Representation Theory

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Representation TheoryCT, Lent 20051 What is Representation Theory ?Groups arise in nature as sets of symmetries (of an object), which are closed under compo-sition and under taking inverses . For example, thesymmetric groupSnis the group of allpermutations (symmetries) of{1,...,n}; thealternating groupAnis the set of all symmetriespreserving the parity of the number of ordered pairs (did you really remember that one?); thedihedral groupD2nis the group of symmetries of the regularn-gon in the plane. TheorthogonalgroupO(3) is the group of distance-preserving transformations of Euclidean space which fix theorigin. There is also the group ofalldistance-preserving transformations, which includes thetranslations along with O(3).

1.9 Definition. An isomorphism φbetween two representations (ρ 1,V 1) and (ρ 2,V 2) of Gis a linear isomorphism φ: V 1 → V 2 which intertwines with the action of G, that is, satisfies φ(ρ 1(g)(v)) = ρ 2(g)(φ(v)). Note that the equality makes sense even if φis not invertible, in …