Normal Supercharacter Theory

NormalSupercharacterTheory

FaridAliniaeifard

arXiv:1503.02734v1 [math.RA] 9 Mar 2015March11,2015AbstractTherearetwomainconstructionsofsupercharactertheoriesforagroupG.The?rst,de?nedbyDiaconisandIsaacs,comesfromtheactionofagroupAviaautomorphismsonourgivengroupG.Thesecond,de?nedbyHendrickson,iscombiningasupercharactertheoriesofanormalsubgroupNofGwithasupercharactertheoryofG/N.InthispaperweconstructasupercharactertheoryfromanarbitrarysetofnormalsubgroupsofG.WeshowthatwhenconsiderthesetofallnormalsubgroupsofG,thecorrespondingsupercharactertheoryisrelatedtoapartitionofGgivenbycertainvaluesonthecentralidempotents.Also,weshowthesupercharactertheoriesthatweconstructcannotbeobtainedviaautomorphismsorasinglenormalsubgroup.1IntroductionLetUn(q)denotethegroupofn×nunipotentuppertriangularmatricesovera?nite?eldFq.Classi?cationoftheirreduciblecharactersofUn(q)isawell-knownwildproblem,provablyintractableforarbitraryn.Inorderto?ndamoretractablewaytounderstandtherepresentationtheoryofUn(q),C.Andr´e[2]de?nesandconstructssupercharactertheory.YanshowshowtoreplaceAndr´e’sconstructionwithmoreelementarymethods.Diaconis

andIsaacs[6]axiomatizetheconceptofsupercharactertheoryforanarbitrarygroup.TheymentionhowtoobtainasupercharactertheoryforGfromtheactionofAonGbyauto-morphisms.TheyalsogeneralizeAndr´e’soriginalconstructiontode?neasupercharactertheoryforalgebragroups,agroupoftheform1+JwhereJisa?nitedimensionalnilpotentassociativealgebraovera?nite?http://wendang.chazidian.comter,inHendricksonshowshowtoconstructothersupercharactertheoriesforanarbitrarygroupGbycombiningcertainsupercharactertheoryforanormalsubgroupNofGwithasupercharactertheoryforG/N.Alsointheauthorsobtainarelationshipbetweenthesupercharactertheoryofallunipo-tentuppertriangularmatricesovera?nite?eldFqsimultaneouslyandthecombinatorialHopfalgebraofsymmetricfunctionsinnon-commutingvariables.LetGbea?nitegroup,wedenotethesetofirreduciblecharactersofGbyIrr(G).TheconjugacyclasscontainingganditscardinalityaredenotedbyCgandmgrespectively.For??=??asubsetSofG,letSs∈Ss.

1

LetN(G)bethesetofallnormalsubgroupofG.Let{N1,...,Nk}?N(G).Wede?neA(N1,...,Nk)tobethesmallestsubsetofN(G)suchthat

(1)e,G∈A(N1,...,Nk).

(2){N1,...,Nk}?A(N1,...,Nk).

(3)A(N1,...,Nk)isclosedunderproductandintersection.

De?ne

?NA(N1,···,Nk)=N\

K∈A(N1,···,Nk),K<N??K.

?Forsimplicityofnotation,wewriteN?insteadofNA(N1,...,Nk)whenitisclearthatNisin

A(N1,...,Nk).Wewillshowthat{N?:N∈A(N1,...,Nk)}isthesetofsuperclassesofasupercharactertheory,andwecallsuchsupercharactertheorythenormalsuperchractertheorygeneratedby{N1,...,Nk}.Ingeneralthissupercharactertheoriescannotbecon-structedbytheprevioussupercharctertheoryconstructions.Remarkthatwhenwehavealargersetofnormalsubgroups,thenormalsupercharactertheoryweobtainwillbe?ner.Inparticularthe?nestnormalsupercharactertheoryisobtainedwhenweconsiderthesetofallnormalsubgroupsofG,andisrelatedtoapartitionofGgivenbycertainvaluesonthecentralidempotents.

InSection2,wereviewde?nitionsandnotationsforsupercharactertheories.Inpartic-ularwementiontheknownconstructionsofsupercharactertheories.NextinSection3,wede?neournormalsupercharactertheoryconstruction.Wealsoshowthatthe?nestnormalsupercharactertheoryisobtainedbyconsideringcertainvaluesofthecentralidempotents.InSection4,weshowthatthenormalsupercharactertheorycannotbeobtainedbythepre-viousgeneralconstructions.Finally,inthelastsectionwelistsomeopenproblemsrelatedtotheconcept.

Acknowledgment

IwishtoexpressmyappreciationtomysupervisorprofessorBergeronwhocarefullyreadanearlierversionofthispaperandmadesigni?cantsuggestionsforimprovement.Also,IwouldliketothankShuXiaoLiforhishelpfulcomments.

2Background

We?rstmentionthede?nitionofsupercharactertheorybyDiaconisandIsaacs[6].

Asupercharactertheoryofa?nitegroupGisapair(X,K)whereXisapartitionofIrr(G)andKisapartitionofGsuchthat:

2

(a)|K|=|X|,

(b)forX∈X,thecharacterXX,anonzerocharacterwhoseirreducibleconstituentslieinX,isconstantonthepartsofKand

(c)theset{1}∈K.

WewillrefertocharactersXXasthesupercharactersandtothememberofKassuperclasses.

Every?nitegrouphastwotrivialsupercharactertheories:theusualirreduciblecharac-tertheoryandthesupercharactertheory({1},{Irr(G)\1}},{{1},G\{1}}),where1istheprincipalcharacterofG.

TheconceptofaSchurringisde?nedbySchurin[11].Hendrickson[7]showsthatthereisabijectionbetweenthesupercharactertheoriesofagroupGandSchurringsoverGcon-tainedinZ(C[G]),thecenterofC[G].

De?nition.LetGbea?nitegroup.AsubringSofthegroupalgebraC[G]iscalledaSchur

??:K∈K},ringoverGifthereisasetpartitionKofGsuchthat{1}∈K,S=C-span{K

and{g?1:g∈K}∈KforallK∈K.

Proposition2.1[7,Proposition2.4]LetGbea?nitegroup.Thenthereisabijection{Supercharactertheories(X,K)ofG}←→{SchurringsoverGcontainedinZ(C[G])}

(X,K)?→??:K∈K}.C-span{K

Intheproofofsurjectivityoftheabovebijection,Hendricksondoesnotneedthecondition{g?1:g∈K}∈Kfromthede?nitionofSchurring.Sowehavethefollowingcorollary.Corollary2.2LetGbea?nitegroupandletKbeapartitionofG.Thenthefollowingstatementsareequivalent.

(1)Kisthesetofsuperclassesofasupercharactertheory.

??:K∈K}isasubringofZ(C[G]).(2){1}∈KandC-span{K

De?nition.AsuperclasstheoryisapartitionKofGsatisfyingoneofthetwoequivalentconditionsinCorollary2.2.

Nowwediscusstwomainmethodsofconstructingsupercharactertheoriesofanarbitrary?nitegroup.

2.1AGroupActsViaAutomorphismsonaGivenGroup

Given?nitegroupsAandG,wesaythatAactsviaautomorphismsonGifAactsonGasaset,andinaddition(gh).x=(g.x)(h.x)forallg,h∈Gandx∈A.Anactionvia

3

automorphismsofAonGdeterminesandisdeterminedbyahomomorphismφ:A→Aut(G).

SupposethatAisagroupthatactsviaautomorphismsonourgivengroupG.ItiswellknownthatApermutesboththeirreduciblecharactersofGandtheconjugacyclassesofG.ByalemmaofR.Brauer,thepermutationcharactersofAcorrespondingtothesetwoactionsareidentical,andsothenumbersofA-orbitsonIrr(G)andonthesetofclassesofGareequal(SeeTheorem6.32andCorollary6.33of[8]).Itiseasytoseethattheseorbitdecompositionsyieldasupercharactertheory(X,K)wheremembersofXaretheA-orbitsonIrr(G)andmembersofKaretheunionsoftheA-orbitsontheclassesofG.Itisclearthatinthissituation,thesumofthecharactersinanorbitX∈XisconstantoneachmemberofK.WedenotebyAutSup(G)thesetofallsuchsupercharactertheoriesofG.

2.2?-Product

SupposethatAisagroupthatactsviaautomorphismsonourgivengroupG.LetSup(G)bethesetofallsupercharactertheoriesofG.Wesaythat(X,K)∈Sup(G)isA-invariantiftheactionofA?xeseachpartK∈Ksetwise.WedenotebySupA(G)thesetofA-invariantsupercharactertheoriesofG.NotethatifNisnormalinG,thenC∈Sup(H)isG-invariantifandonlyifitssuperclassesareunionofconjugacyclassesofG.Also,ifM,NarenormalsubgroupofGandN<M,thenasupercharactertheoryofM/NisG/N-invariantifandonlyifitisG-invariant.

Notation.LetNbeanormalsubgroupofagroupG.IfLisasetofsubsetsofG/N,

={∩Ng∈LNg:L∈L}.Letχ∈Irr(N).WedenotebyIrr(G|ψ)thesetthenwede?neL

ofirreduciblecharactersψofGsuchthattheinnerproductofψandχispositive.IfZisasetofsubsetsofIrr(N),thenwede?neZG={∪ψ∈ZIrr(G|ψ):Z∈Z}.Nowconsider(X,K)∈SupG(N).Since{1N}∈X,onepartofXGis{1N}G={χ∈Irr(G):N?kerχ},whichweidentifywithIrr(G/N)intheusualnaturalway.

Theorem2.3[7,Theorem4.3]LetGbeagroupandNbeanormalsubgroupofG.LetC=(X,K)∈SupG(N)andD=(Y,L)∈Sup(G/N).Then

\{N})(Y∪XG\{Irr(G/N)},K∪L

isasupercharactertheoryofG.

WecallthesupercharactertheoryofGconstructedintheprocendingtheoremthe?-productof(X,K)and(Y,L),andwriteitas(X,K)?(Y,L).Also,letSup?(G)denotethesetofallsupercharactertheoriesofGobtainedby?-product.

3NormalSupercharacterTheory

Inthischapterweconstructasupercharactertheoryfromanarbitrarysetofnormalsubgroups.Wecallsuchsupercharactertheoryanormalsupercharactertheory.

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3.1SupercharacterTheoryFromCentralIdempotents

Inthissection,weconsiderapartitionofconjugacyclassesandirreduciblecharactersgivenbycertainvaluesofcentralidempotent.Inthenextsectionwewillseethatitisasupercharactertheoryandisgivenbythe?nestnormalsupercharactertheory.

By[10,Proposition8.15]everycharacterχ∈Irr(G)hasacorrespondingcentralidempo-tent

eχ=|G|?1χ(1)??χ(g?1)g.

g∈G

Theseidempotentsareorthogonal,i.e,eχeφ=0whenχ=φ.RecallthatC??g=i(1)?1χi(g)eχi.Therefore,??imgχ

mg1?C??g=mg1???mgχi(1)?1χi(g)eχi=mg(1?χi(1)?1χi(g)eχi)=

ii

mg(??eχ?11

i?

i??????χi(1)χi(g)eχi)=mg((1?χi(1)?χi(g))eχi)

ii

?1?C??g

χi(1))eχi=??(1?χi(g)

i

(1?χi(g)

mg=??i

χi(1)=0},Kg=∪Eg=EhCh,andUg=∪Eh?EgCh.

Asinfollowingexamplewewillseethat{Kg:g∈G}isasuperclasstheory.

Example3.1ThecharactertableofS5is

Classes(1)(12)(123)(1234)(12345)(12)(34)(12)(345)