The groups <math> <msub> <mi>G</mi><mrow> <mi>H</mi><mi>,</mi><mn>1</mn><mi>,</mi><mi>n</mi> </mrow> </msub> </math>

The groups G H,H/K,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 H,H/K,n

Let H be a group. Assume that H is abelian so that there is a well defined map φ: G H,1,n H given by φ th w = h1 hn , for h= h1 hn Hn and w Sn . Let K be a subgroup of H and define a normal subgroup G H,H/K,n of G H,1,n by the exact sequence 1 G H,H/K,n G H,1,n H/K 1 th w h1 h2 hn . Thus G H,H/K,n = th w| h1 h2 hn K   with order   GK = H n-1 K n! Let H be an abelian group and let H^ be an index set for the simple H -modules. If H= * then the irreducible representations of H (as an algebraic group) are Xk : * * x xk   for  k,  and   Xk Xl = X k+l , so that H^ . If H= * n then H^ is a lattice, H^ n ,  with   Xλ Xμ = X λ+μ , for λ,μ n . If H is a finite abelian group then H/ r1 / r2 / rl   and   H^ / r1 / r2 / rl , a quotient of the lattice in (???), so that H^ = Xλ | λ= λ1 λn   with   λi / ri .

Let H be abelian and let H^ be the dual group of H. The group H^ acts on the group algebra of Hn by algebra automorphisms, Xλ : Hn Hn t h1 hn Xλ h1 h2 hn t h1 hn and on the group algebra of G H,1,n = Hn Sn by algebra automorphisms, Xλ : G H,1,n G H,1,n t h1 hn w Xλ h1 h2 hn t h1 hn w Let K^ be a subgroup of H^ . Then the subalgebra of G H,1,n fixed by K^ is G H,1,n K^ =span t h1 hn w| Xλ h1 h2 hn =1,  for all   Xλ K^ . Then G H,1,n K^ = span t h1 hn w| h1 h2 hn K = G H,H/K,n ,  where  K= λ K^ ker Xλ .

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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