Representation_are_everywhere

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1、RepresentationsAreEverywhereNanghuaXiMemberofChineseAcademyofSciences1WhatisRepresentationtheoryRepresentationisreappearanceofsomepropertiesorstructuresofoneobjectonanother.I.M.Gelfandhassaid:AllofMathematicsissomekindsofrepresentationtheory."Theaccuratemeanistomakethealg

2、ebraicstructureofanobjecttoreappearonanconcreteobjectwhichismadeupoflineartransformations(ormatrices).Thealgebraicstructureswhichtherepresentationtheoryconcernsin-clude:group,algebra,Liealgebraandsoon.Algebraicstructurecanbedeterminedbyoperation.Representationisonekindofho

3、momorphism.Generally,twomathematicalobjectsarerelatedbymap.Themappreservingtheoperationrepresenttherelationsbetweenstructures,anditiscalledhomomorphism.Arepresentationisjustahomo-morphism,whoseimageiscomposedoflineartransformations.Therepresentationtheorycontainsthreeparts

4、:therepresentationtheoryofgroups,therepresentationtheoryofalgebras,therepresentationtheoryofLiealgebras.²Representationofgroups:homomorphismofgroupGroup¡¡¡¡¡¡¡¡¡¡¡¡¡!finvertiblelineartransformationonalinearspaceg²Representationofalgebras:homomorphismofalgebraAlgebra¡¡¡¡¡¡¡

5、¡¡¡¡¡¡¡!flineartransformationonalinearspaceg²RepresentationofLiealgebras:homomorphismofLiealgebraLiealgebra¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡!flineartransformationonalinearspacegA¯nitedimensionalrepresentationmeansthedimensionallinearspaceis¯nite.1Anotherformofthe¯nitedimensionalrepresentat

6、ion:²Representationofgroups:homomorphismofgroupGroup¡¡¡¡¡¡¡¡¡¡¡¡¡!fn£ninvertiblematricesonsome¯eldg²Representationofalgebras:homomorphismofalgebraAlgebra¡¡¡¡¡¡¡¡¡¡¡¡¡¡!fn£nmatricesonsome¯eldg²RepresentationofLiealgebra:homomorphismofLiealgebraLiealgebras¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡!fn

7、£nmatricesonsome¯eldgTheonedimensionalrepresentationofgroups:homomorphismofgroupGroup¡¡¡¡¡¡¡¡¡¡¡¡¡!fInvertibleelementsonsome¯eldgThisisalsocalledthecharacterofagroup.Example²GL(F)!F¤;A7!detA.HereFisa¯eld,F¤=F¡f0g,nGLn(F)=fn£ninvertiblematricesonFg:³´²TheLegendresymbolxinqu

8、adraticreciprocitylawisacharacterpofacyclicgroupwithdegreenP²Gaussiansuminnumbertheory:G(