A super Frobenius formula for the characters of Iw

arXiv:0708.1065v1[math.RT]8Aug2007AsuperFrobeniusformulaforthecharactersofIwahori-HeckealgebrasHideoMitsuhashi∗DepartmentofMechanicalEngineeringTomakomaiNationalCollegeofTechnology443Nishikioka,Tomakomai,Hokkaido,059-1275,JapanAbstractInthispaper,weestablishasuperFrobeniusformulaforthecharactersofIwahori-Heckealgebras.WedefineHall-Littlewoodsupersymmetricfunctionsinastandardmannertomakesupersymmetricfunctionsfromsymmetricfunctions,andinvestigatesomepropertiesofsupersymmetricfunctions.BasedonSchur-WeylreciprocitybetweenIwahori-Heckealgebrasandthegeneralquantumsuperalgebras,whichwasobtainedin[8],wederivethatoneofseveraltypesofHall-Littlewoodsupersym-metricfunctions,uptoconstant,generatesthevaluesoftheirreduciblecharactersofIwahori-Heckealgebrasattheelementscorrespondingtocyclepermutations.Ourformulainthisarticleincludesboththeordinaryquantumcase(n=0)thatwasobtainedin[10]andtheclassicalsupercase(q→1).1IntroductionTheFrobeniusformulaisoneofpowerfulmethodstocomputetheirreduciblecharactersofsymmetricgroups.ItisbasedonSchur-Weylreciprocity;thereareactionsofsymmetricgroupsandofgenerallineargroupswhichgeneratethefullcentralizerofeachother.Schur-Weylreciprocityhasbeenextendedtovariousgroupsandalgebrasuptothepresent.Amongthem,tworemarkableextensionsforusarethesupertypeextension([2],[12])andthequantumtypeone([4]).Inourpaper[8],weestablishedSchur-WeylreciprocitybetweentheIwahori-HeckealgebraHQ(q),r(q)andthequantumsuperalgebraUσqgl(m,n)
,whichunifiesSchur-Weylreciprocityofsupertypeandthatofquantumtype.In[10],Ramgavea(ordinary)FrobeniusformulaforthecharactersoftheIwahori-HeckealgebrasoftypeA,whichisbasedontheSchur-WeylreciprocitybetweentheIwahori-HeckealgebraoftypeAandthequantumenvelopingalgebraofglnthatwasgivenin[4].AnextensionofFrobeniusformulatoAriki-Koikealgebras,thatareHeckealgebrasassociatedtocomplexreflectiongroupsG(r,1,n),isfoundin[13].Inthispaper,∗E-mail:[e-mail]1wegiveasuperFrobeniusformulaforthecharactersoftheIwahori-HeckealgebrasoftypeAthatextendstheresultofRam.AsintheMacdonald’sbook[6],symmetricfunctionsplayacrucialroleintherepresentationtheoryofsymmetricgroups.Especially,thetransitionmatrixM(p,s)frompowersumfunctionstoSchurfunctionsisthecharactertableofthesymmetricgroup.CombinatorialrulestocomputecharactervaluessuchasMurnaghan-Nakayamaformulahavebeengivenbymakinguseofpropertiesofsymmetricfunctions.Ontheotherhand,variousgeneralizationsofsymmetricfunctionsaredefineduntilnow.Hall-LittlewoodfunctionsaresymmetricfunctionswithoneparameterwhichintermediatebetweenmonomialsymmetricfunctionsandSchurfunctions.AmongseveraltypesofHall-Littlewoodfunctions,Theonewhichisdenotedbyqλ(x,q)yieldstheFrobeniusformulaforthecharactersoftheIwahori-Heckealgebra([10]).InordertogiveasuperFrobeniusformulafortheIwahori-Heckealgebra,wedefinetheHall-LittlewoodsupersymmetricfunctionsPλ(x/y;q)andqλ(x/y,q)andinvestigatesomepropertiesoftheminsection2.Letx=(x1,x2,...,xm)andy=(y1,y2,...,yn)becommutativevariables.UsingapartitionofunityinHQ(q),r(q),thatisacompletesetoforthogonalminimalidempotentswhichisspecializedtoapartitionofunityinC[Sr],wederivethatthetraceoftheproductofπr(h)∈AqandDr∈BqisR.H.S.ofthesuperFrobeniusformula(Definitionsofπr,Aq,Dr,Bqaregiveninsection3).Theorem4.4.Foranyh∈HQ(q)(x,y),r(q),tr(Drπr(h))=Xλ⊢rχλ(h)sλ(x/y),whereχλistheirreduciblecharacterofHQ(q)(x,y),r(q)correspondingtoλ.WeinvestigateL.H.S.ofTheorem4.4insection5.WeobtainthatwhenhistheelementTγkofHQ(q)(x,y),r(q)correspondingtothecyclepermutationoflengthk,L.H.S.coincideswithqk(x/y;q−2)uptoconstant.Theorem5.3.trDkπk(Tγk)
=qkq−q−1qk(x/y;q−2).Finally,fortheproductTγμ=Tγμ1Tγμ2···Tγμlwhereμ=(μ1,μ2,...)⊢r,wehaveTheorem5.5.Forμ⊢r,q|μ|(q−q−1)l(μ)qμ(x/y;q−2)=Xλ⊢rχλ(Tγμ)sλ(x/y).ThoughnotallthevaluesofcharacterscanbecomputedbyTheorem5.5,owingtoourversionofRam’sresultTheorem5.6([10],Theorem5.1).ForeachTσ,σ∈Sr,thereexistsaZ[q,q−1]linearcombinationcσ=Xμ⊢raσμTγμ,2aσμ∈Z[q,q−1],suchthatχ(Tσ)=χ(cσ)forallcharactersχofHQ(q),r(q),showsthatanycharacterofHQ(q),r(q)isdeterminedbyitsvaluesontheelementsTγμ.Ourresultextends[10],andmayyieldnewcombinatorialrulesforcomputingtheirreduciblechar-actersofIwahori-Heckealgebras.Moreover,aswehavepointedoutin[8],thesuperversionoftherepresentationtheoryofthesymmetricgroupandtheIwahori-Heckealgebraaremoresuitabletode-scribetherepresentationtheoryofthealternatinggroupanditsq-analogue.OurresultwillbeusedtoderiveaFrobeniusformulafortheq-analogueofthealternatinggroupwhichwasdefinedin[7].2PreliminariesonsupersymmetricfunctionsSymmetricfunctions,besidestheirowninterest,playimportantrolesinvariousareasinmathematics.Particularly,therelationbetweensymmetricfunctionsandtherepresentationtheoryisintimate.Inthissection,wewillgivesupersymmetrizationsofvariousclassesofsymmetricfunctionsandinvestigaterelationshipsbetweenthem.ThebasicreferenceofsymmetricfunctionsandsupersymmetricfunctionsisMacdonald’sbook[6].Wewillfollow[6]withrespecttoournotationaboutsymmetricfunctionsunlessotherwisestated.WedenotebySmthesymmetricgroupoforderm!.LetΛm=Z[x1,x2,...,xm]Smbetheringofsymmet