请输入您要查询的字词:

 

单词 ProofThatDimensionOfComplexIrreducibleRepresentationDividesOrderOfGroup
释义

proof that dimension of complex irreducible representation divides order of group


Theorem Let G be a finite groupMathworldPlanetmath and V an irreduciblePlanetmathPlanetmathcomplex representation of finite dimensionMathworldPlanetmathPlanetmath d. Then d divides|G|.

Proof: Given any α in the group ring of G (denotedG) we may define a sequence of submodules ofG (regarded as a module over )by Ai equals the linear span of{1,α,α2,,αi}.

G is Noetherian as a module over so wemust have Ai=Ai-1 for some i. Hence αi may beexpressed as a linear combinationMathworldPlanetmath of lower powers ofα. In other α solves a monic polynomial ofdegree i with coefficients in .

Given a conjugacy classMathworldPlanetmathPlanetmath C in G, we may set ϕC=gCg. Then ϕC is central in G, as given hG, we have:

ϕCh=hgCh-1gh=hgCg=hϕC

Hence applying ϕC to V induces a G linear mapVV. By Schur’s lemma this must bemultiplication by somecomplex numberMathworldPlanetmathPlanetmath λC. Then λC is an algebraicintegerMathworldPlanetmath as it solves the same monic polynomial as ϕC.

Also any gG has finite order so the map it induces on Vmust have eigenvaluesMathworldPlanetmathPlanetmathPlanetmathPlanetmath which are roots of unityMathworldPlanetmath and hence algebraicintegers. Hence the sum of the eigenvalues, χV(g), mustalso be an algebraic integer.

Now V is irreducible so,

|G|=gGχV(g)χV(g)*=CGtr(ϕC)χV(C)*=dCGλCχV(C)*

Therefore |G|/d is both rational and an algebraic integer. Henceit is an integer and d divides |G|.

随便看

 

数学辞典收录了18232条数学词条,基本涵盖了常用数学知识及数学英语单词词组的翻译及用法,是数学学习的有利工具。

 

Copyright © 2000-2023 Newdu.com.com All Rights Reserved
更新时间:2025/5/4 6:26:28