infty \right).$
2. $B\left(\infty \right)$ is the crystal graph of $U_v^{-}𝔤.$

The crystals $B\left(\Lambda \right)$

Type $A_{l-1}:$ Let $$\lambda =\sum _{i=1}^l{\lambda }_i{ϵ}_i=\sum _{i\in I}{\gamma }_i{\omega }_i\in P^{+},$$ and identify $\lambda$ with the partition which has ${\lambda }_i$ boxes in row $i.$ Let $$B\left(\lambda \right)=\left\{column\; strict\; tableaux\; of\; shape\; \lambda \right\}$$ and define an imbedding $$\begin{array}{rcl}B\left(\lambda \right)& \to & B\left(\infty \right)\\ P& \mapsto & \left[\begin{array}{cccccccccc}1& 1& \cdots & 1& 2& 2& \cdots & 2& \cdots & n\\ n& \cdots & n& & & & & & & \\ i_1& i_2& \cdots & i_{{\lambda }_1}& i_{{\lambda }_1+1}& & \cdots & i_{{\lambda }_1+{\lambda }_2}& \cdots & \\ & \cdots & i_k& & & & & & & \end{array}\right]\end{array}$$ where the entries $i_1{i}_2\cdots i_k$ are the entries of $P$ read in Arabic reading order.

The tensor product representation

The $l-$dimensional simple $U_q{\mathrm{𝔰𝔩}}_l-$module of highest weight ${\omega }_1$ is given by $$L\left({\omega }_1\right)=ℂ-span\{v_0,...,v_{l-1}\}$$ with $U_q{\mathrm{𝔰𝔩}}_l-$action $$e_i{v}_j=\{\begin{array}{ll}{v}_{i-1},& if\; j=i,\\ 0,& if\; j\ne i,\end{array}\phantom{\}}{f}_i{v}_j=\{\begin{array}{ll}{v}_i,& if\; j=i-1,\\ 0,& if\; j\ne i-1,\end{array}\phantom{\}}{k}_i{v}_j=\{\begin{array}{ll}q{v}_{i-1},& if\; j=i,\\ q^{-1}{v}_i,& if\; j=i-1,\\ v_j,& if\; j\ne i,i-1.\end{array}\phantom{\}}$$ Then $$L{\left({\omega }_1\right)}^{\otimes k}=ℂ-span\{v_{j_1}\otimes \cdots \otimes v_{j_k} | 1\le j_1,j_2,...,j_k\le l\}.$$ If $v=v_{j_1}\otimes \cdots \otimes v_{j_k}$ place

1. +1 over each $v_{i-1}$ in $v,$
2. -1 over each $v_i$ in $v,$
3. 0 over each $v_j,$ $j\ne i,i-1.$

Then the $U_q{\mathrm{𝔰𝔩}}_l-$action on $L{\left({\omega }_1\right)}^{\otimes k}$ is given by $$\begin{array}{rcl}{e}_i\left(v\right)& =& \sum _{v^{-}}{q}^{-(sum\; of\; \pm 1s\; before\; v/v^{-})}{v}^{-},\\ f_i\left(v\right)& =& \sum _{v^{+}}{q}^{-(sum\; of\; \pm 1s\; after\; v^{+}/v)}{v}^{+},\\ k_i\left(v\right)& =& q^{(sum\; of\; \pm 1s\; for\; v)}v,\end{array}$$ where the first sum is over all $v^{-}$ which are obtained from $v$ by changing a $v_i$ to $v_{i-1}$ and the second sum is over all $v^{+}$ which are obtained from $v$ by changing a $v_{i-1}$ to $v_i.$

The Fock space

Let $\mu \in {𝔥}^{*}$ for ${\mathrm{𝔤𝔩}}_n.$ Define $${ℱ}_{\mu }=ℂ-span\{multisegments\; \lambda =\lambda /\mu \}.$$ Define an action of $U_v{\widehat{\mathrm{𝔰𝔩}}}_l$ on ${ℱ}_{\mu }$ by $$\begin{array}{rcl}{e}_{\lambda }& =& \sum _{c(\lambda /{\lambda }^{-})\equiv i}{q}^{(sum\; of\; \pm 1s\; before\; \lambda /{\lambda }^{-})}{\lambda }^{-},\\ f_i\lambda & =& \sum _{c({\lambda }^{+}/\lambda )\equiv i}{q}^{(sum\; of\; \pm 1s\; after\; {\lambda }^{+}/\lambda )}{\lambda }^{+},\\ k_i\lambda & =& q^{(sum\; of\; the\; \pm 1 \; sequence\; for\; \lambda )}\lambda ,\\ D\lambda & =& q^{(\# \; of\; boxes\; of\; content\; 0\; in\; \lambda )}\lambda .\end{array}$$

1. These formulas make ${ℱ}_{\mu }$ into a $U_v{\widehat{\mathrm{𝔰𝔩}}}_l-$module.
2. If $L_{\mu }=\mathbb{Z}[q,q^{-1}]-span\{multisegments\; \lambda =\lambda /\mu \}$ so that the multisegments form a $\mathbb{Z}[q,q^{-1}]$ basis of $L_{\mu }$ then $${\tilde{e}}_i\left[\lambda \right]=\left[{\tilde{e}}_i\lambda \right] \mathrm{mod}q{L}_{\mu }and{\tilde{f}}_i\left[\lambda \right]=\left[{\tilde{f}}_i\lambda \right] \mathrm{mod}q{L}_{\mu }.$$

		Proof.
	The permutations of the sequence $+1,+1,...,+1,-1,-1,...,-1$ are indexed by the elements of $S_t/S_k\times S_{t-k}$ where $t$ is the number of nodes after $(-1,+1)$ pairing. The group ${(\mathbb{Z}/2\mathbb{Z})}^p$ acts on the $(-1,+1)$ pairs by changing a pair $(-1,+1)$ to $(+1,-1).$ For each $1\le k\le r$ define $$u_k=\sum _{\sigma \in S_t/S_k\times S_{t-k}}\sum _{\tau \in {(\mathbb{Z}/2\mathbb{Z})}^t}{q}^{l\left(\sigma \right)}{(-1)}^{l\left(\tau \right)}\left(\sigma \tau \lambda \left[k\right]\right).$$ Then $$u_k=\lambda \left[k\right] \mathrm{mod}q{L}_{\mu }and{e}_i{u}_k=\left[k\right]u_{k-1}.$$ The first statement is clear. To obtain the second statement $$\begin{array}{rcl}{e}_i{u}_k& =& \sum _{\sigma \in S_t/S_k\times S_{t-k}}\sum _{\tau \in {(\mathbb{Z}/2\mathbb{Z})}^t}{q}^{l\left(\sigma \right)}{(-1)}^{l\left(\tau \right)}\left(e_i\sigma \tau \lambda \left[k\right]\right)\\ & =& \sum _{e_i \; changes\; a\; pair}\sum _{\sigma \in S_t/S_k\times S_{t-k}}\sum _{\tau \in {(\mathbb{Z}/2\mathbb{Z})}^t}{q}^{l\left(\sigma \right)}{(-1)}^{l\left(\tau \right)}{\left(\sigma \tau \lambda \left[k\right]\right)}^{-}\\ & & +\sum _{e_i \; changes\; a\; node}\sum _{\sigma \in S_t/S_k\times S_{t-k}}\sum _{\tau \in {(\mathbb{Z}/2\mathbb{Z})}^t}{q}^{l\left(\sigma \right)}{(-1)}^{l\left(\tau \right)}{\left(\sigma \tau \lambda \left[k\right]\right)}^{-}\\ & =& 0+\sum _{\tau \in {(\mathbb{Z}/2\mathbb{Z})}^t}\sum _{\sigma \in S_t/S_k\times S_{t-k}}\sum _{e_i \; changes\; a\; node}{q}^{l\left(\sigma \right)}{(-1)}^{l\left(\tau \right)}{\left(\sigma \tau \lambda \left[k\right]\right)}^{-}\\ & =& \sum _{\tau \in {(\mathbb{Z}/2\mathbb{Z})}^t}\left[k\right]\left(\sum _{\sigma \in S_t/S_k\times S_{t-k}}{q}^{l\left(\sigma \right)}{(-1)}^{l\left(\tau \right)}{\left(\sigma \tau \lambda \left[k\right]\right)}^{-}\right)\end{array}$$ $\square$

A Schur-Weyl duality connection to affine Hecke algebras

A multisegment is a collection of rows of boxes (segments) placed on graph paper. We can label this multisegment by a pair of weights $\lambda ={\lambda }_1{\epsilon }_1+\cdots +{\lambda }_{n+1}{\epsilon }_{n+1}$ and $\mu ={\mu }_1{\epsilon }_1+\cdots +{\mu }_{n+1}{\epsilon }_{n+1}$ by setting $$\begin{array}{rcl}{(\lambda +\rho )}_i& =& content\; of\; the\; last\; box\; in\; row\; i,and\\ {(\mu +\rho )}_i& =& \left(content\; of\; the\; last\; box\; in\; row\; i\right)-1.\end{array}$$ For example $$\begin{array}{ll}\begin{array}{c} 3 4 5 6 7 3 4 5 6 7 5 6 7 1 2 3 4 5 3 4 5 \end{array}corresponds\; to\begin{array}{rcl}\lambda +\rho & =& \left(7,7,7,5,5\right) \; and\\ \mu +\rho & =& \left(2,2,4,0,2\right)\end{array}& (MS\; 3)\end{array}$$ (the numbers in the boxes in the picture are the contents of the boxes). The construction forces the condition

1. ${(\lambda +\rho )}_i-{(\mu +\rho )}_i\in {\mathbb{Z}}_{\ge 0},$
   and since we want to consider unordered collections of boxes it is natural to take the following pseudo-lexicographic ordering on the segments,
2. ${(\lambda +\rho )}_i\ge {(\lambda +\rho )}_{i+1},$
3. ${(\mu +\rho )}_i\le {(\mu +\rho )}_{i+1}$ if ${(\lambda +\rho )}_i={(\lambda +\rho )}_{i+1},$

when we denote the multisegment $\lambda /\mu$ by a pair of weights $\lambda ,\mu .$ In terms of weights the conditions (a), (b) and (c) can be restated as (note that in this case both $\lambda$ and $\mu$ are integral)

1. $\lambda -\mu$ is a weight of $V^{\otimes k},$ where $k$ is the number of boxes in $\lambda /\mu ,$
2. $\lambda$ is integrally dominant,
3. $\mu =w\circ \nu$ with $\nu$ integrally dominant and $w$ maximal length in the coset $W_{\lambda +\rho }w{W}_{\nu +\rho }.$

Let $\lambda /\mu$ be a multisegment with $k$ boxes and number the boxes of $\lambda /\mu$ from left to right (like a book). Define $${\tilde{H}}_{\lambda /\mu }= \; subalgebra\; of\; {\tilde{H}}_k \; generated\; by\; \{X^{\lambda },T_j | \lambda \in L, {\mathrm{box}}_j \; is\; not\; at\; the\; end\; of\; its\; row\},$$ so that ${\tilde{H}}_{\lambda /\mu }$ is the "parabolic" subalgebra of ${\tilde{H}}_k$ corresponding to the multisegment $\lambda /\mu .$ Define a one-dimensional ${\tilde{H}}_{\lambda /\mu }$ module ${ℂ}_{\lambda /\mu }=ℂv_{\lambda /\mu }$ by setting $$\begin{array}{ll}{X}^{{\epsilon }_i}{v}_{\lambda /\mu }=q^{2c\left({\mathrm{box}}_i\right)}{v}_{\lambda /\mu },and{T}_j{v}_{\lambda /\mu }=q{v}_{\lambda /\mu },& (MS\; 4)\end{array}$$ for $1\le i\le k$ and $j$ such that ${\mathrm{box}}_j$ is not at the end of its row.

Let $𝔤$ be of type $A_n$ and let $F_{\lambda }$ be the functor ${\mathrm{Hom}}_{U_h𝔤}(M\left(\lambda \right),\cdot \otimes V^{\otimes k})$ where $V=L\left({\omega }_1\right).$ The standard module for the affine Hecke algebra ${\tilde{H}}_k$ is $$\begin{array}{ll}{ℳ}^{\lambda /\mu }=F_{\lambda }\left(M\left(\mu \right)\right)& (MS\; 5)\end{array}$$ as defined in (4.1). It follows from the above discussion that these modules are naturally indexed by multisegments $\lambda /\mu .$ The following proposition shows that this standard module coincides with the usual standard module for the affine Hecke algebra as considered by Zelevinsky [Ze2] (see also [Ar], [CG] and [KL]).

Let $\lambda /\mu$ be a multisegment determined by a pair of weights $(\lambda ,\mu )$ with $\lambda$ integrally dominant. Let ${ℂ}_{\lambda /\mu }$ be the one dimensional representation of the parabolic subalgebra ${\tilde{H}}_{\lambda /\mu }$ of the affine Hecke algebra ${\tilde{H}}_k$ defined in (???). Then $${ℳ}^{\lambda /\mu }\cong {\mathrm{Ind}}_{{\tilde{H}}_{\lambda /\mu }}^{{\tilde{H}}_k}\left({ℂ}_{\lambda /\mu }\right).$$

		Proof.
	To remove the constants that come from the difference between ${\mathrm{𝔤𝔩}}_n$ and ${\mathrm{𝔰𝔩}}_n$ the affine braid group action in Theorem 6.17a should be normalized so that ${\Phi }_k\left(X^{{\epsilon }_1}\right)=q^{2\left|\mu \right|/(n+1)}{\check{R}}_0^2$ and ${\Phi }_k\left(T_i\right)=q^{\frac{1}{(n+1)}}{\check{R}}_i.$ By Proposition 4.3a, $c{M}^{\lambda /\mu }\cong {\left(V^{\otimes k}\right)}_{\lambda -\mu }$ as a vector space. Let $\{v_1,v_2,...,v_{n+1}\}$ be the standard basis of $V=L\left({\omega }_1\right)$ with $\mathrm{wt}\left(v_i\right)={\epsilon }_i.$ If we let the symmetric group $S_k$ act on $V^{\otimes k}$ by permuting the tensor factors then $${\left(V^{\otimes k}\right)}_{\lambda -\mu }=\mathrm{span-}\{\pi \cdot v^{\otimes (\lambda -\mu )} | \pi \in S_k\}=\mathrm{span-}\{\pi \cdot v^{\otimes (\lambda -\mu )} | \pi \in S_k/S_{\lambda -\mu }\},$$ where $$v^{\otimes (\lambda -\mu )}={\displaystyle \underset{{\lambda }_1-{\mu }_1}{\underbrace{v_1\otimes \cdots \otimes v_1}}}\otimes \cdots \otimes {\displaystyle \underset{{\lambda }_n-{\mu }_n}{\underbrace{v_n\otimes \cdots \otimes v_n}}}and{S}_{\lambda -\mu }=S_{{\lambda }_1-{\mu }_1}\times \cdots \times S_{{\lambda }_n-{\mu }_n}$$ is the parabolic subgroup of $S_k$ which stabilizes the vector $v^{\otimes (\lambda -\mu )}\in V^{\otimes k}.$ This shows that, as vector spaces, $$\begin{array}{ll}{ℳ}^{\lambda /\mu }\cong {\mathrm{Ind}}_{{\tilde{H}}_{\lambda /\mu }}^{{\tilde{H}}_k}\left({ℂ}_{\lambda /\mu }\right)=\mathrm{span-}\{T_{\pi }\otimes v_{\lambda /\mu } | \pi \in S_k/S_{\lambda -\mu }\}& (MS\; 6)\end{array}$$ are isomorphic. For notational purposes let $$b_{\lambda /\mu }=v_{\mu }^{+}\otimes v^{\otimes (\lambda -\mu )}=v_{\mu }^{+}\otimes v_{i_1}\otimes \cdots \otimes v_{i_k}$$ and let ${\stackrel{\_}{b}}_{\lambda /\mu }$ be the image of $b_{\lambda /\mu }$ in ${(M\otimes V^{\otimes k})}^{\left[\lambda \right]}.$ Since $\lambda$ is integrally dominant and ${\stackrel{\_}{b}}_{\lambda /\mu }$ has weight $\lambda$ it must be a highest weight vector. We will show that $X^{{\epsilon }_l}$ acts on ${\stackrel{\_}{b}}_{\lambda /\mu }$ by the constant $q^{c\left({\mathrm{box}}_l\right)},$ where $c\left({\mathrm{box}}_l\right)$ is the content of the $l^{\mathrm{th}}$ box of the multisegment $\lambda /\mu$ (read left to right and top to bottom like a book). Consider the projections $${\mathrm{pr}}_l:M\left(\mu \right)\otimes V^{\otimes k}\to {(M\left(\mu \right)\otimes V^{\otimes l})}^{\left[{\lambda }^{\left(l\right)}\right]}\otimes V^{\otimes (k-l)}where{\lambda }^{\left(l\right)}=\mu +\sum _{j\le l}\mathrm{wt}\left(v_{i_l}\right)$$ and ${\mathrm{pr}}_i$ acts as the identity on the last $k-i$ factors of $M\left(\mu \right)\otimes V^{\otimes k}.$ Then $${\stackrel{\_}{b}}_{\lambda /\mu }={\mathrm{pr}}_k{\mathrm{pr}}_{k-1}\cdots {\mathrm{pr}}_1{b}_{\lambda /\mu },$$ and for each $1\le l\le k,$ (the first $l$ components of) ${\mathrm{pr}}_{l-1}\cdots {\mathrm{pr}}_1\left(b_{\lambda /\mu }\right)$ form a highest weight vector of weight ${\lambda }^{\left(l\right)}$ in $M\otimes V^{\otimes l}.$ It is the "highest" highest weight vector of $$\begin{array}{ll}{({(M\left(\mu \right)\otimes V^{\otimes (l-1)})}^{\left[{\lambda }^{(l-1)}\right]}\otimes V)}^{\left[{\lambda }^{\left(l\right)}\right]}& (MS\; 7)\end{array}$$ with respect to the ordering in Lemma 4.2 and thus it is deepest in the filtration constructed there. Note that the quantum Casimir element acts on the space in (6.29) as the constant $q^{⟨{\lambda }^{\left(l\right)},{\lambda }^{\left(l\right)}+2\rho ⟩}$ times a unipotent transformation, and the unipotent transformation must preserve the filtration coming from Lemma 4.2. Since ${\mathrm{pr}}_l\left(b_{\lambda /\mu }\right)$ is the highest weight vector of the smallest submodule of this filtration (which is isomorphic to a Verma module by Lemma 4.2b) it is an eigenvector for the action of the quantum Casimir. Thus, by (2.11) and (2.13), $X^{{\epsilon }_l}$ acts on ${\mathrm{pr}}_l\left(b_{\lambda /\mu }\right)$ by the constant $$q^{⟨{\lambda }^{\left(l\right)},{⟨}^{\left(l\right)}+2\rho ⟩-⟨{\lambda }^{(l-1)},{\lambda }^{(l-1)}+2\rho ⟩-⟨{\omega }_1,{\omega }_1+2\rho ⟩}=q^{2c\left({\mathrm{box}}_l\right)},$$ (see [LR]). Since $X^{{\epsilon }_l}$ commutes with ${\mathrm{pr}}_j$ for $j>l$ this also specifies the action of $X^{{\epsilon }_l}$ on ${\stackrel{\_}{b}}_{\lambda /\mu }={\mathrm{pr}}_l\left(b_{\lambda /\mu }\right).$ The explicit $R-$matrix ${\check{R}}_{VV}:V\otimes V\to V\otimes V$ for this case ($𝔤$ of type $A$ and $V=L\left({\omega }_1\right)$) is well known (see, for example, the proof of [LR, Prop. 4.4]) and given by $$(v_i\otimes v_j)q^{\frac{1}{(n+1)}}{\check{R}}_{VV}=\{\begin{array}{ll}{v}_j\otimes v_i,& if\; i(LESS\; THAN?)j,\\ (q-q^{-1})v_i\otimes v_j+v_j\otimes v_i,& if\; i(GREATER\; THAN?)j,\\ q{v}_i\otimes v_j,& if\; i=j.\end{array}\phantom{\}}$$ Since $T_i$ acts by ${\check{R}}_{VV}$ on the $i^{\mathrm{th}}$ and ${(i+1)}^{\mathrm{st}}$ tensor factors of $V^{\otimes k}$ and commutes with the projection ${\mathrm{pr}}_{\lambda }$ it follows that $T_j\left({\stackrel{\_}{b}}_{\lambda /\mu }\right)=q{\stackrel{\_}{b}}_{\lambda /\mu },$ if ${\mathrm{box}}_j$ is not a box at the end of a row of $\lambda /\mu .$ This analysis of the action of ${\tilde{H}}_{\lambda /\mu }$ on ${\stackrel{\_}{b}}_{\lambda /\mu }$ shows that there is an ${\tilde{H}}_k-$homomorphism $$\begin{array}{ccc}{\mathrm{Ind}}_{{\tilde{H}}_{\lambda /\mu }}^{{\tilde{H}}_k}\left(ℂv_{\lambda /\mu }\right)& \to & {ℳ}^{\lambda /\mu }\\ v_{\lambda /\mu }& \mapsto & {\stackrel{\_}{b}}_{\lambda /\mu }.\end{array}$$ This map is surjective since ${ℳ}^{\lambda /\mu }$ is generated by ${\stackrel{\_}{b}}_{\lambda /\mu }$ (the ${ℬ}_k$ action on $v^{\lambda -\mu }$ generates all of ${\left(V^{\otimes k}\right)}_{\lambda -\mu }$). Finally, (6.28) guarantees that it is an isomorphism. $\square$

In the same way that each weight $\mu \in {𝔥}^{*}$ has a normal form $$\mu =w\circ \tilde{\mu },with\begin{array}{l}\tilde{\mu } \; integrally\; dominant,\; and\\ w \; maximal\; length\; in\; the\; coset\; w{W}_{\tilde{\mu }+\rho },\end{array}$$ every multisegment $\lambda /\mu$ has a normal form $$\lambda /\mu =\nu /(w\circ \tilde{\nu }),with\begin{array}{l}\nu +\rho \; the\; sequence\; of\; contents\; of\; boxes\; of\; \lambda /\mu ,\\ \tilde{\nu }=\nu -\left(1,1,...,1\right),and\\ w \; maximal\; length\; in\; W_{\nu +\rho }w{W}_{\nu +\rho }.\end{array}$$ The element $w$ in the normal form $\nu /(w\circ \tilde{\nu })$ of $\lambda /\mu$ can be constructed combinatorially by the following scheme. We number (order) the boxes of $\lambda /\mu$ in two different ways.

First ordering: To each box $b$ of $\lambda /\mu$ associate the following triple $$(content\; of\; the\; box\; to\; the\; left\; of\; b,-\left(content\; of\; b\right),-\left(row\; number\; of\; b\right))$$ where, if a box is the leftmost box in a row "the box to its left" is the rightmost box in the same row. The lexicographic ordering on these triples induces an ordering on the boxes of $\lambda /\mu .$

Second ordering: To each box $b$ of $\lambda /\mu$ associate the following pair $$(content\; of\; b,-\left(the\; number\; of\; box\; b \; in\; the\; first\; ordering\right)).$$ The lexicographic ordering of these pairs induces a second ordering on the boxes of $\lambda /\mu .$

If $v$ is the permutation defined by these two numberings of the boxes then $w=w_{0}v{w}_0.$ For example, for the multisegment $\lambda /\mu$ displayed in (6.24) the numberings of the boxes are given by $$\begin{array}{c} 21 6 10 13 18 20 5 9 12 17 19 11 16 15 1 2 4 8 14 3 7 \\ first\; ordering\; of\; boxes\end{array}and\begin{array}{c} 3 7 12 16 19 4 8 13 17 20 11 18 21 1 2 6 9 14 5 10 15 \\ second\; ordering\; of\; boxes\end{array}$$ and the normal form of $\lambda /\mu$ is $$\begin{array}{rcl}\nu & =& \left(7,7,7,6,6,6,5,5,5,5,5,4,4,4,4,3,3,3,3,2,1\right),\\ \tilde{\nu }& =& \left(6,6,6,5,5,5,4,4,4,4,4,3,3,3,3,2,2,2,2,1,0\right), \; and\; w=w_{0}v{w}_0 \; where\end{array}$$ $$v=\left(\begin{array}{cccccccccc}1& 2& 3& 4& 5& 6& 7& 8& 9& 10\\ 11& 12& 13& 14& 15& 16& 17& 18& 19& 20\\ 21& & & & & & & & & \\ 15& 1& 21& 20& 14& 2& 6& 5& 4& 3\\ 19& 10& 9& 8& 7& 13& 12& 11& 18& 17\\ 16& & & & & & & & & \end{array}\right).$$ Let $𝔤$ be of type $A_n$ and $V=L\left({\omega }_1\right)$ and let $$\begin{array}{ll}{ℒ}^{\lambda /\mu }=F_{\lambda }\left(L\left(\mu \right)\right),& (MS\; 8)\end{array}$$ as defined in (4.1). It is known (a consequence of Proposition 6.27 and Proposition 4.3c) that ${ℒ}^{\lambda /\mu }$ is always a simple ${\tilde{H}}_k-$module or 0. Furthermore, all simple ${\tilde{H}}_k$ modules are obtained by this construction. See [Su] for proofs of these statements. The following theorem is a reformulation of Proposition 4.12 in terms of the combinatorics of our present setting.

Let $\lambda /\mu$ and $\rho /\tau$ be multisegments with $k$ boxes (with $\mu$ and $\tau$ assumed to be integral) and let $$\lambda /\mu =\nu /(w\circ \tilde{\nu })and\rho /\tau =\gamma /(v\circ \tilde{\gamma })$$ be their normal forms. Then the multiplicities of ${ℒ}^{\rho /\tau }$ in a Jantzen filtration of ${ℳ}^{\lambda /\mu }$ are given by $$\sum _{j\ge 0}[\frac{{\left({ℳ}^{\lambda /\mu }\right)}^{\left(j\right)}}{{\left({ℳ}^{\lambda /\mu }\right)}^{(j+1)}}:{ℒ}^{\rho /\tau }]v^{\frac{1}{2}(l\left(y\right)-l\left(w\right)+j)}=\{\begin{array}{ll}{P}_{wv},& if\; \nu =\gamma ,\\ 0,& if\; \nu \ne \gamma ,\end{array}\phantom{\}}$$ where $P_{wv}\left(v\right)$ is the Kazhdan-Lusztig polynomial for the symmetric group $S_k.$

Theorem 6.31 says that every decomposition number for affine Hecke algebra representations is a Kazhdan-Lusztig polynomial. The following is a converse statement which says that every Kazhdan-Lusztig polynomial for the symmetric group is a decomposition number for affine Hecke algebra representations. This statement is interesting in that Polo [Po] has shown that every polynomial in $1+v{\mathbb{Z}}_{\ge 0}\left[v\right]$ is a Kazhdan-Lusztig polynomial for some choice of $n$ and permutations $v,w\in S_n.$ Thus, the following proposition also shows that every polynomial arises as a generalized decomposition number for an appropriate pair of affine Hecke algebra modules.

Let $\lambda =\left(r,r,...,r\right)=\left(r^r\right)$ and $\mu =\left(0,0,...,0\right)=\left(0^r\right).$ Then, for each pair of permutations $v,w\in S_r,$ the Kazhdan-Lusztig polynomial $P_{vw}\left(v\right)$ for the symmetric group $S_r$ is equal to $$P_{vw}\left(v\right)=\sum _{j\ge 0}[\frac{{\left({ℳ}^{\lambda /w\circ \mu }\right)}^{\left(j\right)}}{{\left({ℳ}^{\lambda /w\circ \mu }\right)}^{(j+1)}}:{ℒ}^{\lambda /v\circ \mu }]v^{\frac{1}{2}(l\left(y\right)-l\left(w\right)+j)}.$$

		Proof.
	Since $\mu +\rho$ and $\lambda +\rho$ are both regular, $W_{\lambda +\rho }=W_{\mu +\rho }=1$ and the standard and irreducible modules ${ℒ}^{\lambda /(w\circ \mu )}$ and ${ℳ}^{\lambda /(v\circ \mu )}$ ranging over all $v,w\in S_k.$ Thus, this statement is a corollary of Proposition 4.12. $\square$

References

[GL] S. Gaussent, P. Littelmann, LS galleries, the path model, and MV cycles, Duke Math. J. 127 (2005), no. 1, 35-88.

[GR] S. Griffeth and A. Ram, Affine Hecke algebras and the Schubert calculus, European J. Combin. 25 (2004), no. 8, 1263-1283.

[Ka] J. Kamnitzer, Mirkovic-Vilonen cycles and polytopes, math.AG/050136.5.

[KM] M. Kapovich and J.J. Millson, A path model for geodesics in Euclidean buildings and its applications to representation thoery, math.RT/0411182.

[LP] C. Lenart and A. Postnikov, Affine Weyl groups in K-theory and representation theory, math.RT/0309207.

[PR] H. Pittie and A. Ram, A Pieri-Chevalley formula in the K-theory of a $G/B-$bundle, Electronic Research Announcements 5 (1999), 102-107, and A Pieri-Chevalley formula for $K(G/B),$ math.RT/0401332.

[1] A. Ram, Affine Hecke algebras and generalized standard Young tableaux, J. Algebra, 230 (2003), 367-415.

[2] A. Ram, Skew shape representations are irreducible, in Combinatorial and Geometric representation theory, S.-J. Kang and K.-H. Lee eds., Contemp. Math. 325 Amer. Math. Soc. 2003, 161-189, math.RT/0401326.

[3] RR A. Ram and J. Ramagge, Affine Hecke algebras, cyclotomic Hecke algebras and Clifford theory, in A tribute to C.S. Seshadri: Perspectives in Geometry and Representation theory, V. Lakshimibai et al eds., Hindustan Book Agency, New Delhi (2003), 428-466, math.RT/0401322.

[Sc] C. Schwer, Ph.D. Thesis, University of Wuppertal, 2006.

page history