Question

Please Complete 4.1.

Exercises Exercise 4.1. Lete: G → GL(U), ψ: G → GL(V) and : representations of a group G. Suppose that Te HomG(φ, ψ) and Se Prove that ST Homc(p.,p). p: G GL(U Xp. Prove tha Exercise 4.2. Let o be a representation of a group G with character Exercise 4.3. Let p: GGL(V) be an irreducible representation Let be the center of G. Show that if a e Z(G), then p(a) Exercise 4.4. Let G be a group. Show th only if f(gh) f(hg) for all g,he G. AI for some f: G-C is a class function if and Exercise 4.5. For ) (2/22, let a(v) fi l c1. For each Y SI...,m), define a function xy: (Z/22)C by 1. Show that xy is a character 2. Show that every irreducible character of (Z/2zyn is of the form XY for some 3. Show that if X, Y Exercise 4.6. Let sgn: Sn → C* be the representation given by subset Y of (1,...,m). XAY = X UY \ (X n Y) is the symmetric difference. {1, mb, then χχ (v)XY(v) -XXAY(t) where sgn(σ)-j 1 o is even sgn(r)--1 σǐs odd. -1 σ is odd.

Answer 1

DAT s 2.í.4:6-9 GIU) +-)-G-, GIN) IEG ST 51 EHomg(61)