Transcription of Representation Theory - James Lingard
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Representation Theory - James Lingard
Representation TheoryRepresentationsLetGbe a group andVa vector space over a fieldk. ArepresentationofGonVis a grouphomomorphism :G Aut(V). Thedegree(ordimension) of is just representationsLet :G Aut(V) and :G Aut(V ) be two representations ofG. Then aG-linear mapfrom to is a linear map :V V such that ( (g)) = ( (g)) for allg G, or equivalently such that the following diagram commutes:V (g)// V V (g)//V If additionally is an isomorphism of vector spaces then we say that is anisomorphismfrom to , and that isisomorphicto . Notice that :V V is an isomorphism from to iff 1is an isomorphism from to , so isomorphism is an equivalence Gandv V, we often writegvinstead of (g)v. In this notation, :V V is anisomorphism iffg (v) = (gv)for allg Gand allv onV, andWis a subspace ofVsuch thatg(W) Wfor allg G, then we say thatWis subrepresentation ofVistrivialif it is 0 orV, and indecomposible representationsA Representation is calledirreducibleif it has no non-trivial adirect sumofWandW , writtenV=W W , ifWandW are subrepresentations ofVandV=W W as vector spaces.
Representation Theory Representations Let G be a group and V a vector space over a field k.A representation of G on V is a group homomorphism ‰: G ! Aut(V).The degree (or dimension) of ‰ is just dimV.
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