The groups <math> <msub> <mi>G</mi><mrow> <mi>r</mi><mi>,</mi><mi>p</mi><mi>,</mi><mi>n</mi> </mrow> </msub> </math>

The groups G r,p,n

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: 20 June 2010

The groups G r,p,n

The group G H,H/K,n is denoted   G r,p,n   if  H  is a cyclic group of order  r  and  H/K  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×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 r/p 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 = Sn , the symmetric group, or Weyl group of type A,
  4. G ,1,n is the affine symmetric group or the affine Weyl group of type A,
  5. G 2,1,n =W Bn , the Weyl group of type Bn ,
  6. G 2,2,n =W Dn , the Weyl group of type Dn .
All of these are subgroups of the group N= * n Sn of monomial matrices in GL n (the normaliser of the torus of diagonal matrices in GL n ).

Let /r= 01 r-1 and let ξ= e 2πi/r . The groups G r,p,n are complex reflection groups (generated by reflections). The reflections in G r,p,n are the elements tik tj -k s ij , and

ti lp = t 00 lp 00 = ith ξ lp ,

for ????? i<jn,0kr-1, and 0l r/p -1.


t1 = ξ
s1 = ξ ξ -1
si = i-1 i =


The group G r,p,n has a presentation by generators t1p , s1 , s2 ,, 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 .

For G r,1,n the generator s1 is unnecessary and, for G r,r,n the generator t1r =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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