The tower ${\hat{A}}_k\left(r,p,n\right)$

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

Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA
ram@math.wisc.edu

Last updates: 20 June 2010

The tower ${\hat{A}}_k\left(r,p,n\right)$

Let $1_n$ be the trivial representation of $G=G_{r,1n}$ and let $V=ℂ-\mathrm{span}\left\{v_1,\dots ,v_n\right\}$ be the reflection representation. Let $$G=G_{r,p,n}=G_{r,1,n}\cap G_{r,p,n}\text{ and }{L}_{n-1}=\left(G_{r,1,n-1}\times \left(\mathbb{Z}/r\mathbb{Z}\right)\right)\cap G_{r,p,n}.$$ The $G_{r,1,n-1}\times \left(\mathbb{Z}/r\mathbb{Z}\right)$ module $$\chi =1_{n-1}\otimes {\chi }_1,\text{where }{\chi }_1:\begin{array}{rcl}\mathbb{Z}/r\mathbb{Z}& \to & {ℂ}^{*}\\ \xi & \mapsto & \xi \end{array}$$ is, by restriction, an $L_{n-1}$ module and for any $G$-module $M,$$${\mathrm{Ind}}_{L_{n-1}}^G\left({\mathrm{Res}}_{L_{n-1}}^G\left(M\right)\otimes \chi \right)\cong M\otimes {\mathrm{Ind}}_{L_{n-1}}^G\left(\chi \right)\cong M\otimes V,$$ where the first isomorphism comes from the tensor identity,

$$\begin{array}{rcl}{\mathrm{Ind}}_{L_{n-1}}^G\left({\mathrm{Res}}_{L_{n-1}}^G\left(M\right)\otimes N\right)& \tilde{\to }& M\otimes {\mathrm{Ind}}_{L_{n-1}}^{G}N\\ g\otimes \left(m\otimes n\right)& \mapsto & gm\otimes \left(g\otimes n\right)\end{array},$$

for $g\in G,m\in M,n\in N,$ and the fact that ${\mathrm{Ind}}_{L_{n-1}}^G\left(W\right)=ℂG{\otimes }_{L_{n-1}}W.$ Iterating (????) it follows that

$${\left({\mathrm{Ind}}_{L_{n-1}}^G\left({\mathrm{Res}}_{L_{n-1}}^G\left(M\right)\otimes \chi \right)\right)}^k\left(1\right)\cong V^{\otimes k}\text{ and }{\mathrm{Res}}_{L_{n-1}}^G{\left({\mathrm{Ind}}_{L_{n-1}}^G{\mathrm{Res}}_{L_{n-1}}^G\right)}^k\left(1\right)\cong V^{\otimes k}$$

as $G$ modules and $L_{n-1}$-modules respectively.

This analysis allows us to build the Bratteli diagram of $A_k\left(r,1,n\right).$ This graph is constructed inductively as followsL $${\hat{A}}_0\left(r,1,n\right)=\left\{\left(n\right),\varnothing ,\dots ,\varnothing \right\}.$$ If $k\in {\mathbb{Z}}_{>0}$ then there are edges $$\begin{array}{rc}{\hat{A}}_k\left(r,1,n\right):& \lambda \\ & \downarrow \\ {\hat{A}}_{k+\frac{1}{2}}:& \left(\mu ,{\square }_i\right)\end{array}\text{ if }\mu \text{ is obtained from }\lambda \text{ by removing a box from }{\lambda }^{\left(i\right)},$$ and $$\begin{array}{rc}{\hat{A}}_k\left(r,1,n\right):& \left(\nu ,{\square }_i\right)\\ & \downarrow \\ {\hat{A}}_{k+\frac{1}{2}}:& \gamma \end{array}\text{ if }\gamma \text{ is obtained from }\nu \text{ by adding a box to }{\nu }^{\left(i+1\right)},$$ where we make the convention that ${\mu }^{\left(r\right)}={\mu }^{\left(0\right)}.$

The Bratteli diagram of the algebra $ℂA_{k,r,p}\left(n\right)$

Recall that the simple $G_{r,1,n}$ modules are given by $$G_{r,1,n}^{\lambda }=\mathrm{span}\left\{v_T|T\text{ is a standard tableau of shape }\lambda \right\}$$ with action $$t_i{v}_T=s\left(T\left(i\right)\right)v_T\text{ and }{s}_i{v}_T={\left(s_i\right)}_{TT}{v}_T+\left(1+{\left(s_i\right)}_{TT}\right)v_{s_{i}T}.$$ Then $\mathbb{Z}/p\mathbb{Z}$ acts on the $G_{r,1,n}$ modules by $$\begin{array}{rcll}\sigma :& ℂ{G_{r,1,n}}^{\lambda }& \to & ℂG_{r,1,n}^{\lambda }\\ & v_T& \mapsto & v_{\sigma T}\end{array}$$ and this action lifts to an action of $\mathbb{Z}/p\mathbb{Z}$ on $ℂG_{r,1,n}$ by automorphisms $$\begin{array}{rcll}\sigma :& ℂ{G_{r,1,n}}^{\lambda }& \to & ℂG_{r,1,n}^{\lambda }\\ & t_{\lambda }w& \mapsto & {\zeta }^{\left|\lambda \right|}{t}_{\lambda }w\end{array}\text{where }\zeta =e^{2\pi i/p}.$$ Then $V^{\otimes k}$ is a $G_{r,1,n}$ module and we can twist the action by any automorphism. So $$t_i\circ v_j=\left\{\begin{array}{rc}\xi \zeta v_i,& \text{if }j=i,\\ v_i,& \text{if }j\ne i.\end{array}\right.$$ Then $\sigma :V^{\otimes k}\to {\left({\sigma }^{*}V\right)}^{\otimes k}$ as $G_{r,1,n}$ modules and this operation commutes with the $G_{r,p,n}$ action. So $\sigma \in {\mathrm{End}}_{G_{r,p,n}}\left(V^{\otimes k}\right).$

References [PLACEHOLDER]

[BG] A. Braverman and D. Gaitsgory, Crystals via the affine Grassmanian, Duke Math. J. 107 no. 3, (2001), 561-575; arXiv:math/9909077v2, MR1828302 (2002e:20083)

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