Let H ⊂Gbe a subgroup, and let (V, σ) be a finite dimensionalk-linear representation of H

Algebra III/Introduction to Algebra III: Representation Theory Due: Please upload solutions to NUCT by Tuesday, June 30, 2020.

Problem 1. Let k be an algebraically closed field, and let G be a finite group, whose order is not divisible by the characteristic of k. Let H ⊂Gbe a subgroup, and let (V, σ) be a finite dimensionalk-linear representation of H. Given a∈ G, we defineH^a ⊂Gbe the subgroupH^a =aHa⁻¹, and we define (V, σ^a) to be the representation ofH^a given byσ^a(g) =σ(a⁻¹ga) forg∈H^a.

Suppose that σ is irreducible. Show that the induced representation Ind^G_H(σ) is irreducible if and only if for alla∈Gsuch thata /∈H,

dimkHom(Res^H_H∩Ha(σ),Res^H_H∩H^a a(σ^a)) = 0.

[Hint: By Schur’s lemma, a finite dimensional k-linear representation π of G is irreducible if and only if dimkHom(π, π) = 1.]

Problem 2. Letkbe a field, letGbe a finite group, and letH ⊂Gbe a subgroup.

Letσbe a finite dimensionalk-linear representation ofH, letπ= Ind^G_H(τ) be the inducedk-linear representation of G, and let χσ:H →k andχπ:G→k be their characters. Giveng∈G, we denote by

(G/H)^g={aH ∈G/H|gaH=aH} ⊂G/H the subset fixed by left multiplication byg. Show that

χπ(g) = X

aH∈(G/H)^g

χσ(a⁻¹ga).

Note that the summandχ_σ(a⁻¹ga) corresponding to aH ∈(G/H)^g only depends onaH and not on the choice ofa∈aH, sinceχσ:H →k is a class function.

[Hint: One possibility is to use that Ind^G_H =f_∗ 'p_∗◦i_∗ and that i_∗ 'r^∗, where r: [G\(G/H)]→BHis a quasi-inverse ofi:BH→[G\(G/H)].]

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