The groups $G\left(r,p,n\right)$

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: 28 May 2010

The groups $G\left(r,p,n\right)$

Let $r,p,n$ be positive integers such that $p$ divides $r.$ The group $G\left(r,p,n\right)$ is the group of $n\times n$ matrices such that

1. There is exactly one nonzero entry in each row and column,
2. The nonzero entries are $r$th roots of unity,
3. The $\left(r/p\right)$th power of the product of the nonzero entries is 1.

The group $G\left(r,1,n\right)$ has order $r^{n}n!$ since condition c) is trivially satisfied, there are $r$ choices for each of $n$ roots of unity and there are $n!$ permutation matrices. The group $G\left(r,p,n\right)$ is the normal subgroup of $G\left(r,1,n\right)$ of index $p$ given by the exact sequence $$\begin{array}{rclllllll}1& \to & G\left(r,p,n\right)& \hookrightarrow & G\left(r,1,n\right)& \to & \mathbb{Z}/p\mathbb{Z}& \to & 1\\ & & & & t_{\lambda }w& \mapsto & {\lambda }_1+\dots +{\lambda }_n& & \end{array}$$

Some special cases of these groups are:

1. $G\left(r,1,1\right)=\mathbb{Z}/r\mathbb{Z},$ the cyclic group of order $r.$
2. $G\left(r,r,2\right)=W{I}_r,$ the dihedral group of order $2r,$
3. $G\left(1,1,n\right)=S_n=W{A}_{n-1},$ the symmetric group of $n\times n$ permutation matrices,
4. $G\left(2,1,n\right)=W{B}_n,$ the hyperoctahedral group, or Weyl group of type $B_n,$
5. $G\left(2,2,n\right)=W{D}_n,$ they Weyl group of type $D_n,$
6. $G\left(r,1,n\right)=\left(\mathbb{Z}/r\mathbb{Z}\right)\wr S_n={\left(\mathbb{Z}/r\mathbb{Z}\right)}^n⋉S_n,$ where $S_n$ acts on ${\left(\mathbb{Z}/r\mathbb{Z}\right)}^n$ by permuting the factors.

Let $\mathbb{Z}/r\mathbb{Z}=\left\{0,1,\dots ,r-1\right\}$ and let $\xi =e^{2\pi i/r}.$ For each $\lambda =\left({\lambda }_1,\dots ,{\lambda }_n\right)\in \mathbb{Z}/r\mathbb{Z},$ let $t_{\lambda }$ be the diagonal matrix with diagonal entries ${\left(t_{\lambda }\right)}_{ii}={\xi }^{{\lambda }_i}.$ Then $$\begin{array}{rcl}G\left(r,1,n\right)& =& \left\{t_{\lambda }\sigma |\lambda \in {\left(\mathbb{Z}/r\mathbb{Z}\right)}^n,\sigma \in S_n\right\}\end{array}$$ and $$\begin{array}{rcl}G\left(r,p,n\right)& =& \left\{t_{\lambda }\sigma \in G\left(r,1,n\right)|{\lambda }_1+\dots +{\lambda }_n=0\left(\mathrm{mod}p\right)\right\}.\end{array}$$ The multiplication in $G\left(r,1,n\right)$ is determined by the relations $$t_{\lambda }{t}_{\mu }=t_{\lambda +\mu }\lambda ,\mu \in {\left(\mathbb{Z}/r\mathbb{Z}\right)}^n,$$$$w{t}_{\lambda }=t_{w\lambda }w,\lambda \in {\left(\mathbb{Z}/r\mathbb{Z}\right)}^n,w\in S_n,$$ where $w\lambda =\left({\lambda }_{w\left(1\right)},\dots ,{\lambda }_{w\left(n\right)}\right)$ if $\lambda =\left({\lambda }_1,\dots ,{\lambda }_n\right).$

The groups $G\left(r,p,n\right)$ are complex reflection groups. The reflections in $G\left(r,1,n\right)$ are the elements $$t_i^k{t}_j^{-k}\left(i,j\right),k\in \left(\mathbb{Z}/r\mathbb{Z}\right),1\le i<j\le n,\text{and}$$

,

$k\in \mathbb{Z}/r\mathbb{Z},1\le i\le n.$

The reflections in $G\left(r,p,n\right)$ are those reflections in $G\left(r,1,n\right)$ which are also in $G\left(r,p,n\right).$ These are $$t_i^{kp},0\le k<\left(r/p\right)-1,\text{and all}{t}_i^k{t}_j^{-k}\left(i,j\right),k\in \left(\mathbb{Z}/r\mathbb{Z}\right),1\le i<j\le n.$$

Define

$2\le i\le n.$

The group $G\left(r,1,n\right)$ can be presented by generators $t_1,s_1,\dots ,s_{n-1}$ and relations $$s_i{s}_j=s_j{s}_i,\text{if }\left|i-j\right|>1,$$$$s_i{s}_{i+1}{s}_1=s_{i+1}{s}_i{s}_{i+1},2\le i\le n-1,$$$$t_1{s}_1{t}_1{s}_1=s_1{t}_1{t}_1{t}_1,$$$$t_1^r=1\text{ and }{s}_i^2=1,2\le i\le n.$$

The group $G\left(r,p,n\right)$ has a presentation by generators $t_1^p,s_1,s_2,\dots ,s_n$ and relations $$t_1^{r/p}=1\text{and}{s}_i^2=1,1\le i\le n,$$$$s_i{s}_j=s_j{s}_i,\text{for }\left|i-j\right|>1\text{ and }i,j\ge 2,$$$$s_i{s}_{i+1}{s}_i=s_{i+1}{s}_i{s}_{i+1},\text{for }2\le i\le n-1,$$$$s_1{s}_3{s}_1=s_3{s}_1{s}_3,\text{and }{s}_1{s}_j=s_j{s}_1,\text{ for }j>3,$$$$t_1^p{s}_2{t}_1^p{s}_2=s_2{t}_1^p{s}_2{t}_1^p,\text{and}{s}_0{t}_1^p=t_1^p{s}_0,\text{ for all }j>2,$$$$t_1^p{s}_1{t}_1^p\dots \left(2\left(r/p\right)\text{ factors}\right)=s_1{t}_1^p{s}_1{t}_1^p\dots \left(2\left(r/p\right)\text{ factors}\right)\text{ and }{s}_1{s}_2{s}_1\dots \left(r\text{ factors}\right)=s_2{s}_1{s}_2\dots \left(r\text{ factors}\right).$$

Note that only the groups ??? can be generated by $n$ reflections.

Let $\lambda =\left({\lambda }^{\left(0\right)},{\lambda }^{\left(1\right)},\dots ,{\lambda }^{\left(r,-,1\right)}\right)$ can be an $r$-tuple of partitions with $n$ boxes in total. A standard tableau of shape $\lambda$ is a filling of the boxes of $\lambda$ with $1,2,\dots ,n$ such that, in each partition ${\lambda }^{\left(j\right)},$

1. the rows increase from left to right,
2. the columns increase from top to bottom.

The rows and columns of each partition ${\lambda }^{\left(j\right)}$ are numbered as for matrices and $$T\left(i\right)\text{ is the box containing }i\text{ in }T,$$
c b =j-i,  if  b  is in position   ij ,  and  
s b = ξj ,  if  b  is in   λ j , where $\xi =e^{2\pi i/r}.$ The numbers $c\left(b\right)$ and $s\left(b\right)$ are the content and sign of the box $b,$ respectively.

1. The irreducible representations $S^{\lambda }$ of the group $G\left(r,1,n\right)$ are indexed by $r$-tuples of partitions $\lambda =\left({\lambda }^{\left(0\right)},\dots ,{\lambda }^{\left(r-1\right)}\right)$ with $n$ boxes in total.
2. $\dim S^{\lambda }=$ # of standard tableaux of shape $\lambda .$
3. The irreducible $G\left(r,1,n\right)$-module $S^{\lambda }$ is given by $$S^{\lambda }=ℂ-\mathrm{span}\left\{v_T|T\text{ is a standard tableau of shape }\lambda \right\}$$ with basis $\left\{v_T\right\}$ and with $G\left(r,1,n\right)$ action given by $$\begin{array}{rcl}{s}_1{v}_T& =& s\left(T\left(1\right)\right)v_T,\\ s_i{v}_T& =& {\left(s_i\right)}_{TT}{v}_T+\left(1+{\left(s_i\right)}_{TT}\right)v_{s_{i}T},i=2,3,\dots ,n,\end{array}$$ where $${s_i}_{TT}=\left\{\begin{array}{rc}\frac{1}{c\left(T\left(i\right)\right)-c\left(T\left(i-1\right)\right)},& \text{if }s\left(T\left(i\right)\right)=s\left(T\left(i-1\right)\right),\\ 0,& \text{if }s\left(T\left(i\right)\right)\ne s\left(T\left(i-1\right)\right),\end{array}\right.$$ $c\left(T\left(i\right)\right)$ is the content of the box containing $i$ in $T,$
   $s_{i}T$ is the same as $T$ except that $i$ and $i-1$ are switched,
   $v_{s_{i}T}=0$ if $s_{i}T$ is not standard.

For each $0\le k\le r-1$ the elements $t_i^k,1\le i\le n,$ form a conjugacy class in $G\left(r,1,n\right)$ and the elements $t_i^k{t}_j^{-k},1\le i<j\le n,0\le k\le r-1,$ form another conjugacy class in $G\left(r,1,n\right).$ Thus the elements $$z_s\left(k\right)=\sum _{i=1}^n{t}_i^k,1\le k\le r-1,\text{ and }{z}_l=\frac{1}{r}\sum _{i<j,1\le k\le r-1}{t}_i^k{t}_j^{-k}\left(i,j\right),$$ are elements of $Z\left(ℂG\left(r,1,n\right)\right).$ So $z_s\left(k\right)$ and $z_l$ must act by a constant on any irreducible representation $S^{\lambda }$ of $G\left(r,1,n\right).$ Define $x_1=0,$ $$x_k=\left(\sum _{1\le i<j\le k}\right)$$
0≤l≤r-1 til tj -l ij - ∑ 1≤i<j≤k-1
0≤l≤r-1 til tj -l ij = 1 r ∑ 1≤i<k,0≤l≤r-1 ti l tj -l ij ,for  2≤k≤n, and $$y_k=\left(\sum _{i=1}^k{t}_i\right)-\left(\sum _{i=1}^{k-1}{t}_i\right)=t_k,\text{for }1\le k\le n.$$

The elements $x_1,\dots ,x_n$ and $y_1,\dots ,y_n$ all commute with each other and the action of these elements on the irreducible representation $S^{\lambda }$ of $G\left(r,1,n\right)$ is given by $$y_k{v}_T=s\left(T\left(k\right)\right)v_T\text{ and }{x}_k{v}_T=c\left(T\left(k\right)\right)v_T,$$ for all standard tableaux $T.$

		Proof.
	The proof is by induction on $k$ using the relations $$x_k=s_k{x}_{k-1}{s}_k+\sum _{l=0}^{r-1}{y}_{k-1}^{-l}\text{ and }{y}_k=s_k{y}_{k-1}{s}_k.$$ The base cases $$x_1{v}_T=0=c\left(T\left(1\right)\right)v_T\text{ and }{y}_1{v}_T=s\left(T\left(1\right)\right)v_T$$ are immediate from the definitons. Then $$\begin{array}{rcl}{y}_k{v}_T& =& s_k{y}_{k-1}{s}_k\\ & =& s_k\left(s\left(T\left(k-1\right)\right){\left(s_k\right)}_{TT}{v}_T+\left(1+{\left(s_k\right)}_{TT}\right)s\left(T\left(k\right)\right)v_{s_{k}T}\right)\\ & =& s_k\left(s\left(T\left(k\right)\right)s_k{v}_T+\left(s\left(T\left(k-1\right)\right)-s\left(T\left(k\right)\right)\right){\left(s_k\right)}_{TT}{v}_T\right)\\ & =& s\left(T\left(k\right)\right)v_T+0=s\left(T\left(k\right)\right)v_T,\end{array}$$ and $$\begin{array}{rcl}{x}_k{v}_T& =& \left(s_k{x}_{k-1}{s}_k+\frac{1}{r}\sum _{l=0}^{r-1}{s}_k{y}_{k-1}^{-l}{y}_k^l\right)v_T\\ & =& s_k\left(c\left(T\left(k-1\right)\right){\left(s_k\right)}_{TT}{v}_T+c\left(T\left(k\right)\right)\left(1+{\left(s_k\right)}_{TT}\right)v_{s_{k}T}+\frac{1}{r}\sum _{l=0}^{r-1}s{\left(T\left(k-1\right)\right)}^{-l}s{\left(T\left(k\right)\right)}^l{v}_T\right)\\ & =& s_k\left(c\left(T\left(k\right)\right)s_k{v}_T+\left(c\left(T\left(k-1\right)\right)-c\left(T\left(k\right)\right)\right){\left(s_k\right)}_{TT}{v}_T+\frac{1}{r}\sum _{l=0}^{r-1}{\left(s{\left(T\left(k\right)\right)}^{-1}s\left(T\left(k\right)\right)\right)}^l{v}_T\right)\\ & =& \left\{\begin{array}{rc}{s}_k\left(c\left(T\left(k\right)\right)s_k{v}_T+\left(\left(-1\right)+1\right)v_T\right),& \text{if }s\left(T\left(k\right)\right)=s\left(T\left(k-1\right)\right),\\ s_k\left(c\left(T\left(k\right)\right)s_k{v}_T+\left(\left(0\right)+0\right)v_T\right),& \text{if }s\left(T\left(k\right)\right)\ne s\left(T\left(k-1\right)\right),\end{array}\right.\\ & =& c\left(T\left(k\right)\right)v_T.\square \end{array}$$

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