The groups $G_{r,p,n}$

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 groups $G_{r,p,n}$

The group $G_{H,H/K,n}$ is $$\text{denoted }{G}_{r,p,n}\text{ if }H\text{ is a cyclic group of order }r\text{ and }H/K\text{ is order }p.$$ Note that $p$ is not necessarily prime and that $p$ divides $r.$ The group $G_{r,p,n}$ can be realised as the group of $n\times n$ matrices such that

1. There is exactly one nonzero entry in each row and each 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.

Special cases of these groups are

1. $G_{r,1,1},$ the cyclic group of order $r,$
2. $G_{r,r,2},$ the dihedral group of order $2r,$
3. $G_{1,1,n}=S_n,$ the symmetric group, or Weyl group of type $A,$
4. $G_{\infty ,1,n}$ is the affine symmetric group or the affine Weyl group of type $A,$
5. $G_{2,1,n}=W{B}_n,$ the Weyl group of type $B_n,$
6. $G_{2,2,n}=W{D}_n,$ the Weyl group of type $D_n.$

All of these are subgroups of the group $N={\left({ℂ}^{*}\right)}^n⋊S_n$ of monomial matrices in ${\mathrm{GL}}_n\left(ℂ\right)$ (the normaliser of the torus of diagonal matrices in ${\mathrm{GL}}_n\left(ℂ\right)$).

Let $\mathbb{Z}/r\mathbb{Z}=\left\{0,1,\dots ,r-1\right\}$ and let $\xi =e^{2\pi i/r}.$ The groups $G_{r,p,n}$ are complex reflection groups (generated by reflections). The reflections in $G_{r,p,n}$ are the elements $$t_i^k{t}_j^{-k}{s}_{ij},$$ and

,

for ????? $\le i<j\le n,0\le k\le r-1,$ and $0\le l\le \left(r/p\right)-1.$

Define

$2\le i\le n.$

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

For $G_{r,1,n}$ the generator $s_1$ is unnecessary and, for $G_{r,r,n}$ the generator $t_1^r=1$ and is irrelevant. Note that only the groups ??? can be generated by $n$ reflections.

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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