Advanced Algebra I

Advanced Algebra I

Homework 8 due on Nov.14, 2003 Part A.

(1) Let G = Sn and S = {1, ..., n}. We can have the permutation representation ρ : G × C[S] → C[S]. ρ factors as 1 ⊕ ρ⁰. Prove or disprove that ρ⁰ is irreducible.

(2) Determine the character table of the group G =< x, y|x⁷ = y⁶ = e, yxy⁻¹ = x² >

Part B.

(1) Let V be a n-dimensional vector space with basis {e₁, ..., e_n}.

One can consider the vector space V ⊗V as the n²-dimensional vector space with basis {e_i⊗e_j}_i,j=1...n.

(a) Consider θ : V ⊗V → V ⊗V by θ(e_i⊗e_j) = e_j⊗e_i. Show that the space Sym²V := {z ∈ V ⊗V |θ(z) = z} has ba- sis {e_i⊗e_j + e_j⊗e_i}_i≤j. And the space Alt²V := {z ∈ V ⊗V |θ(z) = −z} has basis {e_i⊗e_j−e_j⊗e_i}_i<j. And V ⊗V = Sym²V ⊕ Alt²V .

(b) If ρ : G → GL(V ) is an representation with character χ. Show that the induced representation ρ⊗ρ : G → GL(V ⊗V ) is a representation with character χ².

(c) Show that Alt²V and Sym²V gives subrepresentations.

(d) Show that the induced representation ρ_Alt² : G → GL(Alt²V ) has character χ_Alt² = ¹₂(χ_ρ(g)²− χ_ρ(g²)).

(e) Show that the induced representation ρ_Sym² : G → GL(Sym²V ) has character χ_Sum² = ¹₂(χ_ρ(g)²+ χ_ρ(g²)).

(2) Let ρ₁, .., ρ_r be representatives of isomorphic classes of irre- ducible representation of a finite group G. Let ρ : G → GL(V ) be an arbitrary representation which factors as ρ ∼= n1ρ₁⊕ ... ⊕ n_rρ_r. Prove that the vector space Hom_G(V, V_i) of G-invariant linear transformation has dimension ni. And so is HomG(Vi, V ).

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