On the U-module structure of unipotent Specht modules of finite general linear groups [Elektronische Ressource] / vorgelegt von Qiong Guo

OntheUmoduleStructure
ofUnipotentSpechtModules
GroupsLinearGeneralFiniteof

VonderFakultatMathematikundPhysikderUniversitatStuttgart
zurErlangungderWurdeeinesDoktorsder
Naturwissenschaften(Dr.rer.nat)genehmigteAbhandlung

MitbHauptbericerichhter:ter:

VonvorgelegtGuoQiong

Chinaaus

Prof.Dr.rer.nat.R.Dipper
Prof.Dr.rer.nat.S.Konig
LyleS.Dr.

TagdermundlichenPrufung:21.April2011

InstitutfurAlgebraundZahlentheoriederUniversitatStuttgart

2011

D93

Diss.

atersitUniv

Stuttgart

tstenCon

ductiontroIn

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1Preliminaries11.1Basicsetting...........................1
1.2Thesymmetricgroup......................3
1.3Tableaux.............................6
1.4RootsubgroupsofGL(q)...................7
n

β2TheSpechtmoduleS11
β2.1ThepermutationmoduleM.................11
β2.2TheSpechtmoduleS.....................13
2.3RelationswithIwahori-Heckealgebras............15

)WλWn(3ThepermutationmodulesM17
)WλWn(3.1AdifferentdescriptionMof..............18
)WλWn(3.2TheidempotentbasisMof...............21
wβ3.3StructureofMasanF(U\U)module..........25
3.4Patternmatricesandconditionsets..............34
3.5TheirreducibilitMyof....................45
O3.6UinvarianceMof......................47
O

)WλWn(4TheSpechtmodulesS55
4.1Thehomomorphism....................55
W)WλWn(4.2SpecialorbitsMin...................78
)WλWn(4.3StandardbasisSof...................81
4.4Rankpolynomialsr(q).....................100
J4.5Mainresults...........................104
4.6Examples............................105

summaryGerman5

Notation

yBibliograph

107

115

117

ductiontroIn

In[12],GordonJamesinvestigatedtheSpechtmodulesofthesymmetric
groups:ForeachpartitionλofnthereisaSpechtmoSdule(λ),defined
intermsoftheintersectionofthekernelsofcertainhomomorphisms.The
dimensionofS(λ)forSncanbedeterminedintwodifferentways.
(1)LethβQRbethehooklengthfori,the)j(nodein[λ].Then
(QλR)2[β]hQR
dimS(λ)=Qn!β.
(2)dimS(λ)equalsthenumberofstandardλtableaux.Indeed,there
ThisexistsabasisbasisisS(λof)calledwhicahisstandardindexedS(λbasis).byoftheλstandardtableaux.
FollodefinedwingthetheunipphilosophotentySpthatecGLhtn(moq)isdulesSβaqoveranalogatofieldSFn,forGordonGLn(qJames)in
β[13].indepIfendenthetofFc.IndeedharacteristicFoneisexpofectscoprimethetoqthenrepresenthetationdimensiontheoryGLSof(forq)is
totranslateintothatforSnbysettingq=1.ForGLn(q)wehaven
(QλR)2[β][hQR]
dimSβ=qP(S1)βUQ[n]!β
wherer[]=1+q+q2+··+·qb1.Inthesenseofthefollowingconjecture
byRichardDipperandGordonJames,wehavetheanalogousconceptof
basis.standardConjecture.ForC2Std(λ),thereexistrC(t)2Z[St]withconstantterm1
andBCSβofsizejBCj=rC(q)suchthatB=C2Std(β)BCisabasisSβof.
BiscalledthestandardSbasisβ.of
ThisconjecturewasprovedbyMarcoBrandt,RichardDipper,Gordon
JamesandSin´eadLyleforthecaseλisthata2-partpartition.Butthe
wproorkofforisanratherarbitrarycomλ.binationalThisishenceouritmotivationseemstothefindamethonewdmethotheredwillnot

iv

whichismorerelatedtorepresentationtheory.Infact,wegiveanew
proofofthisconjectureforthecaseλisthata2-partpartitionandthe
characteristicofthefieldiszero.Unfortunatelywhenwemovetothe
arbitrarycharacteristiccase,weappealtoBDJL’sresultatthemoment.
However,wedothinkwecanprovideanindependentproofinthenear
future.WedecomposethepermutationmoduleM(nWλW)intoirreducible
FUnmodulesforthe2-partpartitioncaseUnwhereisthegroupof(lower)
unitriangularmatricesGLinn(q),denotedbyUforconvenience.Thus
wegetamethodtoinvestigatethekernelofthehomomorphismsbetween
permutationmodules,whichgivestheunipotentSpechtmodule.Thisdoes
notonlygiveusthehopetosolvethisconjectureforgeneralpartitions,but
alsointroduceawaypossiblytosolvesomemoreproblems.Forexample,
wehavefoundeveryirreduciblecomponentofthepermutationmodule
M(nWλW)hasadimensionqofpower,therefore,ifitistrueforanarbitrary
partition,wehaveagoodchancetogiveanewproofofatheoremgiven
byIsaacs:EveryirreduciblecomplexcharacterUhasofqpowerdegree.
ThereisalsoagoodchancetosolveaconjectureofHigman:Thenumber
ofconjugateclassesUofisapolynomialinq.
OurnewstrategyistoinvestigateFUthemodulestructureofSβ.The
advantageofrestrictionFUtomoduleisthatFUissemisimple,sinceby
generalassumptionthecharacteristicofthefieldisq.coprimeIndeed,towe
giveacompletedecompositionMof(nWλW)intoirreducibleFUmodules.
WefindeveryirreducibleFUsubmoduleofM(nWλW)islabeledbysome
setS.Wecallitconditionset,andthecorrespondingirreduciblemodule
hasdimensionqL(06c2Z)wherecisfixedbythepositionsoftheentries
intheconditionsetS.Hencethenumberofirreducibledirectsummands
ofResππUGM(nWλW)toafixeddimensionqLisapolynomialinq.
Chapter1setsthesceneandgivesanoverviewofthefundamentaldefini-
tionsandpropositionsforcompositions,partitions,λ-tableauxandBruhat-
osition.decomptheoriginaldefinitionsofthepermutationmoMduleβ=PβFGLn(q)
Inchapter2,foranarbitrarycomposition,weintroλduceflags.We)give
andtheunipotentSpechtmoduleSβ=MβEτ+(λ0)FGLn(q)whereEτ+(λ0)
isanidempotentinU.Butβinfactinthefollowingchapter,weuseanequiv-
alentdefinition.βDefineMasvectorspaceovFerwithλflagsasitsbasis
anddefineSastheintersectionofthekernelsofcertainhomomorphisms.
Fromchapter3,wefocusourattentionon2-partpartitionλ=(nm,m).
WestartwiththeintroductionofanotationWλnΞofthesetλofflags.
Aswecanassigntoeacλhflagλatableauandwehaveatotalorder-
ingon∑thesetofrowstandardλtableaux.Wedefine,foranelement
v=X2YλZCXXinMβ,last(v)asthelastλtableauwhichcanbeas-
signedtoλaflagXoccurringinthissumwithnonzerocoecienCtX;
top(v)asthecollectionofallλtheflagXoccurringinthissumwith

v

tab(X)=last(v).MotivatedbythefactSβthatisasubmoduleofMβ,we
carefullyinvestigatetheoperationUonMofβ.WefirstdecomposeMβinto
JbatchesMJwhereJ2RStd(λ)byusingMackeyDecompositionthenwe
decomposeeachbatchintodirectsummandofirreduciblesubmodules.In
fact,MJhasabasisoforthogonalprimitiveidempEJ=oten{tse/jL2XJ}
whichismoreadaptabletoFtheUmodulestructure.Weshowthatthe
subgroupUw\UofUactsmonomiallyonthesetEJwhereJ=Jβw.Then
weprovetheUw\UorbitmoduleisanirreducibleFUmodule.Moreover
wefindeachirreducibleorbitmodulehasadimensionqofpowsomeer;
andthereisauniquelydeterminedmatrixineachorbit,calledapattern
matrix;andeachorbitcanbeattachedtoauniqueset,calledacondition
set.Moreover,wecanprovewhentheconditionsetisthesame,thenthe
correspondingirreducibleorbitmodulesareisomorphic.
FindingastandardbasisSβofforatwopartpartitionλ=(nm,m)is
thegoalofchapter4.Whenλisa2-partpartition,wehave
Sβ=kerϕ(λR;dimSβ=mm1.
W\1[n[n
=0QSin´eadLyleprovesin[15]thatforeveryelemen≠vt20Sβ,last(v)isa
standardλtableau.WeshowincharacteristiczeroSβ=case,kerϕ1λW1.
Thusaftercomparingthedimensionsweϕ1obtainλW1isanepimorphism.

Finallywegetthefollowingtheorem:
Theorem.(4.4.12)Letλ=(nm,m).ForL2ΞWλn,thereexistsv/2Sβ
suchthatlast(v/)=tab(L)andtop(v/)=e/ifandonlyif
tab(L)n{bQU,Sjj16k6s}isashiftedµstandardtableau,
wheree/2O,S=S(O)={lNRUSUj16k6s},µ=(nms,ms).
ForeveryLsatisfyingtheconditionsabove,wefixonev/elemen(nott
letanddetermined)uniquelyBSβ:={v/je/2OMβ,S(O)=S,tab(L)n(S.∪S,)isstandard}
andBβ=S˙BSβ.ThenBSβisastandardbasisofSthecomponentSβ#S
SandBβisastandardbasisSβ.of

last(Moreov)v=erJ2weStd(shoλ)wisthethenumbrankerpofolynomialtherJ(q)basisgivenelemenbyvtsBDJLsuchinthat[4],
whicdegreehproofvidestheaunipnewotentproSpofecofhtthemoSβdulefactwhereλthat=(ntheym,madd).uptothegeneric

tswledgmenknoAc

IManspyenptweopleorkinghavonethissupported,thesis.Iencouragedwishtoandexpresshelpmyedmegratitudeduringtoalltheoftimethem.

HeFirsthasofball,eenIwaouldgreatliketosourcethankofmysupmotivervisorationandProf.IamDr.RicgratefulhardtoDipphimer.for
havingintroducedmetothefascinatingresearchareaofrepresentation
thattheoryledoftothethisfinitethesis.generalInadditionlinearIgroupamsoandforgratefulguidingforhismemanyresearchsuggestionwork
forformulationoftheresultsofthisthesis,whichhelpedsubstantiallyto
maketheoriginalmanuscripteasiertoread.

FKonigurthermore,andDr.ISinwould´eadlikeLyletoforthankmreadingyco-supthiservisorsthesis.Prof.Dr.Steffen

Manythankstomycolleaguesandfriendsatthe“AbteilungfurDarstel-
lungstheorie"andthe“FachbereichMathematik"whohavemademefeel
verycomfortableattheUniversityofStuttgart.Inparticular,Iwouldlike
tothankmycollegeBerndAckermannforproof-readingthisthesisand
mycollegeMathiasWerthforhelpingmewiththeGermanpartofthis
thesis.

ForfinancialsupportIamgratefultotheChineseChinaScholarship
Councilandthe“FachbereichMathematik".

invFinallyaluable,Iwouldsupplikorteovtoerthankthemylastparenytsearsforwhictheirhallowedencouragemenmetotfullyandconcen-their
trateonmyresearchandthussignificantlycontributedtothesuccessful
thesis.thisofcompletion

vi

1Chapter

Preliminaries

settingBasic1.1Throughoutthisthesis,pbleteaprime,qbeafixedpowerpof;in
particular,itisnever1.FLetbeafieldwhosecharacteristiciscoprime
topandwhichcontainsaprimitivptherootofunity.Fdenotesthe
mtheultiplicativgroupofeinvgroupertiblenFof×.nLetnbmatriceseaovnaturalerGF(nqum),bertheandfieldGLnq(ofq)elemendenotets.

LetVbeavectorspaceovGFer(q)withbasisv(,)v,··,·Zv.Thenwecan
freelyidentifyGLn(q)withthegroupofallautomorphismsVofacting
fromtheright.TheautomorphismgivenbythegRSmatrix)is:(
nvR7!vSgRS,16i6n.
∑=1RIfv(,)v,··,·UvarevectorsinV,welet
⟨v(,)v,··,·Uv⟩