Subalgebras of partition algebras

Representations of the groups GH,H/K,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

Representations of the groups GH,H/K,n

The irreducible representations of the group GH,1,n ca be derived from Clifford theory or via the tower 1G12G1G32G2G52G3, where Gn=GH,1,n=HnSn  and  Gn+12=GH,1,n×GH,1,1, so that Gn+12 is a Levi "subgroup" of Gn+1. The tower of G^ has

  1. vertices on level n: mutlipartitions λ=λααH^ with n boxes total,
  2. vertices on level n+12: pairs λα, where λG^n,αH^.
  3. edges from level n to level n+12:λλα for each αH^.
  4. edges from level n+12 to level n:μαν if ν is obtained from μ by adding a box to μα.

For each 0mr-1 the elements tim,1in, form a conjugacy class in Gr1n and the elements timtj-mij,1i<jn,0mr-1, form another conjugacy class in Gr1n. Thus the elements zsm=i=1ntim,0mn-1,  and  zl=1rm=0r-11i<jntimtj-mij, are elements of ZGr1n. So zsm and zl must act by a constant on any irreducible representation Sλ of Gr1n. Define x1=0, xk=1i<jk0lr-1tiltj-lij-1i<jk-10lr-1tiltj-lij=1r1i<k0lr-1tiltk-lik,for  2kn and yk=i=1kti-i=1k-1ti=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 Gr1n is given by ykvT=sTkvT  and  xkvT=cTkvT, for all standard tableaux T.

Proof.

Define an action of H on H^ by hα=χhα, and extend this to an action of H on the simple GH,1,n modules by mPvTmhPvhT.

  1. The simple GH,H/K,n modules are indexed by pairs λ_μ where λ_K\G^n,μK^n, where Kλ is the stabiliser of λ in K.
  2. The simple GH,H/K,n module, for any fixed representative λ of the coset λ_, GK,nλ_μ=pμGnλ,where  pμ=kKλχμk-1k, is the minimal idempotent of Kλ corresponding to the module Kλμ.
  3. As a GH,H/K,nKλ bimodule Gλ=μK^λGnλμKλμ.

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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