Bessis and Reiner---Cyclic sieving of noncrossing partitions for complex reflection groups.pdf

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1、Ann.Comb.15(2011)197–222AnnalsofCombinatoricsDOI10.1007/s00026-011-0090-9PublishedonlineMay15,2011©SpringerBaselAG2011CyclicSievingofNoncrossingPartitionsforComplexReflectionGroupsDavidBessis1andVictorReiner2∗1DMA-Ecolenormalesup´´erieure,45rued’Ulm,75230Pariscedex

2、05,Francedavid.bessis@ens.fr2SchoolofMathematics,UniversityofMinnesota,Minneapolis,MN55455,USAreiner@math.umn.eduReceivedJuly28,2008MathematicsSubjectClassification:20F55,51F15Abstract.Weproveaninstanceofthecyclicsievingphenomenon,occurringinthecontextofnoncrossing

3、parititionsforwell-generatedcomplexreflectiongroups.Keywords:complexreflectiongroup,unitaryreflectiongroup,noncrossingpartition,cyclicsievingphenomenon,rationalCherednikalgebra1.IntroductionOurgoalisTheorem1.1below,whoseterminologyisexplainedbrieflyhere,andmorefullyin

4、thenextsection.LetV=CnandletW⊂GL(V)beafiniteirreduciblecomplexreflectiongroup,thatis,WisgeneratedbyitssetRofreflections.ShephardandTodd[30]classi-fiedsuchgroups,andusedthistoshowthatwhenWactsonthesymmetricalgebraS:=Sym(V∗),itsinvariantsubringSWisitselfapolynomialalgeb

5、ra,generatedbyhomogeneouspolynomialsf1,...,fnwhosedegreesd1≤d2≤···≤dnareuniquelydetermined;thiswasalsoprovenwithoutusingtheclassificationbyChevalley[14].WewillassumefurtherthatWiswell-generatedinthesensethatitcanbegener-atedbynofitsreflections.DefinetheCoxeternumberh

6、:=dnandtheW-q-Catalannumbern[h+di]qCat(W,q):=∏,(1.1)[di]qi=12n−1qn−1where[n]q:=1+q+q+···+q=.LetcbearegularelementofWintheq−1senseofSpringer[33],oforderh;sucharegularelementoforderhwillexistbecauseWiswell-generated;seeSubsection2.3.Inotherwords,chasaneigenvectorv∈V

7、∗SecondauthorsupportedbyNSFgrantDMS–9877047.198D.BessisandV.Reinerfixedbynoneofthereflections,andwhosec-eigenvalueζhisaprimitivehthrootofunity.LetNC(W):=w∈W:−1R(w)+Rwc=n,(1.2)whereRistheabsolutelengthfunctionR(w):=min{:w=r1r2···rforsomeri∈R}.(1.3)Theinitia

8、ls“NC”inNC(W)aremotivatedbythespecialcasewhereWistheWeylgroupoftypeAn−1,andthesetNC(W)bijectswiththenoncrossingpartitionsof{1,2,...,n};seeSubsection2.1b