RSK BASES AND KAZHDAN-LUSZTIG CELLS K.N.RAGHAVAN,PREENASAMUEL,ANDK.V.SUBRAHMANYAM 0 1 0 Abstract. From the combinatorial characterizations of the right, left, and 2 two-sidedKazhdan-Lusztigcellsofthesymmetricgroup,‘RSKbases’arecon- structed for certain quotients by two-sided ideals of the group ring and the l u Heckealgebra. Applicationstoinvarianttheory,overvariousbaserings,ofthe J generallineargroupandrepresentationtheory,bothordinaryandmodular,of thesymmetricgrouparediscussed. 7 2 ] T 1. Introduction: summary and organization of results R . h The starting point of the work described in this paper is a question in classical t invariant theory (§1.1). It leads naturally to questions about representations of a m the symmetric group over the complex numbers (§1.2, §1.3) and over algebraically closedfields of positive characteristic(§1.5), and in turn to the computationof the [ determinantofacertainmatrixencodingthemultiplicationofKazhdan-Lusztigba- 2 sis elements of the Hecke algebra (§1.6), using which one can recovera well-known v criterionfor the irreducibility of Specht modules over fields of positive characteris- 2 tic (§1.7). 4 8 For the sake of readability, we have tried, to the extent possible, to keep the 2 proofsofourresultsindependentofeachother. Sosections4–6canbereadwithout . 2 reference to one another. This has of course come at the price of some repetition. 0 9 1.1. Motivation from invariant theory. We beginby recallinga basictheorem 0 of classical invariant theory. Let k be a commutative ring with identity and V a : v free k-module of finite rank d. Let GL(V) denote the group of k-automorphisms i of V, and consider the diagonal action of GL(V) on V⊗n. Let S denote the X n symmetric group of bijections of the set {1,...,n} and kS the group ring of S r n n a with coefficients in k. There is a natural action of Sn on V⊗n by permuting the factors: more precisely, (v ⊗···⊗v )·σ :=v ⊗···⊗v (all actions are on the 1 n 1σ nσ right by convention). This action commutes with the action of GL(V), and so the k-algebra map Θ : kS → End V⊗n defining the action of S has image in the n n k n space End V⊗n of GL(V)-endomorphisms of V⊗n. GL(V) We have the following result (see [5, Theorems 4.1, 4.2]): Assumethefollowing: iff(X)isanelementofdegreenofthepoly- nomial ring k[X] in one variable over k that vanishes as a function on k, then f(X) is identically zero. (This holds for example when 2000 Mathematics Subject Classification. Primary: 05E10,05E15,20C08,20C30. Key words and phrases. symmetric group, Hecke algebra, Kazhdan-Lusztig basis, RSK correspondence, RSK-shape, Kazhdan-Lusztig cells, multilinear invariants, picture invariants, cell module,Specht module,Gramdeterminant,Carterconjecture. 1 2 K.N.RAGHAVAN,PREENASAMUEL,ANDK.V.SUBRAHMANYAM k is an infinite field, no matter what n is.) Then the k-algebra homomorphism Θ maps onto End V⊗n and its kernel is the n GL(V) two-sided ideal J(n,d) defined as follows: • J(n,d):=0 if d≥n; • ifdr. t Partitionsofnareinbijectionwithshapes ofYoungdiagrams (orsimplyshapes) with n boxes: the partition λ ≥... ≥λ corresponds to the shape with λ boxes 1 r 1 inthefirstrow,λ inthesecondrow,andsoon,theboxesbeingarrangedleft- and 2 top-justified. Here for example is the shape corresponding to the partition (3,3,2) of 8: Partitions are thus identified with shapes and the two terms are used interchange- ably. 2.2. Dominance order on partitions. Given partitions µ = (µ ,µ ,...) and 1 2 λ=(λ ,λ ,...) of n, we say µ dominates λ, and write µDλ, if 1 2 µ ≥λ , µ +µ ≥λ +λ , µ +µ +µ ≥λ +λ +λ , .... 1 1 1 2 1 2 1 2 3 1 2 3 We write µ⊲λ if µDλ and µ6=λ. The partialorder D onthe set of partitions (or shapes) of n will be referred to as the dominance order. 2.2.1. The partition λ(n,d). Given integers n and d, there exists a unique partition λ(n,d) ⊢ n that has at most d parts and is smallest in the dominance order among those with at most d parts. For example, λ(8,2)=(3,3,2). 2.3. Tableaux and standard tableaux. A Young tableau, or just tableau, of shape λ ⊢ n is an arrangement of the numbers 1, ..., n in the boxes of shape λ. Thereare,evidently,n!tableauxofshapeλ. Atableauisrowstandard (respectively, column standard) if in every row (respectively, column) the entries are increasing left to right (respectively, top to bottom). A tableau is standard if it is both row standardandcolumnstandard. Anexampleofastandardtableauofshape(3,3,2): 1 3 5 2 6 8 4 7 RSK BASES AND KAZHDAN-LUSZTIG CELLS 7 2.3.1. Thenumberofstandardtableaux. Thenumberofstandardtableauxofagiven shapeλ⊢nisdenotedd(λ). Thereisawell-known‘hooklengthformula’forit[11]: d(λ)=n!/ h , where β runs overall boxesof shape λ andh is the hook length β β β of the box βQwhich is defined as one more than the sum of the number of boxes to the right of β and the number of boxes below β. The hook lengths for the shape (3,3,2) are shown below: 5 4 2 4 3 1 2 1 Thus d(3,3,2)=8!/(5.4.2.4.3.1.2.1)=42. 2.4. The RSK-correspondenceand the RSK-shapeofa permutation. The Robinson-Schensted-Knuth correspondence (RSK correspondence for short) is a well-known procedure that sets up a bijection between the symmetric group S n andorderedpairsofstandardtableauxofthe same shapewithn boxes. We donot recallhere the procedure, referringthe readerinstead to [12, Chapter 4]. It will be convenient for our purposes to modify slightly the procedure described in [12]. Denoting by (A(w),B(w)) ↔ w the bijection of [12], what we mean by RSK correspondence is the bijection (B(w),A(w)) ↔ w; since A(w) = B(w−1) and A(w−1) = B(w) (see [12, Corollary on page 41]), we could equally well define our RSK correspondenceas (A(w),B(w)) ↔w−1. The RSK-shape ofa permutation w is defined to be the shape of either of A(w), B(w). 2.4.1. An example. The permutation (1542)(36) (written as a product of disjoint cycles)hasRSK-shape(3,2,1). IndeeditismappedundertheRSKcorrespondence in our sense to the ordered pair (A,B) of standard tableaux, where: 1 3 5 1 2 3 A= 2 4 B = 4 6 6 5 2.4.2. Remark. We have modified the standard definition of RSK-correspondence, for it was the simplest way we could think up of reconciling the notationalconflict amongthetwosetsofpapersuponwhichwerely:[6,21]and[14]. Permutationsact ontherightintheformer—aconventionwhichwetoofollow—butontheleftinthe latter. The direction in which they act makes a difference to statements involving the RSK correspondence: most importantly for us, to the characterization of one- sidedcells(see§3.4.1below). Thiscreatesaproblem: wecannotbequotingliterally from both sets of sources without changing something. Altering the definition of RSKcorrespondenceasaboveisthepathofleastresistance,andallowsustoquote more or less verbatim from both sets. 2.5. Tabloids and tabloid representations. Letλ=(λ ,λ ,...)⊢n. Atabloid 1 2 of shape λ is a partition of the set [n] := {1,...,n} into an ordered r-tuple of subsets, the first consisting of λ elements, the second of λ elements, and so on. 1 2 8 K.N.RAGHAVAN,PREENASAMUEL,ANDK.V.SUBRAHMANYAM Depicted below are two tabloids of shape (3,3,2): 1 3 5 3 5 8 7 8 9 1 6 4 6 2 7 The members of the first subset are arranged in increasing order in the first row, those of the second subset in the second row, and so on. Given a tableau T of shape λ, it determines, in the obvious way, a tabloid of shape λ denoted {T}: the first subset consists of the elements in the first row, the second of those in the second row, and so on. The defining action of S on [n] induces, in the obvious way, an action on the n setT oftabloidsofshapeλ. The free Z-module ZT withT as aZ-basisprovides λ λ λ therefore a linear representation of S over Z. By base change we get such a n representation over any commutative ring with unity k: kTλ :=ZTλ⊗Zk. We call it the tabloid representation corresponding to the shape λ. 2.6. Specht modules. The Specht module corresponding to a partition λ ⊢ n is a certain S -submodule of the tabloid representation ZT just defined. For a n λ tableau T of shape λ, define e in ZT by T λ e := sgn(σ){Tσ} T X where the sum is taken over permutations σ of S in the column stabiliser of T, n sgn(σ) denotes the sign of σ, and {Tσ} denotes the tabloid corresponding to the tableau Tσ in the obvious way (see §2.5). The Specht module Sλ is the linear span ofthee asT runsoveralltableauxofshapeλ. ItisanS -submoduleofZT with T n λ Z-basise ,asT variesoverstandardtableaux(see,forexample,[12,§7.2]). Bybase T change we get the Specht module Sλ over any commutative ring with identity k: k Skλ :=Sλ⊗Zk. Evidently,Skλisafreek-moduleofrankthenumberd(λ)ofstandard tableaux of shape λ (§2.3.1). 3. Set up: Hecke algebra and Kazhdan-Lusztig cells Letn denotea fixedpositiveintegerandS the symmetricgrouponn letters. Let n S be the subset consisting of the simple transpositions (1,2), (2,3), ..., (n−1,n) of the symmetric group S . Then (S ,S) is a Coxeter system in the sense of [3, n n Chapter 4]. Let A:=Z[v,v−1], the Laurent polynomial ring in the variable v over the integers. 3.1. The Hecke algebra and its T-basis. Let H be the Iwahori-Hecke algebra corresponding to (S ,S), with notation as in [14]. Recall that H is an A-algebra: n it is a free A-module with basis T , w ∈S , the multiplication being defined by w n T if ℓ(sw)=ℓ(w)+1 T T = sw s w (cid:26) (v−v−1)Tw+Tsw if ℓ(sw)=ℓ(w)−1 for s∈S and w ∈S and ℓ is the length function. We put n ǫ(w):=(−1)ℓ(w) and v :=vℓ(w) for w ∈S . w n RSK BASES AND KAZHDAN-LUSZTIG CELLS 9 An induction on length gives (in any case, see [6, Lemma 2.1 (iii)] for (3.1.2)): (3.1.1) TwTw′ =Tww′ if ℓ(w)+ℓ(w′)=ℓ(ww′). (3.1.2) TuTu′ =Tuu′ + awTw for u, u′ in Sn uuX′0 where H := A T , A :=v−1Z[v−1] H := A T , A :=vZ[v] <0 <0 w <0 >0 >0 w >0 wX∈Sn wX∈Sn 10 K.N.RAGHAVAN,PREENASAMUEL,ANDK.V.SUBRAHMANYAM The anti-automorphismh7→h∗ and the ring involutionh7→h commute with each other, so that, by the characterization(3.2.1): (3.2.2) (Cx)∗ =Cx−1 (Cx′)∗ =Cx′−1 We have by [18, Theorem 1.1]: (3.2.3) C′ =T + p T C =T + ǫ ǫ p T w w y,w y w w y w y,w y y∈SXn,y

RSK bases and Kazhdan-Lusztig cells

Similar RSK bases and Kazhdan-Lusztig cells