Subalgebras of partition algebras

Representations of the groups G H,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 G H,H/K,n

The irreducible representations of the group G H,1,n ca be derived from Clifford theory or via the tower 1 ⊆ G 1 2 ⊆ G1 ⊆ G 3 2 ⊆ G2 ⊆ G 5 2 ⊆ G3 ⊆…, where Gn = G H,1,n = Hn ⋊ Sn   and   G n+ 1 2 = G H,1,n × G H,1,1 , so that G n+ 1 2 is a Levi "subgroup" of G n+1 . The tower of G^ has

  1. vertices on level n: mutlipartitions λ= λ α α∈ H^ with n boxes total,
  2. vertices on level n+ 1 2 : pairs λ □α , where λ∈ G^ n ,α∈ H^ .
  3. edges from level n to level n+ 1 2 : λ→ λ □α for each α∈ H^ .
  4. edges from level n+ 1 2 to level n: μ □α →ν if ν is obtained from μ by adding a box to μ α .

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

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.

Proof.

The proof is via induction on k using the relations xk = sk x k-1 sk + ∑ l=0 r-1 y k-1 l sk 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 defintions. Then yk vT = sk y k-1 sk = sk s T k-1 sk TT + 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 -l s T k l vT = sk c T k sk vT + c T k-1 -c T k sk TT vT + 1 r ∑ 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 .□

Define an action of H on H^ by hα= χh ⊗α, and extend this to an action of H on the simple G H,1,n modules by mP ⊗ vT ↦ m hP ⊗ v hT .

  1. The simple G H,H/K,n modules are indexed by pairs λ_ μ where λ_ ∈K\ G^ n ,μ∈ K^ n , where Kλ is the stabiliser of λ in K.
  2. The simple G H,H/K,n module, for any fixed representative λ of the coset λ_ , G K,n λ_ μ = pμ Gnλ ,where   pμ = ∑ k∈ Kλ χμ k -1 k, is the minimal idempotent of Kλ corresponding to the module Kλμ .
  3. As a G H,H/K,n Kλ bimodule Gλ = ⊗ μ∈ K^ λ Gn λμ ⊗ Kλμ .

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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