Combinatorial Gelfand Models Ron M. Adin, Alex Postnikov, Yuval Roichman To cite this version: Ron M. Adin, Alex Postnikov, Yuval Roichman. Combinatorial Gelfand Models. 20th Annual Inter- national Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2008), 2008, Viña del Mar, Chile. pp.717-730, 10.46298/dmtcs.3612. hal-01185146 HAL Id: hal-01185146 https://hal.inria.fr/hal-01185146 Submitted on 19 Aug 2015 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. FPSAC2008,Valparaiso-VinãdelMar,Chile DMTCSproc.AJ,2008,717–730 Combinatorial Gelfand Models Ron M. Adin1 and Alex Postnikov2 and Yuval Roichman3 1DepartmentofMathematics,Bar-IlanUniversity,Ramat-Gan52900,Israel radin@math.biu.ac.il 2Dept.ofAppliedMathematics,MassachusettsInstituteofTechnology,MA02139,USA apost@math.mit.edu 3DepartmentofMathematics,Bar-IlanUniversity,Ramat-Gan52900,Israel yuvalr@math.biu.ac.il Abstract.AcombinatorialconstructionofGelfandmodelsforthesymmetricgroup, foritsIwahori-Heckealgebra andforthehyperoctahedralgroupispresented. Keywords: symmetricgroup,hyperoctahedralgroup,Iwahori-Heckealgebra,descents,inversions,characterformu- las,Gelfandmodel 1 Introduction AcomplexrepresentationofagrouporanalgebraAiscalledaGelfandmodelforA,orsimplyamodel, ifitisequivalenttothemultiplicityfreedirectsumofallA-irreduciblerepresentations. Models(forcompactLiegroups)werefirstconstructedbyBernstein,GelfandandGelfand[8]. Con- structionsofmodelsforthesymmetricgroup,usinginducedrepresentationsfromcentralizers,werefound by Klyachko[13, 14] and by Inglis, Richardsonand Saxl [11]; see also [6, 21, 4, 3, 5]. Our goalis to determineanexplicitandsimplecombinatorialactionwhichgivesamodelforthesymmetricgroupand its Iwahori-Heckealgebra and for the hyperoctahedralgroup. The descent set of a permutationplays a crucialroleintheconstructionandinaresultingcharacterformula. Therestofthepaperisorganizedasfollows.MainresultsaregiveninSections2–4.Proofsaresketched inSections5–7. Section8concludeswithremarksandopenproblems. Thisisanextendedabstract. Foramoredetailedversionsee[1]. 2 The Symmetric Group Let S be the symmetric group on n letters, S = {s ,...,s } the set of simple reflections in S , n 1 n−1 n I = {π ∈ S |π2 = id}itsthesetofinvolutions,andV := span {C |w ∈ I }avectorspaceover n n n Q w n Qformallyspannedbytheinvolutions. Recallthestandardlengthfunctiononthesymmetricgroup ℓ(π):=min{ℓ|π=s s ···s , s ∈S(∀j)}, i1 i2 iℓ ij thedescentset Des(π):={s∈S|ℓ(πs)<ℓ(π)}, 1365–8050(cid:13)c 2008DiscreteMathematicsandTheoreticalComputerScience(DMTCS),Nancy,France 718 RonM.Adin andAlexPostnikov andYuvalRoichman andthedescentnumberdes(π):=#Des(π). Defineamapρ:S →GL(V )by n ρ(s)C :=sign(s; w)·C (∀s∈S,w ∈I ) w sws n where −1, ifsws=wands∈Des(w); sign(s; w):= (1) (1, otherwise. Theorem2.1 ρdeterminesanS -representation. n Theorem2.2 ρdeterminesaGelfandmodelforS . n 3 Hecke Algebra Action ConsiderH (q),theHeckealgebraofthesymmetricgroupS (sayoverthefieldQ(q1/2)),generatedby n n {T |1≤i1, i j j i T T T =T T T (1≤iπ >π >···>π . aj+1 aj+2 aj+dj aj+dj+1 aj+dj+2 aj+1 Thecharacterofρ maybeexpressedasageneratingfunctionforthedescentnumberoverµ-unimodal q involutions. Proposition3.3 Tr(ρ (T ))= (−q)des(w) q µ {w∈In|wXisµ-unimodal} whereT := T T ···T T ···T isthesubproductofT T ···T omittingthefac- µ 1 2 µ1−1 µ1+1 µ1+...+µt−1 1 2 n−1 torsT forall1≤i0. If|{i,i+1}∩Supp(u)|=1then t− t=±1 t∈SXupp(v) t∈SXupp(u) andinv(v )=inv(u ). If|{i,i+1}∩Supp(u)|=2then |Supp(v) |Supp(u) t= t t∈SXupp(v) t∈SXupp(u) andinv(v )−inv(u )∈{2,0,−2}.Thusinbothcases|f(v)−f(u)|≤1. Thisproves,by |Supp(v) |Supp(u) inductiononℓˆ,thatf(w)≤ℓˆ(w)foreveryinvolutionw. On the other hand, if w is an involution with f(w) > 0 then either t > 2k+1 , or t∈Supp(w) 2 t = 2k+1 and inv(w ) > k. In the first case there exists i+1 ∈ Supp(w) such t∈Supp(w) 2 |Supp(w) P (cid:0) (cid:1) that i 6∈ Supp(w). Then f(s ws ) = f(w)−1. In the second case Supp(w) = {1,...,2k}. Since P (cid:0) (cid:1) i i inv(w )>k,w 6=s s ···s . Thustheremustexistaminimalisatisfyingw(i)>i+1.De- |Supp(w) 1 3 2k−1 notingj :=w(i)−1>iwecannothavew(j)i=w(j+1) andf(s ws )=f(w)−1. Weconcludethatℓˆ(w)≤f(w)foreveryinvolutionw. j j 2 724 RonM.Adin andAlexPostnikov andYuvalRoichman 7 The Hecke Algebra 7.1 Proof of Theorem 3.2(1). Weprovethatρ isanH (q)-representationbyverifyingthedefiningrelations. q n First, consider the braid relation T T T = T T T . To verify this relation observe that there i i+1 i i+1 i i+1 are six possibletypesof orbitsof aninvolutionw underconjugationby hs ,s i - the subgroupin S i i+1 n generatedbys ands . Theseorbitsdifferbytheactionofwonthelettersi,i+1,i+2: i i+1 1. i,i+1,i+26∈Supp(w). 2. Exactlyoneofthelettersi,i+1,i+2isinSupp(w). 3. Exactlytwoofthelettersi,i+1,i+2areinSupp(w),andthesetwolettersforma2-cycleinw. 4. Exactlytwoofthelettersi,i+1,i+2areinSupp(w),andthesetwolettersdonotforma2-cycle inw. 5. i,i+1,i+2∈Supp(w),andtwooftheselettersforma2-cycleinw. 6. i,i+1,i+2∈Supp(w),andnotwooftheselettersforma2-cycleinw. Notethatanorbitofthefirsttypeisoforderone;orbitsofthesecond,thirdandfifthtypeareoforder three;andorbitsofthefourthandsixthtypeareofordersix. Moreover,byLemma6.1,orbitsofthesame orderformisomorphicintervalsintheweakinvolutiveorder(seeDefinition3.1). Inparticular,allorbits ofordersixhavearepresentativewofminimalinvolutivelength,suchthattheorbithastheform: s s s ws s s i i+1 i i i+1 i ւ ց s s ws s s s ws s i i+1 i+1 i i+1 i i i+1 ↓ ↓ (5) s ws s ws i+1 i+1 i i ց ւ w Allorbitsoforderthreearelinearposets: w ≺ s ws ≺ s s ws s , (6) I i i I i+1 i i i+1 or w ≺ s ws ≺ s s ws s . (7) I i+1 i+1 I i i+1 i+1 i Thustheanalysisisreducedintothreecases. Case(a). Anorbitofordersix. By(2)and(5),therepresentationmatricesofthegeneratorswithrespect totheorderedbasisC ,C ,C ,C ,C ,C are: w siwsi si+1siwsisi+1 sisi+1siwsisi+1si si+1wsi+1 sisi+1wsi+1si 1−q 1 0 0 0 0 q 0 0 0 0 0   0 0 1−q 1 0 0 ρ (T )= q i  0 0 q 0 0 0    0 0 0 0 1−q 1    0 0 0 0 q 0     CombinatorialGelfandModels 725 and 1−q 0 0 0 1 0 0 1−q 1 0 0 0   0 q 0 0 0 0 ρ (T )= q i+1  0 0 0 0 0 q     q 0 0 0 0 0     0 0 0 1 0 1−q     Itiseasytoverifythatindeed ρ (T )ρ (T )ρ (T )=ρ (T )ρ (T )ρ (T ). q i q i+1 q i q i+1 q i q i+1 Case (b). An orbit of order three. With no loss of generality, the orbit is of type (6); the analysis of type (7) is analogous. Then s ws = w and s (s s ws s )s = s s ws s . It is shown i+1 i+1 i i+1 i i i+1 i i+1 i i i+1 in[1]thats ∈Des(w)ifandonlyifs ∈Des(s s ws s ). i+1 i i+1 i i i+1 Giventheabove,by(2),therepresentationmatricesofthegeneratorswithrespecttotheorderedbasis w ≺ s ws ≺ s s ws s are I i i I i+1 i i i+1 1−q 1 0 ρ (T )= q 0 0 q i   0 0 x   and x 0 0 ρ (T )= 0 1−q 1 , q i+1   0 q 0   wherex∈{1,−q}.Thesematricessatisfythebraidrelation. Case (c). An orbit of order one. Then s ws = w, s ws = w and s ,s 6∈ Des(w). By (2), i i i+1 i+1 i i+1 ρ (T )ρ (T )ρ (T )C = ρ (T )ρ (T )ρ (T )C = C , completingtheproofofthethirdrela- q i q i+1 q i w q i+1 q i q i+1 w w tion. Theproofoftheothertworelationsiseasierandwillbelefttothereader. 2 Remark7.1 Substitutingq =1provesTheorem2.1. 7.2 Proof of Theorem 3.2(2). WeapplyLusztig’sversionofTits’deformationtheoremtoprovethatρ isaGelfandmodel. q Consider the Hecke algebra H (q) as the algebra over Q(q1/2) spanned by {T |v ∈ S } with the n v n multiplicationrules T T =T ifℓ(vu)=ℓ(v)+ℓ(u) v u vu and (T +q)(T −1)=0 (∀s∈S). s s

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