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
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

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 $φ : G_{H, 1, n} \to H$ given by $φ (t_{h} w) = h_{1} \dots h_{n},$ for $h = (h_{1} \dots h_{n}) \in H^{n}$ and $w \in S_{n} .$ 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 $\begin{array}{rcl} \{1\} & \to & G_{H, H / K, n} & \to & G_{H, 1, n} & \to & H / K & \to & \{1\} \\ t_{h} w & \mapsto & h_{1} h_{2} \dots h_{n} . \end{array}$ Thus $G_{H, H / K, n} = \{t_{h} w | h_{1} h_{2} \dots h_{n} \in K\} with order |G^{K}| = {|H|}^{n - 1} |K| n!$ Let $H$ be an abelian group and let $\hat{H}$ be an index set for the simple $H$ -modules. If $H = ℂ^{*}$ then the irreducible representations of $H$ (as an algebraic group) are $\begin{array}{rcl} X^{k} : & ℂ^{*} & \to & ℂ^{*} \\ x & \mapsto & x^{k} \end{array} for k \in ℤ, and X^{k} X^{l} = X^{k + l},$ so that $\hat{H} ≅ ℤ .$ If $H = {(ℂ^{*})}^{n}$ then $\hat{H}$ is a lattice, $\hat{H} ≅ ℤ^{n}, with X^{λ} X^{μ} = X^{λ + μ},$ for $λ, μ \in ℤ^{n} .$ If $H$ is a finite abelian group then $H ≅ ℤ / r_{1} ℤ \otimes ℤ / r_{2} ℤ \otimes \dots \otimes ℤ / r_{l} ℤ and \hat{H} ≅ ℤ / r_{1} ℤ \otimes ℤ / r_{2} ℤ \otimes \dots \otimes ℤ / r_{l} ℤ,$ a quotient of the lattice in (???), so that $\hat{H} = \{X^{λ} | λ = (λ_{1} \dots λ_{n}) with λ_{i} \in ℤ / r_{i} ℤ\} .$

Let $H$ be abelian and let $\hat{H}$ be the dual group of $H .$ The group $\hat{H}$ acts on the group algebra of $H^{n}$ by algebra automorphisms, $\begin{array}{rcl} X^{λ} : & ℂ H^{n} & \to & ℂ H^{n} \\ t_{(h_{1} \dots h_{n})} & \mapsto & X^{λ} (h_{1} h_{2} \dots h_{n}) t_{(h_{1} \dots h_{n})} \end{array}$ and on the group algebra of $G_{H, 1, n} = H^{n} ⋊ S_{n}$ by algebra automorphisms, $\begin{array}{rcl} X^{λ} : & ℂ G_{H, 1, n} & \to & ℂ G_{H, 1, n} \\ t_{(h_{1} \dots h_{n})} w & \mapsto & X^{λ} (h_{1} h_{2} \dots h_{n}) t_{(h_{1} \dots h_{n})} w \end{array}$ Let $\hat{K}$ be a subgroup of $\hat{H} .$ Then the subalgebra of $ℂ G_{H, 1, n}$ fixed by $\hat{K}$ is ${(ℂ G_{H, 1, n})}^{\hat{K}} = span \{t_{(h_{1} \dots h_{n})} w | X^{λ} (h_{1} h_{2} \dots h_{n}) = 1, for all X^{λ} \in \hat{K}\} .$ Then $\begin{array}{rcl} {(ℂ G_{H, 1, n})}^{\hat{K}} & = & span \{t_{(h_{1} \dots h_{n})} w | h_{1} h_{2} \dots h_{n} \in K\} \\ = & ℂ G_{H, H / K, n}, where K = \underset{λ \in \hat{K}}{\cap} \ker (X^{λ}) . \end{array}$

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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The groups G H,H/K,n

The groups G H,H/K,n

References [PLACEHOLDER]

The groups $G_{H, H / K, n}$

The groups $G_{H, H / K, n}$