Representation Theory of Groups - Blatt 3
11:30-12:15, Seminarraum 9, Oskar-Morgenstern-Platz 1, 2.Stock
http://www.mat.univie.ac.at/~gagt/rep_theory2017
Goulnara Arzhantseva
goulnara.arzhantseva@univie.ac.at
Martin Finn-Sell
martin.finn-sell@univie.ac.at
Question 1. Let G = S3 be the permutation group on the setX = {1, 2, 3}and letρ : G → GL(CX)be the corresponding permutation representation. Use Maschke’s theorem to find the projection onto the vector subspace U := span(δ1 +δ2 + δ3), and construct the G-stable compliment ofUinsideCX. What is the corresponding projection onto this compliment?
Question 2. Let G = C3 = hg | g3 = 1i be the cyclic group of order 3. Let V be the 2 dimensional vector space on the lettersv1 andv2. Let Gact on V by extending the following formulae linearly:
ρ(g)(v1) =v2, ρ(g)(v2) = −(v1+v2).
a) Show thatρdefines a representation ofGonV;
b) ExpressV as a sum ofG-stable irreducible subspaces.
Question 3. LetGbe a finite group and letρ:G→GL(2,C)be a representation of degree2.
Prove that if there exists elementsg, h ∈ Gsuch thatρ(g)andρ(h)do not commute, thenρ is irreducible.
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