Corrigendum to “Generators of the Hecke algebra of (S2,B)” [Adv. Math. 231 (2012) 2465–2483] 2017 Mahir Bilen Can.pdf

该【Corrigendum to “Generators of the Hecke algebra of (S2,B)” [Adv. Math. 231 (2012) 2465–2483] 2017 Mahir Bilen Can 】是由【探春文档】上传分享，文档一共【3】页，该文档可以免费在线阅读，需要了解更多关于【Corrigendum to “Generators of the Hecke algebra of (S2,B)” [Adv. Math. 231 (2012) 2465–2483] 2017 Mahir Bilen Can 】的内容，可以使用淘豆网的站内搜索功能，选择自己适合的文档，以下文字是截取该文章内的部分文字，如需要获得完整电子版，请下载此文档到您的设备，方便您编辑和打印。AdvancesinMathematics308(2017)1337–1339ContentslistsavailableatScienceDirectAdvancesinMathematicsate/aimCorrigendumCorrigendumto“GeneratorsoftheHeckealgebraof(S2n,Bn)”[(2012)2465–2483]MahirBilenCan?,?afak?zdenarticleinfoArticlehistory:Availableonline11January2017CommunicatedbyEzraMillerKeywords:Farahat–HigmanringsStructureconstantsBn-conjugacyclassesIn[1],amongotherthings,weobservedthatthestructureconstantsoftheHeckealgebraoftheGel’fandpair(S2n,Bn),thesymmetricgroupanditshyperoctahedralsubgroup,[,[1]].Moreover,hefoundanalternativeapproachforshowingthepolynomialityresult,see[3].:ν∈,λ,(x)[x]νn?w(ν)ν∈Zsuchthatbμλ(n)=2n!fμλ(n)[1],themultiplicand2n?w(ν)n!ismissing,,letusremindhowitwaswritten:DOIoforiginalarticle:http://dx./..*-mailaddress:******@().http://dx./.-,?.?zden/AdvancesinMathematics308(2017)1337–1339“Letλ,||||||Athatbλμ(n)=0ifν>λ+,letdenotethesetofpairs(x,y)∈S∞×S∞satisfyingx∈Kλ,y∈Kμ,xy∈∞×B∞.LetA(n)denotetheintersectionA∩(S2n×S2n).ν|A|||Hence,bλμ(n)=(n)/Kν(n).{}×AνLetA1,...,ArdenotethesetoforbitsofB∞B∞in(n).Thenbλμ(n)=|A|||(n)=rAi.”|Kν(n)|i=1|Kν(n)|||pletedbyusingthepolynomiality(inn)oftheexpressionsAi|Kν(n)|···ν(i=1,,r),sincethestructureconstantbμλ(n),as{}AngrowsthenumberoforbitsA1,...,Arin(n),hencethenumberofpolynomial|A(n)|r|Ai|summandsoftherighthandsideof|K(n)|=i=1|K(n)|,letusννassumewithoutlossofgeneralitythatif(x,y)∈A1(n),then|N(xy)|≥|N(xy)|forall(x,y)∈Ai(n)withi≥,itisthecasethatthenumber|N(xy)|,whichwe∈∈denotebycA1,isindependentoftheelement(x,y)(x,y)A1,wede?nea∈A|N|newpair(x1,y1)(n+1)by(x1,y1)=((2n+12n+2)x,y).Since(x1y1)=cA1+2,andcA1ismaximal,(x1,y1)isnotcontainedinanyoftheorbitsA1,...,(n)?xtureofthisproblemdoesnotrequireachangeinourinitialstrategybutm?(ν),λ,=V(Kμ×Kλ;Kν)isthesetofpairsw(ν)(x,y)∈Kμ×Kλsuchthatxy∈Kν,thenVisa?niteunionofrevertedB∞×B∞orbits.νNotethatthestructureconstantb(n)is,byde?nition,thecoe?cientof∈zμλzKν(n)intheproductx·,bν(n)isthenumberofx∈Kμ(n)y∈Kλ(n)μλ∈×||pairs(x,y)Kμ(n)Kλ(n)whoseproductxyliesinKν(n)dividedbyKν(n).SincekKν(n)=k≥1Kν(n)isadisjointunion,weseethat|V(K(n)×K(n);Kw(ν)(n))|bν(n)=μλν.(1)μλw(ν)|Kν(n)|LetLbeanorbitoftherevertedactionofB∞×B∞,andlet(x,y),mL=mL(x,y)bythesmallestvalueoftheequation2mL=|N(xy)|+|t(N(xy))|+|DS(x)|+|DS(y)|,whereDS(x)denotesthenumberofpairs{2i?1,2i}?Nthataremappedtoanon-∩×magnitudemListherightnumbersothattheintersectionL(mL):=L(S2mLS2mL)isnon-,?.?zden/AdvancesinMathematics308(2017)1337–,thenthereexistconstantskL||(2nn!)2|w(ν)|2nn!andkνsuchthatL(n)=n?mandKν(n)=n?w(ν).k(L)(2L(n?mL)!)kν(2(n?w(ν))!)?rstequalityissimilartotheproofof[,[1]].Forw(ν)thesecondequality,weobservethatKν(n)isaBn-||||Bn-conjugacyclasshasthecardinalityBn/CS2n(x),whereCS2n(x)isthecentralizerofanelementxfromtheorbit,therestoftheprooffollowseasilyfromtheresultsof[2].,...,Lrbethelistofallrevertedorbitscontainedw(ν)inV=V(Kμ×Kλ;Kν).,weknowthatV(n):=V(Kμ(n)×w(ν)∪···∪Kλ(n);Kν(n))=L1(n)Lr(n).Thus,byeqn.(1),puteνr|Li(n)|bμλ(n)=i=1w(ν).Fori=1,...,r|Kν(n)||L|(2nn!)2k(2n?w(ν)(n?w(ν))!)i=·νw(ν)n?mL?n|K(n)|k(Li)(2(nmL)!)2n!νmLi?···??···?2n(n1)(nmLi+1)n(n1)(nw(ν)+1)=2n?w(ν)n!,k(Li)wherekνandk(Li),fori=1,...,r,theexpressions||f(n):=Li(n)arepolynomialsinn,hence,(n)=i|w(ν)|(n?w(ν))μλKν(n)2n!n?w(ν)r22n!i=1fi(n),[1],,GeneratorsoftheHeckealgebraof(S2n,Bn),(2012)2465–2483.[2],,Thecentresofthesymmetricgrouprings,(1959)212–221.[3],Structurecoe?cientsoftheHeckealgebraof(S2n,Bn),(4)(2014).