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Letπ:G→GL(V) be a unitary representation of the groupGon a finite dimensional complex vector spaceV

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Algebra III/Introduction to Algebra III: Representation Theory Due: Please upload solutions to NUCT by Tuesday, May 12, 2020.

Problem 1. Letπ:G→GL(V) be a unitary representation of the groupGon a finite dimensional complex vector spaceV. Prove that for allg, the eigenvalues of the linear automorphismπ(g) :V →V have absolute value 1.

Problem 2. LetG={1, ζ, ζ2}be a cyclic group of order 3, and letπ:G→GL2(R) be the representation ofGonR2 defined by

π(ζ) =

0 −1

1 −1

.

Find an inner product onR2 that is invariant with respect toπin the sense that hπ(g)(x), π(g)(y)i=hx, yi

for allg∈Gandx, y∈R2.

1

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