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LetGbe an abelian group, and letπ:G→GL(V) be an irreducible real representation

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Algebra III/Introduction to Algebra III: Representation Theory Due: Please upload solutions to NUCT by Tuesday, June 2, 2020.

Problem 1. We consider the additive groupG= (R,+) of real numbers and the represenationπ:G→GL(V) onV =R2 defined by

π(t) =

cost −sint sint cost

.

Sinceπ is irreducible, the ringD = End(π)⊂EndR(V) is a real division algebra.

Determine the structure ofD.

Problem 2. LetGbe an abelian group, and letπ:G→GL(V) be an irreducible real representation. Show that either dimR(V) = 1 or dimR(V) = 2.

Problem 3. Letk=k be an algebraically closed field,1letπ:G→GL(V) be a finite dimensional irreduciblek-linear representation, and let

A= span(π(g)|g∈G)⊂Endk(V).

Show thatA= Endk(V).

[Hint: Consider the representationρ:G×G→GL(Endk(V)) defined by ρ(g1, g2)(f)(x) =π(g1)(f(π(g−12 )(x))).

Show thatA⊂Endk(V) isρ-invariant and that ρ'ππ. Then use a theorem.]

1You are welcome to assume thatk=C.

1

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