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