The groups G(r,p,n)

The groups G rpn

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA

Last updates: 28 May 2010

The groups G rpn

Let r,p,n be positive integers such that p divides r. The group G rpn is the group of n×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 r/p th power of the product of the nonzero entries is 1.
The group G r1n has order rn 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 rpn is the normal subgroup of G r1n of index p given by the exact sequence 1 G rpn G r1n /p 1 tλ w λ1 ++ λn

Some special cases of these groups are:

  1. G r11 =/r, the cyclic group of order r.
  2. G rr2 =W Ir , the dihedral group of order 2r,
  3. G 11n = Sn =W A n-1 , the symmetric group of n×n permutation matrices,
  4. G 21n =W Bn , the hyperoctahedral group, or Weyl group of type Bn ,
  5. G 22n =W Dn , they Weyl group of type Dn ,
  6. G r1n = /r Sn = /r n Sn , where Sn acts on /r n by permuting the factors.

Let /r= 01 r-1 and let ξ= e 2πi/r . For each λ= λ1 λn /r, let tλ be the diagonal matrix with diagonal entries tλ ii = ξ λ i . Then G r1n = tλ σ| λ /r n ,σ Sn and G rpn = tλ σG r1n | λ1 ++ λn =0 modp . The multiplication in G r1n is determined by the relations tλ tμ = t λ+μ λ,μ /r n , w tλ = t wλ w,λ /r n,w Sn , where wλ= λ w 1 λ w n if λ= λ1 λn .

The groups G rpn are complex reflection groups. The reflections in G r1n are the elements tik tj -k ij ,k /r ,1i<jn,and

tik = t 00k00 = ith ξk ,


The reflections in G rpn are those reflections in G r1n which are also in G rpn . These are ti kp ,0k< r/p -1,and all tik tj -k ij ,k /r ,1i<jn.


t1 = ξ
s1 = ξ ξ -1
si = i-1 i =


The group G r1n can be presented by generators t1 , s1 ,, s n-1 and relations si sj = sj si ,if   i-j >1, si s i+1 s1 = s i+1 si s i+1 ,2in-1, t1 s1 t1 s1 = s1 t1 t1 t1 , t1r =1  and   s i 2 =1,2in.

The group G rpn has a presentation by generators t 1 p , s1 , s 2 ,, sn and relations t1 r/p =1and s i 2 =1,1in, si sj = sj s i ,for   i-j >1  and  i,j2, si s i+1 si = s i+1 si s i+1 ,for  2in-1, s1 s3 s1 = s3 s1 s3 ,and   s1 sj = sj s1 ,  for  j>3, t1p s2 t1p s2 = s2 t1p s2 t1p ,and s0 t1p = t1p s0 ,  for all  j>2, t1p s1 t1p 2 r/p   factors = s1 t1p s1 t1p 2 r/p   factors   and   s1 s 2 s 1 r  factors = s2 s 1 s2 r  factors .

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

Let λ= λ 0 λ 1 λ r-1 can be an r -tuple of partitions with n boxes in total. A standard tableau of shape λ is a filling of the boxes of λ with 1,2,,n such that, in each partition λ j ,

  1. the rows increase from left to right,
  2. the columns increase from top to bottom.
The rows and columns of each partition λ j are numbered as for matrices and T i   is the box containing  i  in  T,
c b =j-i,  if  b  is in position   ij ,  and  
s b = ξj ,  if  b  is in   λ j , where ξ= e 2πi/r . The numbers c b and s b are the content and sign of the box b, respectively.

  1. The irreducible representations Sλ of the group G r1n are indexed by r -tuples of partitions λ= λ 0 λ r-1 with n boxes in total.
  2. dim Sλ = # of standard tableaux of shape λ.
  3. The irreducible G r1n -module Sλ is given by Sλ =-span vT | T  is a standard tableau of shape  λ with basis vT and with G r1n action given by s1 vT = s T 1 vT , si vT = si TT vT + 1+ si TT v si T ,i=2,3,,n, where si TT = 1 c T i -c T i-1 , if  s T i =s T i-1 , 0, if  s T i s T i-1 , c T i is the content of the box containing i in T,
    si T is the same as T except that i and i-1 are switched,
    v si T =0 if si T is not standard.

For each 0kr-1 the elements tik ,1in, form a conjugacy class in G r1n and the elements ti k t j -k ,1i<jn,0kr-1, form another conjugacy class in G r1n . Thus the elements zs k = i=1 n tik ,1kr-1,  and   zl = 1 r i<j,1kr-1 tik tj -k ij , are elements of Z G r1n . So zs k and zl must act by a constant on any irreducible representation Sλ of G r1n . Define x1 =0, xk = 1i<jk
0lr-1 til tj -l ij - 1i<jk-1
til tj -l ij
= 1 r 1i<k,0lr-1 ti l tj -l ij ,for  2kn, and yk = i=1 k ti - i=1 k-1 ti = tk ,for  1kn.

The elements x1 ,, xn and y1 ,, yn all commute with each other and the action of these elements on the irreducible representation Sλ of G r1n is given by yk vT =s T k vT   and   xk vT =c T k vT , for all standard tableaux T.

The proof is by induction on k using the relations xk = sk x k-1 sk + l=0 r-1 y k-1 -l   and   yk = sk y k-1 sk . The base cases x1 vT =0=c T 1 vT   and   y1 vT =s T 1 vT are immediate from the definitons. Then yk vT = sk y k-1 sk = sk s T k-1 sk TT vT + 1+ sk TT s T k v sk T = sk s T k sk vT + s T k-1 -s T k sk TT vT = s T k vT +0=s T k vT , and xk vT = sk x k-1 sk + 1 r l=0 r-1 sk y k-1 -l ykl vT = sk c T k-1 sk TT vT +c T k 1+ sk TT v sk T + 1 r l=0 r-1 s T k-1 -l s T k l vT = sk c T k sk vT + c T k-1 -c T k sk TT vT + 1r l=0 r-1 s T k -1 s T k l vT = sk c T k sk vT + -1 +1 vT , if  s T k =s T k-1 , sk c T k sk vT + 0 +0 vT , if  s T k s T k-1 , = c T k vT .

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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