The affine symmetric group

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: 28 May 2010

The affine symmetric group

The affine symmetric group ${\tilde{S}}_n$ is the group of permutations with edges labeled by elements of $\mathbb{Z}.$ For each $\lambda =\left({\lambda }_1,\dots ,{\lambda }_n\right)\in {\mathbb{Z}}^n$ let

Then every element $\tilde{w}\in {\tilde{S}}_n$ can be written uniquely in the form $$\tilde{w}=t_{\lambda }w,\text{with }\lambda \in {\mathbb{Z}}^n\text{ and }w\in S_n.$$

$$=t_{\left(5,32,0,-7,-61,1\right)}\left(\begin{array}{cccccc}1& 2& 3& 4& 5& 6\\ 3& 6& 2& 4& 1& 5\end{array}\right)$$

The multiplication in $S_n$ is determined by $$t_{\lambda }{t}_{\mu }=t_{\lambda +\mu }\text{and}w{t}_{w\lambda }w,w\in S_n,\lambda ,\mu \in {\mathbb{Z}}^n,$$ where $\lambda +\mu =\left({\lambda }_1+{\mu }_1,\dots ,{\lambda }_n+{\mu }_n\right)$ and $S_n$ is acting on ${\mathbb{Z}}^n$ by $w\lambda =w\left({\lambda }_1,\dots ,{\lambda }_n\right)=\left({\lambda }_{w\left(1\right)},\dots ,{\lambda }_{w\left(n\right)}\right).$ So $${\tilde{S}}_n={\mathbb{Z}}^n⋉S_n.$$

The affine symmetric group ${\tilde{S}}_n$ has a presentation by generators

,

,

,

$1\le i\le n-1,$ and relations $$\begin{array}{rcl}{s}_i{s}_j& =& s_j{s}_i,\left|i-j\right|>1,\\ s_i{s}_{i+1}{s}_i& =& s_{i+1}{s}_i{s}_{i+1},\\ s_i^2& =& 1,\\ {\omega }^{-1}{s}_i\omega & =& s_{i+1},\end{array}$$ where the indices on the $s_i$ are interpreted mod n.

		Proof.
	The theorem is proved by using the relations $$\begin{array}{rcl}{s}_0& =& t_{\left(-1,0,0,\dots ,0,1\right)}{s}_{n-1}{s}_{n-2}\dots s_2{s}_1{s}_2\dots s_{n-2}{s}_{n-1},\\ \omega & =& t_{\left(-1,0,0,\dots ,0\right)}{s}_1{s}_2\dots s_{n-2}{s}_{n-1},\end{array}$$ and using these expressions one can write $t_{\lambda }$ in terms of $s_0,\dots ,s_{n-1}$ and $\omega$ and prove that the equation $t_{\lambda }{t}_{\mu }=t_{\lambda +\mu },$ for $\lambda ,\mu \in {\mathbb{Z}}^n,$ is a consequence of the relations in the statement of the theorem. $\square$

For each $r$ there is a surjective homomorphism $$\begin{array}{rcl}{\tilde{S}}_n& \to & G\left(r,1,n\right)\\ t_{\lambda }w& \mapsto & t_{\stackrel{-}{\lambda }}w,\end{array}$$ where $\stackrel{-}{\lambda }=\left({\stackrel{-}{\lambda }}_1,\dots ,{\stackrel{-}{\lambda }}_n\right)$ and $\stackrel{-}{i}=i\left(\mathrm{mod}r\right).$ Thus, if $M$ is a $G\left(r,1,n\right)$-module then there is an action of ${\tilde{S}}_n$ on $M$ given by $$\left(t_{\lambda }w\right)m=\left(t_{\stackrel{-}{\lambda }}w\right)m,\text{for all }{t}_{\lambda }w\in {\tilde{S}}_n.$$ So every irreducible $G\left(r,1,n\right)$ module is an irreducible ${\tilde{S}}_n$-module. Are these all irreducible ${\tilde{S}}_n$-modules? The answer is ????? as well shall see in ????, using Clifford theory.

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)

page history