MTH 410/514/620: REPRESENTATION THEORY SEMESTER 2, 2016-2017 DUE: JAN 23, 2017 HOMEWORK 2 TOTAL MARKS: 30 (1) Let ρ : S3 → GL2 (C) be given by −1 −1 −1 −1 ρ((12)) = and ρ((123)) = 0 1 1 0 Show that this defines a representation of S3 . Note: The goal of the next problem is to give an alternate proof of [BS, Corollary 3.2.5]. Before solving it, it might help to read about projection operators (for instance, [HK, Section 6.6]). (2) Let ϕ : G → GL(V ) be a representation and suppose W ≤ V is a G-invariant subspace. Let P : V → V be the projection operator onto W . ie. P is a linear operator such that • P ◦P =P • Im(P ) = W Let Q = P ] as in [BS, Proposition 4.2.2], and let W 0 = ker(Q). Show that (a) Q ◦ Q = Q (b) Im(Q) = W (c) W 0 is a G-invariant subspace. (d) V = W ⊕ W 0 Note: We will focus on finite groups in this course. To study the representation theory of infinite groups, one typically needs to put a topology on them, and then require that the representations are continuous with respect to that topology. Under these conditions, [BS, Corollary 4.1.8] still holds. The goal of the next problem then is to describe all continuous irreducible unitary representations of R. (3) Fix t ∈ R and define ϕt : R → S 1 by ϕt (s) = e2πist (a) Show that ϕt is a representation of (R, +) Let ϕ : (R, +) → (S 1 , ×) be a group homomorphism that is also a continuous function. Let t ∈ R such that ϕ(1) = e2πit . (b) Show that ϕ(n) = e2πint for all n ∈ Z (c) Show that ϕ(r) = e2πirt for all r ∈ Q (d) Show that ϕ = ϕt References [BS] B. Steinberg, Representation Theory of Finite Groups, Springer 2012 [HK] K. Hoffman, R. Kunze, Linear Algebra (2nd Ed), PHI Learning, 2013

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