Generalized characters of the symmetric group

arXiv:math/0605029v1[math.CO]1May2006GENERALIZEDCHARACTERSOFTHESYMMETRICGROUPEUGENESTRAHOVAbstract.NormalizedirreduciblecharactersofthesymmetricgroupS(n)canbeun-derstoodaszonalsphericalfunctionsoftheGelfandpair(S(n)×S(n),diagS(n)).TheyformanorthogonalbasisinthespaceofthefunctionsonthegroupS(n)invariantwithrespecttoconjugationsbyS(n).InthispaperweconsideradifferentGelfandpaircon-nectedwiththesymmetricgroup,thatisan“unbalanced”Gelfandpair(S(n)×S(n−1),diagS(n−1)).ZonalsphericalfunctionsofthisGelfandpairformanorthogonalbasisinalargerspaceoffunctionsonS(n),namelyinthespaceoffunctionsinvariantwithrespecttoconjugationsbyS(n−1).WerefertothesezonalsphericalfunctionsasnormalizedgeneralizedcharactersofS(n).Themaindiscoveryofthepresentpaperisthatthesegeneralizedcharacterscanbecomputedonthesamelevelastheirreduciblecharactersofthesymmetricgroup.ThepapergivesaMurnaghan-Nakayamatyperule,aFrobeniustypeformula,andananalogueofthedeterminantalformulaforthegeneralizedcharactersofS(n).1.Introduction1.1.Preliminariesandformulationoftheproblem.Oneofthecentralgoalsoftherepresentationtheoryoffinitegroupsisincomputationofcharactersofirreduciblerepresentations.Whenagroupunderconsiderationsisthesymmetricgroup,S(n),theirreduciblecharacterscanbecomputedusingeithertheFrobeniusformula,orthedeter-minantalformula,ortheMurnaghan-Nakayamarule(see,forexample,Macdonald[Mac],Sagan[S],Stanley[St]).LetΛdenotethealgebraofsymmetricfunctions,whichisagradedalgebra,isomorphictothealgebraofpolynomialsinthepowersumsp1,p2,....Ifwedefinepρ=pρ1pρ2...foreachpartitionρ=(ρ1,ρ2,...)1,thenthepρformahomogeneousbasisinΛ.AnothernaturalhomogeneousbasisinΛisformedbytheSchurfunctionssλindexedbyYoungdiagramsλ.TheFrobeniusformulaispρ=Xλ⊢nχλρsλ,whereχλρisthevalueoftheirreduciblecharacterχλofthesymmetricgroupS(n)ontheconjugacyclassinS(n)indexedbythepartitionρofn.ThisformulaisthekeyresultintheclassicaltheoryofcharactersofthesymmetricgroupS(n).ItshowsthatthecharactertableisthetransitionmatrixbetweentwobasespρandsλinthealgebraofsymmetricfunctionsΛ.TheFrobeniusformulafollowsfromthefactthattheSchurfunctionssλareimagesofχλinΛunderacertainmap.Thismapiscalledthecharacteristicmap,see[Mac]I,§7.Thus,ifwedenotethismapbych,wehavesλ=ch(χλ).AnotheravailableresultonirreduciblecharactersofS(n)istheformulawhichrepresentsanirreduciblecharacter,χλ,asanalternatingsumoftheinducedcharacters(i.e.the1AsinMacdonald[Mac]weidentifyeachpartitionwithitsYoungdiagram.12EUGENESTRAHOVdeterminantalformula).Namely,denotebyηktheidentitycharacterofS(k).Ifλ=(λ1,λ2,...)isanypartitionofn,letηλdenoteηλ1·ηλ2·....Herethemultiplication,f·g,betweentwocharacters,fandg,of,say,groupsS(k)andS(m)isdefinedbytheinductionf·g=indS(k+m)S(k)×S(m)(f×g).Withtheabovenotationtheirreduciblecharacterχλisgivenbyχλ=det(ηλi−i+j)1≤i,j≤n.Sincech(ηλ)=hλ,wherehλ=hλ1hλ2...,andhristherthcompletesymmetricfunc-tion,thedeterminantalformulaforirreduciblecharactersisequivalenttotheJacobi-TrudiformulafortheSchursymmetricfunctions,sλ=det(hλi−i+j)1≤i,j≤n.TheMurnaghan-Nakayamaruleisarecursivemethodtocomputetheirreduciblechar-actersofthesymmetricgroups.Itcanbeformulatedasfollows.LetussaythataskewYoungdiagramisaborderstripifitisconnectedanddoesnotcontainany2×2blockofboxes.SupposethatπσisanelementofthesymmetricgroupS(n),whereσisacycleoflengthj,andπisapermutationoftheremainingn−jnumbersofcycle-typeρ,ρisapartitionofn−j.TheMurnaghan-Nakayamarulesaysthatthevalueoftheirre-duciblecharacterofS(n)parameterizedbytheYoungdiagramλontheelementπσ(i.e.ofχλ(πσ))isgivenbyχλ(πσ)=Xν⊆λν⊢n−jφλ/νχνρwhereφλ/νisacombinatorialcoefficient.Thiscombinatorialcoefficientisdefinedbytheformulaφλ/ν=(−1)hλ/νi,ifλ/νisaborderstrip;0,otherwise,wherehλ/νiistheheightofaborderstripdefinedtobeonelessthanthenumberofrowsitoccupies.ThetheoryofcharacterscanbereformulatedintermsofGelfandpairs,see[Mac],VII,§1.Specifically,letGbeafinitegroup,andKbeasubgroupofG.DenotebyC(G,K)thealgebraofcomplexvaluedfunctionsfonG(withconvolutionasthemultiplication)suchthatf(kxk′)=f(x)forallx∈Gandk,k′∈K.IfC(G,K)iscommutative,thepair(G,K)iscalledaGelfandpair,andonecanassociatewith(G,K)thesetofzonalsphericalfunctions.Zonalsphericalfunctionshavemanyremarkableproperties,someofthesepropertiesareanalogoustothoseofgroupcharacters.Inparticular,thesetofzonalsphericalfunctionsdefinesanorthogonalbasisofC(G,K),see[Mac],VII,§1.IfKisafinitegroup,thenthenormalizedirreduciblecharactersofKarecloselyconnectedwiththezonalsphericalfunctionsoftheGelfandpair(K×K,diagK),see[Mac],VII,§1,Ex.9,andSection2below.(HerediagK={(x,x):x∈K}isthediagonalsubgroupofK×K.)Explicitly,letχi(1≤i≤r)betheirreduciblecharactersofK,andωi(1≤i≤r)bethezonalsphericalfunctionsoftheGelfandpair(K×K,diagK).Thenforallelementsx,yofthegroupKthefollowingformulaholds(1.1.1)ωi(x,y)=χi(xy−1)χi(e).GENERALIZEDCHARACTERSOFTHESYMMETRICGROUP3InthissituationC(K×K,diagK)canbeidentifiedwiththealgebraofcentralfunctionsfdefinedonthegroupK,i.e.itconsistsofthefunctionsfsuchthatf(xyx−1)=f(y)forallx,y∈K.Inparticular,ifK=S(n),whereS(n)denotesthesymmetricgroupofsymbols1,2,...,n,thenC(S(n)×S(n),diagS(n))canbeide