Module structure of cells in unequal parameter Hecke algebras
aa r X i v : . [ m a t h . R T ] F e b MODULE STRUCTURE OF CELLS IN UNEQUAL PARAMETERHECKE ALGEBRAS
THOMAS PIETRAHO
Abstract.
A conjecture of C. Bonnaf´e, M. Geck, L. Iancu, and T. Lam pa-rameterizes Kazhdan-Lusztig left cells for unequal parameter Hecke algebrasin type B n by families of standard domino tableaux of arbitrary rank. Rely-ing on a family of properties outlined by G. Lusztig and the recent work ofC. Bonnaf´e, we verify the conjecture and describe the structure of each cell asa module for the underlying Weyl group. Introduction
Consider a Coxeter system (
W, S ), a positive weight function L , and the cor-responding generic Iwahori-Hecke algebra H . As detailed by G. Lusztig in [20], achoice of weight function gives rise to a partition of W into left, right, and two-sided Kazhdan-Lusztig cells, each of which carries the structure of an H - as wellas a W -module. The cell decomposition of W is understood for all finite Coxetergroups and all choices of weight functions with the exception of type B n . We focusour attention on this remaining case and write W = W n . A weight function isthen specified by a choice of two integer parameters a and b assigned to the simplereflections in W n : t b t t tq q q a a a Given a, b = 0, we write s = ba for their quotient. We have the followingdescription of cells due to C. Bonnaf´e, M. Geck, L. Iancu, and T. Lam. It isstated in terms of a family of generalized Robinson-Schensted algorithms G r whichdefine bijections between W n and same-shape pairs of domino tableaux of rank r . Conjecture ([4]) . Consider a Weyl group W n of type B n with a weight function L and parameter s defined as above.(1) When s N , let r = ⌊ s ⌋ . Two elements of W n lie in the same Kazhdan-Lusztig left cell whenever they share the same right tableau in the image of G r .(2) When s ∈ N , let r = s − . Two elements of W n lie in the same Kazhdan-Lusztig left cell whenever their right tableaux in the image of G r are relatedby moving through a set of non-core open cycles. Significant progress has been made towards the verification of the above, whichwe detail in Section 3.3. Most recently, C. Bonnaf´e has shown that if a certain family
Mathematics Subject Classification.
Key words and phrases. unequal parameter Hecke algebras, Kazhdan-Lusztig cells, dominotableaux. of statements conjectured by G. Lusztig is assumed to hold, then the conjectureholds if s N , and furthermore, if s ∈ N , then Kazhdan-Lusztig left cells are unionsof the sets described [2]. We sharpen this result, and verify that the conjecture holdsin the latter case as well.We concurrently describe the structure of Kazhdan-Lusztig left cells as W n -modules. The canonical parameter set for irreducible W n -modules consists of or-dered pairs of partitions ( d, f ) where the the parts of d and f sum to n . As detailedin Section 4.1, there is a natural identification of this parameter set with the set ofpartitions P r ( n ) of a fixed rank r . Since P r ( n ) corresponds exactly to the shapesof rank r domino tableaux, the parametrization of Kazhdan-Lusztig left cells viastandard tableaux of fixed rank in the above conjecture suggests a module structurefor each cell for every choice of weight function. Mainly, the irreducible constituentsof the module carried by each cell should correspond to the shapes of the rank r tableaux of its elements, with r determined from the parameter s as in the conjec-ture. We verify that this suggested module structure is indeed the one carried byeach cell.Our approach is based on M. Geck’s characterization of left cells as constructiblerepresentations; that is, those representations which are obtained by successivetruncated parabolic induction and tensoring with the sign representation, see [10].In Section 2, we detail the general construction of Kazhdan-Lusztig cells in anunequal parameter Hecke algebra and extend a result of G. Lusztig on the inter-section of left and right cells to the unequal parameter setting. In Section 3, wedetail the situation in type B n and the relevant combinatorics. Section 4 examinesconstructible representations and provides a combinatorial description of truncatedparabolic induction and tensoring with sign, mimicking the work of W. M. McGov-ern in the equal parameter case [21]. The final section contains the proof of themain results. 2. Unequal Parameter Hecke Algebras
We briefly recount the definitions of unequal parameter Hecke algebras and thecorresponding Kazhdan-Lusztig cells, following [20].2.1.
Kazhdan-Lusztig Cells.
Consider a Coxeter system (
W, S ) and let ℓ be theusual length function. A weight function L : W → Z satisfies L ( xy ) = L ( x ) + L ( y )whenever ℓ ( xy ) = ℓ ( x ) + ℓ ( y ) and is uniquely determined by its values on S . Wewill consider those weight functions which take positive values on all s ∈ S .Let H be the generic Iwahori-Hecke algebra over A = Z [ v, v − ] with parameters { v s | s ∈ S } , where v x = v L ( x ) for all x ∈ W . The algebra H is free over A and hasa basis { T x | x ∈ W } . Multiplication in H takes the form T s T x = (cid:26) T sx if ℓ ( sx ) > ℓ ( x ), and T sx + ( v s − v − s ) T x if ℓ ( sx ) < ℓ ( x )As in [20](5.2), it is possible to construct a Kazhdan-Lusztig basis of H which wedenote by { C x | x ∈ W } . In terms of it, multiplication has the form C x C y = X z ∈ W h xyz C z . for some h xyz ∈ A . Although we suppress it in the notation, all of these notionsdepend on the specific choice of weight function L . ODULE STRUCTURE OF CELLS IN UNEQUAL PARAMETER HECKE ALGEBRAS 3
Definition 2.1.
Fix (
W, S ) a Coxeter system with a weight function L . We willwrite y ≤ L x if there exists s ∈ S such that C y appears with a non-zero coefficientin C s C x . By taking the transitive closure, this binary relation defines a preorderon W which we also denote by ≤ L . Let y ≤ R x iff y − ≤ L x − and define ≤ LR asthe pre-order generated by ≤ L and ≤ R .Each of the above preorders defines equivalence relations which we denote by ∼ L , ∼ R , and ∼ LR respectively. The resulting equivalence classes are called the left, right, and two-sided Kazhdan-Lusztig cells of W .As described in [20](8.3), Kazhdan-Lusztig cells carry representations of H . If C is a Kazhdan-Lusztig left cell and x ∈ C , then define[ C ] A = M y ≤ L x A C y . M y ≤ L x,y / ∈ C A C y . This is a quotient of two left ideals in H and consequently is itself a left H -module; it does not depend on the specific choice of x ∈ C , is free over A , and hasa basis { e x | x ∈ C } indexed by elements of C with e x the image of C x in the abovequotient. The action of H on [ C ] A is determined by C x e y = X z ∈ C h xyz e z for x ∈ W and y ∈ C . A Kazhdan-Lusztig left cell gives rise to a W -module [ C ]by restricting [ C ] A to scalars. The same construction can be used to define modulestructures on the right and two-sided cells of W .2.2. A Family of Properties.
The main results of this paper rely on a familyof conjectures formulated by G. Lusztig in [20, § L is a multiple of the length function ℓ , a number of results aboutKazhdan-Lusztig cells depend on positivity results derived by geometric methodsof intersection cohomology. Unfortunately, this positivity does not hold for unequalparameter Hecke algebras; for examples see [19, §
6] and [9, 2.7]. As a substitute,G. Lusztig detailed a list properties which both, axiomatize known equal-parameterresults, and outline methods of approaching non-positivity in general.In order to list Lusztig’s conjectures, we must first define two integer-valuedfunctions on W . For any z ∈ W , let a ( z ) be the smallest non-negative integer sothat h xyz ∈ v a ( z ) Z [ v − ] for every x and y in W and write γ xyz − for the constantterm of v − a ( z ) h xyz . If p xy is defined by C y = P x ∈ W p xy T x , then [20](5.4) showsthat p z is non-zero. We write p z = n z v − ∆( z ) + terms of smaller degree in v thereby defining a constant n z and integer ∆( z ) for every z ∈ W . Finally, let D = { z ∈ W | a ( z ) = ∆( z ) } . Lusztig has conjectured the following statements are true in the general setting ofunequal parameter Hecke algebras:
P1.
For any z ∈ W we have a ( z ) ≤ ∆( z ). P2. If d ∈ D and x, y ∈ W satisfy γ x,y,d = 0, then x = y − . P3. If y ∈ W, there exists a unique d ∈ D such that γ y − ,y,d = 0. P4. If z ′ ≤ LR z then a ( z ′ ) ≥ a ( z ). Hence, if z ′ ∼ LR z , then a ( z ) = a ( z ′ ). P5. If d ∈ D , y ∈ W , γ y − ,y,d = 0, then γ y − ,y,d = n d = ± THOMAS PIETRAHO
P6. If d ∈ D , then d = 1. P7.
For any x, y, z ∈ W , we have γ x,y,z = γ y,z,x . P8.
Let x, y, z ∈ W be such that γ x,y,z = 0. Then x ∼ L y − , y ∼ L z − , and z ∼ L x − . P9. If z ′ ≤ L z and a ( z ′ ) = a ( z ), then z ′ ∼ L z . P10. If z ′ ≤ R z and a ( z ′ ) = a ( z ), then z ′ ∼ R z . P11. If z ′ ≤ LR z and a ( z ′ ) = a ( z ), then z ′ ∼ LR z . P12.
Let I ⊆ S and W I be the parabolic subgroup defined by I . If y ∈ W I , then a ( y ) computed in terms of W I is equal to a ( y ) computed in terms of W . P13.
Any left cell C of W contains a unique element d ∈ D . We have γ x − ,x,d = 0for all x ∈ C . P14.
For any z ∈ W , we have z ∼ LR z − . P15. If v ′ is an indeterminate and h ′ xyz is obtained from h xyz via the substitution v v ′ , then whenever a ( w ) = a ( y ), we have X y ′ h ′ wx ′ y ′ h xy ′ y = X y ′ h xwy ′ h ′ y ′ x ′ y . The statements
P1-P15 are known to hold for finite Weyl groups in the equalparameter case by work of Kazhdan-Lusztig [16] and Springer [25]. If the Coxetersystem is of type I ( m ), H , or H , they follow from work of Alvis [1] and DuCloux[5]. In the unequal parameter case, P1-P15 have been verified by Geck in types I ( m ) and F [14], and in the so-called asymptotic case of type B n by Geck-Iancu[13] and Geck [11], [14]. Although the geometric approach from which the abovefollow in the equal parameter case is not available in the general unequal parametercase, it seems that it may not be required. At least in type A , Geck has shown that P1-P15 hold via elementary, purely algebraic methods [12].2.3.
The Asymptotic Ring J . The goal of this section is to verify Lemma 12.15of [18] in our more general setting. We begin with a brief discussion of Lusztig’sring J which can be viewed as an asymptotic version of H . Although originallydefined in the equal parameter case, its construction also makes sense in the settingof unequal parameter Hecke algebras under the the assumption that the conjectures
P1-P15 hold. Using the methods developed in [20], J provides us with a way ofstudying the left-cell representations of H .Recall the integers γ xyz defined for all x, y, and z in W as the constant terms of v a ( z ) h xyz − . Then J is the free abelian group with basis { t x | x ∈ W } . To endowit with a ring structure, define a bilinear product on J by t x · t y = X z ∈ W γ xyz t z − for x and y in W . Conjectures P1-P15 allow us to state the following results.
Theorem 2.2 ([20]) . Assuming conjectures
P1-P15 , the following hold:(1) J is an associative ring with identity element J = P d ∈D n d t d . (2) The group algebra C [ W ] is isomorphic as a C -algebra to J C = C ⊗ Z J. Following [20, § E ♠ for the J C -module corresponding to a C [ W ]-module E . It shares its underlying space with E , while the action of anelement of J C is defined by the action of its image under the isomorphism with C [ W ]. Consider a left cell C of W and define J C C to be ⊕ x ∈ C C t x . By P8 , this is aleft ideal in J C . Furthermore,
ODULE STRUCTURE OF CELLS IN UNEQUAL PARAMETER HECKE ALGEBRAS 5
Theorem 2.3 ([20]) . Assuming that the conjectures
P1-P15 hold, the J C -modules J C C and [ C ] ♠ are isomorphic. We are ready to address Lemma 12.15 of [18]. Its original proof relies on acharacterization of left cells in terms of the dual bases { C x } and { D x } stated in[18](5.1.14). This result in turn relies on positivity properties which do not hold inthe unequal parameter case and therefore a new approach to the lemma is required.We owe the idea of using J in the present proof to M. Geck. Lemma 2.4.
Assume that conjectures
P1-P15 hold. If C and C ′ are two left cellsin W with respect to a weight function L , then dim Hom W ([ C ] , [ C ′ ]) = | C ∩ C ′− | . Proof.
Let x ∈ C − ∩ C ′ and define a map φ x on J C C via φ x ( t y ) = t y t x . With x and y as above, we can write t y t x = X γ yxz t z − . For γ yxz = 0, P8 implies x ∼ L z − . Since x ∈ C ′ , this forces t y t x to lie in J C ′ C , andwe have in fact defined a map φ x : J C C → J C ′ C . We will show that as x runs over the set C − ∩ C ′ , the maps φ x are linearlyindependent. So assume that for some constants a x we have X x ∈ C − ∩ C ′ a x φ x = 0 and, consequently X x ∈ C − ∩ C ′ a x t y t x = 0for all y ∈ C . In particular, if d is the unique element in D ∩ C guaranteed by P13 then we also have X x ∈ C − ∩ C ′ a x t d t x = X y ∈ C − ∩ C ′ ± a x t x = 0 , where the first equality follows from P2, P5, P7, and
P13 . But this means that a x = 0 for all relevant x , or in other words, that the φ x are linearly independent. Wecan therefore conclude that dim Hom J C ( J C C , J C ′ C ) ≥ | C − ∩ C ′ | . Since this inequalityis true for all pairs of left cells C and C ′ in W , we have X C , C ′ dim Hom J C ( J C C , J C ′ C ) ≥ X C , C ′ | C − ∩ C ′ | . The right side of this inequality is just the order of W since each of its elements liesin a unique left and a unique right cell. On the other hand, by the correspondenceresulting from Theorem 2.3 the left side isdim Hom J C (cid:16) X C J C C , X C ′ J C ′ C (cid:17) = dim Hom W (Reg W , Reg W ) = | W | . Hence the original inequality must be in fact an equality and the lemma follows. (cid:3)
We immediately obtain the following corollary, whose proof is identical to thatof [18](12.17).
Corollary 2.5.
Assume that conjectures
P1-P15 hold and that the left cell modulesof W with respect to a weight function L are multiplicity-free. Then C ∩ C − is theset of involutions in C . THOMAS PIETRAHO Type B n The goal of this section is to detail the combinatorics of arbitrary rank standarddomino tableaux necessary to describe Kazhdan-Lusztig cells in type B n .3.1. Domino Tableaux.
Consider a partition p of a natural number n . We willview it as a Young diagram Y p , a left-justified array of squares whose row lengthsdecrease weakly. The square in row i and column j of Y p will be denoted s ij anda pair of squares in Y p of the form { s ij , s i +1 ,j } or { s ij , s i,j +1 } will be called a domino . A domino is removable from Y p if deleting its underlying squares leaveseither another Young diagram containing the square s or the empty set.Successive deletions of removable dominos from a Young diagram Y p must even-tually terminate in a staircase partition containing (cid:0) r +12 (cid:1) squares for some non-negative integer r . This number is determined entirely by the underlying partition p and does not depend on the sequence of deletions of removable dominos. We willwrite p ∈ P r and say that p is a partition of rank r . The core of p is its underlyingstaircase partition. Example . The partition p = [4 , ,
1] lies in the set P . Below are its Youngdiagram Y p and a domino tiling resulting from a sequence of deletions of removabledominos exhibiting the underlying staircase partition.Consider p ∈ P r . It is a partition of the integer 2 n + (cid:0) r +12 (cid:1) for some n . A standard domino tableau of rank r and shape p is a tiling of the non-core squares of Y p by dominos, each of which is labeled by a unique integer from { , . . . , n } in sucha way that the labels increase along its rows and columns. We will write SDT r ( p )for the set of standard domino tableaux of rank r of shape p and SDT r ( n ) for theset of standard domino tableaux of rank r which contain exactly n dominos.For T ∈ SDT r ( n ), we will say that the square s ij is variable if i + j ≡ r mod 2and fixed otherwise. As discussed in [6] and [22], a choice of fixed squares on atableau T allows us to define two notions, a partition of its dominos into cycles andthe operation of moving through a cycle. The moving through map, when appliedto a cycle c in a tableau T yields another standard domino tableau M T ( T, c ) whichdiffers from T only in the labels of the variable squares of c . If c contains D ( l, T ), thedomino in T with label l , then M T ( T, c ) is in some sense the minimally-affectedstandard domino tableau in which the label of the variable square in D ( l, T ) ischanged. We refer the reader to [22] for the detailed definitions.If the shape of M T ( T, c ) is the same as the shape of T , we will say that c isa closed cycle . Otherwise, one square will be removed from T (or added to itscore) and one will be added. In this case, we will say the c is open and denote theaforementioned squares as s b ( c ) and s f ( c ) , respectively. Finally, if s b ( c ) is adjacentto the core of T , we will say that c is a core open cycle . We will write OC ( T ) forthe set of all open cycles of T and OC ∗ ( T ) the subset of non-core open cycles.3.2. Generalized Robinson-Schensted Algorithms.
The Weyl group W n oftype B n consists of the set of signed permutations on n letters, which we write in ODULE STRUCTURE OF CELLS IN UNEQUAL PARAMETER HECKE ALGEBRAS 7 one-line notation as w = ( w w . . . w n ). For each non-negative integer r , there isan injective map G r : W n → SDT r ( n ) × SDT r ( n )which is onto the subset of domino tableaux of the same-shape, see [6] and [26].We will write G r ( x ) = ( S r ( x ) , T r ( x )) for the image of a permutation x and refer tothe two components as the left and right tableaux of x . Definition 3.2.
Consider x, y ∈ W n and fix a non-negative integer r . We will saythat(1) x ≈ ι L y if T r ( y ) = T r ( x ), and(2) x ≈ L y if T r ( y ) = M T ( T r ( x ) , C ) for some C ⊂ OC ∗ ( T r ( x )) . We will call the equivalence classes defined by ≈ ι L irreducible combinatorial leftcells of rank r in W , and those defined by ≈ L its reducible combinatorial left cellsof rank r . In the irreducible case, we will say that the combinatorial left cellis represented by the tableau T r ( x ). In the reducible case, we will say that thecombinatorial left cell is represented by the set { M T ( T r ( x ) , C ) | C ⊂ OC ∗ ( T r ( x )) } of standard domino tableaux.3.3. Cells in type B n . Consider the generators of W n as in the following diagram: t t t t tq q q s s s n − Define the weight function L by L ( t ) = b and L ( s i ) = a for all i and set s = ba .The following is a conjecture of Bonnaf´e, Geck, Iancu, and Lam, and appears asConjectures A, B, and D in [4]: Conjecture 3.3.
Consider a Weyl group of type B n with a weight function L andparameter s defined as above.(1) When s N , the Kazhdan-Lusztig left cells coincide with the irreduciblecombinatorial left cells of rank ⌊ s ⌋ .(2) When s ∈ N , the Kazhdan-Lusztig left cells coincide with the reduciblecombinatorial left cells of rank s − . This conjecture is well-known to be true for s = 1 by work of Garfinkle [8],and has been verified when s > n − s when n ≤
6, see [4]. Furthermore, assuming
P1-P15 , C. Bonnaf´e has shown the conjecture to be true in the irreducible case, andthat in the reducible case, Kazhdan-Lusztig left cells are unions of the reduciblecombinatorial left cells [2].4.
Constructible Representations in Type B n M. Geck has shown that if Lusztig’s conjectures
P1-P15 hold, then the W -modules carried by the Kazhdan-Lusztig left cells of an unequal parameter Heckealgebra are precisely the constructible ones [10]. Defined in the unequal parametersetting by Lusztig in [20](20.15), constructible modules arise via truncated induc-tion and tensoring with the sign representation. The goal of this section is to givea combinatorial description of the effects of these two operations on W -modules intype B n . Our approach is based on the equal-parameter results of [21].
THOMAS PIETRAHO
Irreducible W n -modules. Let us restrict our attention to type B n , write W n for the corresponding Weyl group, and define constants a , b , and s, as in Section3.3. We begin by recalling the standard parametrization of irreducible W n -modules.Let P be the set of ordered pairs of partitions and P ( n ) be the subset of P wherethe combined sum of the parts of both partitions is n . Theorem 4.1.
The set of irreducible representations of W n is parametrized by P ( n ) . If we write [( d, f )] for the representation corresponding to ( d, f ) ∈ P ( n ) ,then [( f t , d t )] ∼ = [( d, f )] ⊗ sgn , where p t denotes the transpose of the partition p . In this form, the connection between irreducible W n -modules and the descriptionof left cells in Conjecture 3.3 is not clear. To remedy this, we would like to restateTheorem 4.1 in terms of partitions of arbitrary rank which arise as shapes of thestandard domino tableaux in this conjecture. Thus let r = ⌊ s ⌋ if s N , r = s − ǫ = s − ⌊ s ⌋ . As an intermediary to this goal, we definethe notion of a symbol of defect t and residue ǫ for a non-negative integer t and0 ≤ ǫ <
1. It is an array of non-negative numbers of the formΛ = (cid:18) λ + ǫ λ + ǫ . . . λ N + t + ǫµ µ . . . µ N (cid:19) where the (possibly empty) sequences { λ i } and { µ i } consist of integers and arestrictly increasing. If we define a related symbol by lettingΛ ′ = (cid:18) ǫ λ + 1 + ǫ λ + 2 + ǫ . . . λ N + t + N + t + ǫ µ + 1 . . . µ N + N (cid:19) then the binary relation defined by setting Λ ∼ Λ ′ generates an equivalence relation.We will write Sym ǫt for the set of its equivalence classes.We describe two maps between symbols and partitions. A partition can beused to construct a symbol in the following way. If p = ( p , p , . . . , p k ), form p ♯ = ( p , p , . . . , p k ′ ) by adding an additional zero term to p if the rank of p hasthe same parity as k . Dividing the set { p i + k ′ − i } k ′ i =1 into its odd and even partsyields two sequences { µ i + 1 } Ni =1 and { λ i } N + ti =1 for some non-negative integer t . A symbol Λ p of defect t and residue ǫ correspondingto p can now be defined by arranging the integers λ i and µ i into an array as above.Given a symbol of defect t and residue ǫ , it is also possible to construct anordered pair of partitions. With Λ as above, let d Λ = { λ i − i + 1 } N + ti =1 and f Λ = { µ i − i + 1 } Ni =1 . Both constructions are well-behaved with respect to the equivalence on symbolsdefined above. The next theorem follows from [15](2.7).
Theorem 4.2.
The maps p Λ p and Λ ( d Λ , f Λ ) define bijections P r → Sym ǫr +1 → P for all values of r and ǫ . Consequently, their composition yields a bijection between P r ( n ) and P ( n ) . ODULE STRUCTURE OF CELLS IN UNEQUAL PARAMETER HECKE ALGEBRAS 9
This result allows us to custom tailor a parametrization of irreducible W n -modules to each value of the parameter s by defining r and ǫ as above. Together withLusztig’s Lemma 22.18 of [20], the present theorem implies the following alternateparametrization of the representations of W n in terms of symbols. A parametriza-tion in terms of partitions of rank r follows. Corollary 4.3.
If we fix values of the defect r and residue ǫ , then the set of ir-reducible representations of W n is parametrized by the set of equivalence classes ofsymbols { Λ ∈ Sym ǫr +1 | parts of d Λ and f Λ sum to n } . Writing [Λ] for the repre-sentation corresponding to Λ , we have [¯Λ] = [Λ] ⊗ sgn where the symbol ¯Λ is defined from Λ by the following procedure. Write Λ as aboveand let τ be the integer part its largest entry. Then the integer parts of the top andbottom rows of ¯Λ consist of the complements of { τ − µ i } i and { τ − λ i } i in [0 , τ ] ∩ Z ,respectively. Corollary 4.4.
If we fix a non-negative integer r , then the set of irreducible rep-resentations of W n is parametrized by P r ( n ) . Writing [ p ] for the representationcorresponding to p ∈ P r ( n ) , we have [ p t ] ∼ = [ p ] ⊗ sgn , where p t is the transpose of the partition p .Example . Let s = 3 , so that r = 3 and ǫ = , and consider the irreduciblerepresentation [((1 ) , (1))] of W . Then according to the above parametrizations,[((1 ) , (1))] = [(4 , , )] = [Λ (4 , , ) ] whereΛ [(4 , , )] = (cid:18) (cid:19) is a symbol of defect 3 and residue . Note that ((1 ) , (1)) ∈ P (4), (4 , , ) ∈P (4), and Λ (4 , , ) is a representative of a class in Sym ǫ for ǫ = 1 /
2. Furthermore,[((1 ) , (1))] ⊗ sgn = [((1) , (3))] = [(4 , , )] ⊗ sgn = [(4 , , (4 , , ) ] ⊗ sgn =[Λ (4 , , ] , where Λ [(4 , , = (cid:18) (cid:19) . We will need the following lemma, which holds for finite W whenever P1-P15 hold. It is a combination of [20](11.7) and [20](21.5).
Lemma 4.6.
Consider a Kazhdan-Lusztig left cell C ⊂ W and let w be the longestelement of W . Then C w is also a left cell in W , and [ C w ] ∼ = C ⊗ sgn as W -modules. Truncated Induction.
We now turn to a combinatorial description of trun-cated induction in terms of the above parameter sets. If π is a representation of W I ,a parabolic subgroup of W n , Lusztig defined a representation J WW I ( π ) of W = W n ,[20](20.15). Its precise definition depends of the parameters of the underlying Heckealgebra, so it is natural to expect that this is manifested in the combinatorics stud-ied above. Following [21, §
2] and [17], we note that due to the transitivity oftruncated induction and the fact that the situation in type A is well-understood,we need to only understand how truncated induction works when W I is a maximalparabolic subgroup whose type A component acts by the sign representation on π . Henceforth, let W I be a maximal parabolic subgroup in W n with factors W ′ oftype B m and S l of type A l − , where m + l = n ; furthermore, write sgn l for the signrepresentation of S l .Truncated induction behaves well with respect to cell structure. In fact, thefollowing lemma holds for general W . Lemma 4.7 ([9]) . Let C ′ be a left cell of W I . Then we have J WW I ([ C ′ ]) ∼ = [ C ] , where C is the left cell of W such that C ′ ⊂ C . We first provide a description of the situation in type B n in terms of symbols.Consider a symbol Λ ′ of defect r + 1 and residue ǫ ; via the equivalence on symbols,we can assume that it has at least l entries. If the set of l largest entries of Λ ′ is uniquely defined, then let Λ be the symbol obtained by increasing each of theentries in this set by one. If it is not, then let Λ I and Λ II be the two symbolsobtained by increasing the largest l − ′ and then each of the two l thlargest entries in turn by one. Proposition 4.8 ([20](22.17)) . The representation J WW I ([Λ ′ ] ⊗ sgn l ) is [Λ] if theset of l largest entries of Λ ′ is uniquely defined, and [Λ I ] + [Λ II ] if it is not. Theformer is always the case if [Λ ′ ] is a symbol of residue ǫ = 0 . It is not difficult to reformulate this result in terms of partitions of rank r .Consider a partition p = ( p , p , . . . p k ) ∈ P r . We can assume that k ≥ l by addingzero parts to p as necessary. Let k ′ be the number of parts of p ♯ . Define p I = ( p + 2 , . . . , p l + 2 , p l +1 , . . . , p k ) , and p II = ( p + 2 , . . . , p l − + 2 , p l + 1 , p l +1 + 1 , p r +2 , . . . , p k ) . Note that both p I and p II are again partitions of rank r . Corollary 4.9.
The representation J WW I ([ p ] ⊗ sgn l ) produced by truncated inductionis [ p I ] whenever p l > p l +1 , p l + r − l is odd, or ǫ = 0 . Otherwise, J WW I ([ p ] ⊗ sgn l ) = [ p I ] + [ p II ] . Proof.
Using the results of the preceding proposition, we have to check under whatconditions the set of l largest entries in a symbol Λ ′ is uniquely defined and thendetermine the preimages of the symbols Λ I and Λ II under the map of Theorem4.2. When ǫ = 0, the l largest entries in Λ ′ are uniquely determined since all of itsentries must be distinct. When ǫ = 0, there will be an ambiguity in determiningthe l largest entries iff p l + k ′ − l and p l +1 + k ′ − l − k ′ is always of theopposite parity from r , this gives us the conditions of the proposition. Determiningthe partitions corresponding to Λ I and Λ II is then just a simple calculation. (cid:3) Note that the parity conditions of the proposition imply that in the case when J WW I ([ p ] ⊗ sgn l ) is reducible, the square s l,p l +1 of the Young diagrams of p I and p II is fixed. In particular, this means that when endowed with the maximal label, thedomino { s l,p l +1 , s l,p l +2 } constitutes an open cycle in a domino tableau of shape p I . Its image under the moving through map is { s l +1 ,p l +1 , s l,p l +1 } with underlyingpartition p II . This observation leads to the following lemma:
ODULE STRUCTURE OF CELLS IN UNEQUAL PARAMETER HECKE ALGEBRAS 11
Lemma 4.10.
Let n = m + l and consider w ′ = ( w w . . . w m ) ∈ W m . Write T ′ = T r ( w ′ ) for its right tableau of rank r and define a set of partitions P ′ = { shape M T ( T ′ , C ) | C ⊂ OC ∗ ( T ′ ) } ⊂ P r ( m ) . Define the set P = { p I | p ∈ P ′ } ∪ { p II | p ∈ P ′ and p l = p l +1 with p l + r − l even } . If w = ( w w . . . w m n n − . . . m + 1) ∈ W n with right tableau T = T r ( w ) , then P = { shape M T ( T, C ) | C ⊂ OC ∗ ( T ) } ⊂ P r ( n ) . Proof.
The lemma relates the non-core open cycles in T ′ to the non-core open cyclesin T , hence it follows from the description of the behavior of cycles under dominoinsertion in [22](3.6). However, things are really simpler than that, and we describethe situation fully. Note that T is obtained from T ′ by placing horizontal dominoswith labels m + 1 through n at the end of its first l rows. Essentially, there are fourpossibilities. We write s ij for the left square of the domino added to row i and let p = shape T ′ .(1) s ij = S f ( c ) for a cycle c of T ′ . Then the domino joins the cycle c and thefinal square of the new cycle is s i,j +2 .(2) s ij − = S b ( c ) for a cycle c of T ′ . Then the domino joins the cycle c andthe beginning square of the new cycle is s i,j +1 .(3) p i − = p i with p i + r − i odd. Then the dominos with labels m + i − m + i in T form a closed cycle in T .(4) p l = p l +1 with p l + r − l even. Then the domino with label n forms asingleton non-core open cycle in T which does not correspond to a cycle in T ′ .If C ⊂ OC ∗ ( T ′ ) and e C is the set of the corresponding cycles in T , then it is clearfrom the above description that { shape M T ( T, e C ) | C ⊂ OC ∗ ( T ′ ) } = { p I | p ∈ P ′ } .If case (4) arises and T has an additional non-core open cycle c = { n } , then { shape M T ( T, e C ∪ c ) | C ⊂ OC ∗ ( T ′ ) } = { p II | p ∈ P ′ } . The lemma follows. (cid:3)
Example . Let s = 3, so that r = 2 and ǫ = 0, and consider the partition(4 , , ) ∈ P (5). It corresponds to the symbolΛ [(4 , , )] = (cid:18) (cid:19) ∈ Sym For l = 4, we have J WW I ([(4 , , )] ⊗ sgn ) = [(6 , , , )] + [(6 , , , . Note thatboth partitions lie in P (9). In terms of symbols, J WW I ([Λ (4 , , ) ] ⊗ sgn ) = (cid:20)(cid:18) (cid:19)(cid:21) + (cid:20)(cid:18) (cid:19)(cid:21) W n -module structure and standard domino tableaux Viewing cells as constructible representations allows us to examine their structureinductively. Using the description of truncated induction and tensoring with signderived in the previous section we describe the W n -module carried by each cell interms of the parametrization of irreducible W n -modules of Section 4.1. We beginwith a few facts about combinatorial cells. Lemma 5.1.
Consider two combinatorial left cells C and C ′ in W n of rank r rep-resented by sets T and T ′ of rank r standard domino tableaux. Then | C ∩ C ′− | = M where M is the number of tableaux in T whose shape matches the shape of a tableauin T ′ .Proof. Suppose first that C and C ′ are irreducible so that T = { T } and T ′ = { T ′ } . Ifthey are of the same shape, then the intersection C ∩ C ′− = G − r ( T ′ , T ); otherwise,it is empty.On the other hand, if C and C ′ are reducible, then let J consist of the tableauxin T whose shape matches the shape of a tableau in T ′ and define | J | = M . Recallthat by the definition of a combinatorial left cell, T = { M T ( T, C ) | C ∈ OC ∗ T } for some tableau T and therefore T consists of only tableaux of differing shapes. If T ∈ J , write T ′ for the the unique tableau in T ′ of the same shape as T . Then C ∩ C ′− = [ T ∈ J G − r ( T ′ , T ) . (cid:3) We can obtain a slightly better description of the intersection of a combinatorialleft cell and a combinatorial right cell by recalling the definition of an extendedopen cycle in a tableau relative to another tableau of the same shape. See [7](2.3.1)or [23](2.4) for the details. In general, an extended open cycle is a union of opencycles.
Corollary 5.2.
Consider two reducible combinatorial left cells C and C ′ in W n ofrank r represented by sets T and T ′ of rank r standard domino tableaux. If T ∈ T and T ′ ∈ T ′ are of the same shape and m is the number of non-core extended opencycles m in T relative to T ′ , then | C ∩ C ′− | = 2 m . Proof.
An extended open cycle in T relative to T ′ is a minimal set of open cyclesin T and T ′ such that moving through it produces another pair of tableaux of thesame shape. Consequently, moving through two different extended open cycles areindependent operations. Noting that T = { M T ( T, C ) | C ⊂ OC ∗ ( T ) } and T ′ = { M T ( T ′ , C ) | C ⊂ OC ∗ ( T ′ ) } , we have that a tableau-pair ( S, S ′ ) ∈ T × T ′ is same-shape iff it differs from ( T, T ′ )by moving through a set of non-core extended open cycles in T relative to T ′ . Thus,if E is the set of non-core extended open cycles in T relative to T ′ , then C ∩ C ′− = [ D ⊂ E G − r (cid:0) M T (( T ′ , T ) , D ) (cid:1) , from which the result follows. (cid:3) Recall the parameter s derived from a weight function L in type B n . We willcall a Kazhdan-Lusztig left cell in this setting a left cell of weight s . C. Bonnaf´e [2]has shown that: • under the assumption that statements P1-P15 of Section 2.1 hold, when s N , left cells of weight s are precisely the irreducible combinatorial leftcells of rank r = ⌊ s ⌋ , and • when s ∈ N , left cells of weight s are unions of reducible combinatorial leftcells of rank r = s − ODULE STRUCTURE OF CELLS IN UNEQUAL PARAMETER HECKE ALGEBRAS 13
In this way, as in Definition 3.2, we can say that a left cell of weight s is rep-resented by a set of standard domino tableaux of rank r . For non-integer s , thisset consists of the unique tableau representing the irreducible combinatorial leftcell, and in the latter, it is the union of the sets of tableaux representing each ofthe reducible combinatorial cells in the Kazhdan-Lusztig cell. In what follows, weassume that statements P1-P15 hold.
Lemma 5.3.
Suppose that C is a left cell of weight s and C = ` i D i is its decom-position into combinatorial left cells of rank r . If we let T i be the set of dominotableaux representing D i , then the set of shapes of tableaux in T i is disjoint fromthe set of shapes of tableaux in T j whenever i = j .Proof. By Corollary 2.5, C ∩ C − consists of the involutions in C . The set of involu-tions in each combinatorial cell D i consists of D i ∩ D − i . This forces D i ∩ D − j = ∅ whenever i = j , which can only occur if the set of shapes of tableaux in T i is disjointfrom the set of shapes of tableaux in T j , by Lemma 5.1. (cid:3) We first show that the shapes of the standard domino tableaux of rank r repre-senting a left cell of weight s determine its W n -module structure: Definition 5.4.
Suppose T is a set of standard domino tableaux of rank r . For T ∈ T , we will write p T ∈ P r ( n ) for its underlying partition, and define[ T ] = M T ∈ T [ p T ] . Lemma 5.5.
Suppose that C and C ′ are left cells of weight s in W n and C = a i ≤ c D i as well as C ′ = a i ≤ d D ′ i are their decompositions into combinatorial left cells of rank r . Suppose that each D i and D ′ i is represented by the set of rank r tableaux T i and T ′ i , respectively. Then [ C ] ∼ = [ C ′ ] iff c = d and, suitably ordered, [ T i ] ∼ = [ T ′ i ] for all i .Proof. For clarity, we treat the integer and non-integer values of s separately. Firstassume s N so that c = d = 1 and take { T } = T and { T ′ } = T ′ . By Lemmas2.4 and 5.1, we have dim Hom W ([ C ] , [ C ]) = dim Hom W ([ C ′ ] , [ C ′ ]) = 1 . Furthermore,we have that dim Hom W ([ C ] , [ C ′ ]) = | C ∩ C ′− | = 1 if and only if the shapes of T and T ′ coincide; otherwise, dim Hom W ([ C ] , [ C ′ ]) = 0. The lemma follows.Next, assume s ∈ N . Suppose first that [ C ] ∼ = [ C ′ ] . Then dim Hom( C , C ) =dim Hom( C ′ , C ′ ) = dim Hom( C , C ′ ), and by Lemma 2.4, | C ∩ C − | = | C ′ ∩ C ′− | = | C ∩ C ′− | . By Lemma 5.3, we have X i ≤ c | D i ∩ D − i | = X i ≤ d | D ′ i ∩ D ′ i − | = X i,j | D i ∩ D ′ j − | . We can now use Corollary 5.2 to examine the terms of this equality. For a com-binatorial cell D i , there is at most one cell D ′ i ′ such that there are T i ∈ T i and T ′ i ′ ∈ T ′ i ′ of the same shape, by Lemma 5.3. Let I be the set of i for which thisoccurs. Let c i and d i be the numbers of non-core open cycles in T i and T ′ i ′ andfor each i ∈ I , let m i be the number of non-core extended open cycles in T i rela-tive to T ′ i ′ . Then m i ≤ c i , d i ′ with equality iff the non-core extended open cyclesare just the non-core open cycles. By Corollary 5.2, P i ≤ c | D i ∩ D − i | = P i ≤ c c i , P i ≤ d | D ′ i ∩ D ′ i − | = P i ≤ d d i , and P I | D i ∩ D ′ i ′ − | = P I m i . But the previousequation now implies that m i = c i = d i ′ , c = d , I = { , . . . , c } and by the definitionof a combinatorial left cell in our setting, that [ T i ] ∼ = [ T ′ i ′ ] for all i ∈ I .Conversely, assume that c = d and [ T i ] ∼ = [ T ′ i ] for all i and choose tableaux T i ∈ T i and T ′ i ∈ T ′ i of the same shape. By the definition of combinatorial cells,there is a correspondence between the non-core open cycles of T i and those of T ′ i such that their beginning and final squares coincide, implying that the set of non-core extended open cycles in T i relative to T ′ i is precisely the set of open cyclesof T i . Therefore, for each i we have | D i ∩ D − i | = | D i ∩ D ′ i − | . Consequently, byLemmas 5.3 and 2.4, and Corollary 5.2:dim Hom( C , C ′ ) = X i | D i ∩ D ′ i − | = X i | D i ∩ D − i | = dim Hom( C , C ) . Reversing the roles of C and C ′ above implies the desired result. (cid:3) Theorem 5.6.
Suppose that C is a left cell of weight s in W n represented by a set T of standard domino tableaux of rank r . Then [ C ] ∼ = [ T ] as W n -modules.Proof. In light of the result from Lemma 5.5, we can prove the theorem by verifyingit holds for a representative of each isomorphism class of left cells. Under ourassumptions, the results of [10] hold and left cell modules coincide with constructiblerepresentations of W n . Therefore, a representative of each isomorphism class of leftcells can be obtained by repeated truncated induction and tensoring with sign.Recall our description of irreducible W n -modules by partitions of rank r . ViaCorollaries 4.4 and 4.9, we have a description of both operations on the level ofpartitions. We verify that the effect of truncated induction and tensoring with signon the shapes of the tableaux representing a left cell is the same, and the theoremfollows by induction.We treat the integer and non-integer values of s separately. First assume s N ,so that each left cell is represented by a single tableau. We begin by investigatingthe effect on tensoring with sign. If [ C ] is a left cell module and w ∈ C , then C isrepresented by the tableau T r ( w ) of shape p . By Lemma 4.6, C w is also a left celland [ C w ] ∼ = [ C ] ⊗ sgn . It is represented by the tableau T r ( ww ) = T r ( w ) t of shape p t . By Corollary 4.4, if we assume that [ C ] carries the irreducible module associatedto the shape of its representative tableau, then so does [ C w ] ∼ = [ C ] ⊗ sgn . For the case of truncated induction, consider a maximal parabolic subgroup W I = W m × S l of W n . Choose w ′ = ( w w . . . w m ) ∈ W m and let C ′ be its leftcell, represented by the tableau T ′ = T r ( w ′ ). Let p = shape T ′ . By Lemma4.7, J WW I ([ C ′ ] ⊗ sgn l ) = [ C ] for a left cell C ⊂ W n and furthermore, the element w = ( w w . . . w m n n − . . . m + 1) ∈ C . The left cell C is represented by thetableau T r ( w ) whose shape is p I , using the notation of (4.9). By Corollary 4.9, ifwe assume that [ C ′ ] carries the irreducible module associated to the shape of itsrepresentative tableau, then so does [ C ] ∼ = J WW I ([ C ′ ] ⊗ sgn l ) . Next assume s ∈ N , so that each left cell is represented by a family of rank r standard domino tableaux. Again, we begin by investigating the effect on tensoringwith sign. Suppose C is a left cell represented by the set T and for each T ∈ T , w T ∈ W n is chosen so that T r ( w T ) = T . By Lemma 4.6, C w is also a left cell and[ C w ] ∼ = [ C ] ⊗ sgn . It is represented by the set of tableaux T r ( w T w ) = T r ( w T ) t (for T ∈ T ), which we write as T t . By Corollary 4.4, if we assume that [ C ] carries themodule [ T ] then [ C w ] ∼ = [ C ] ⊗ sgn carries the module [ T t ] . ODULE STRUCTURE OF CELLS IN UNEQUAL PARAMETER HECKE ALGEBRAS 15
For the case of truncated induction, again consider a maximal parabolic subgroup W I = W m × S l of W n . Let C ′ be a left cell of W m and let C ′ = ` i D ′ i be itsdecomposition into combinatorial left cells. Suppose that D ′ i is represented by theset T ′ i of domino tableaux and let T ′ = ` i T ′ i . By definition of combinatorial leftcells, every T ′ i = { M T ( T ′ i , C ) | C ⊂ OC ∗ ( T ′ i ) } for some rank r standard dominotableau T ′ i . For each i , choose e w i = ( w i w i . . . w im ) ∈ W m with T ′ i = T r ( e w i ) so that e w i ∈ D ′ i . By Lemma 4.7, J WW I ([ C ′ ] ⊗ sgn l ) = [ C ] for a left cell C ⊂ W n . Furthermore, w i = ( w i w i . . . w im n n − . . . m + 1) ∈ C and if T i = T r ( w i ), then C is representedby the set of tableaux T = ` i { M T ( T i , C ) | C ⊂ OC ∗ ( T i ) } . Lemma 4.10 describesthe shapes of the tableaux in T in terms of the shapes of the tableaux in T ′ . This,together with Corollary 4.9 shows that if we assume that [ C ′ ] carries the module[ T ′ ], then [ C ] ∼ = J WW I ([ C ′ ] ⊗ sgn l ) carries the module [ T ] . (cid:3) Corollary 5.7.
Consider a Weyl group of type B n with a weight function L andparameter s defined as above. If statements P1-P15 hold, then(1) When s N , the Kazhdan-Lusztig left cells of weight s coincide with theirreducible combinatorial left cells of rank ⌊ s ⌋ .(2) When s ∈ N , the Kazhdan-Lusztig left cells of weight s coincide with thereducible combinatorial left cells of rank s − .If the set T of standard domino tableaux represents the left cell C in W n , then [ C ] ∼ = [ T ] as W n -modules. Furthermore, if T ∈ T , then the number of elements of C with right tableau T is the dimension of the irreducible constituent [ p T ] of [ C ] .Proof. The first part in the case s N is a result of C. Bonnaf´e [2]. To verify it inthe case s ∈ N , write a Kazhdan-Lusztig left cell C in terms of combinatorial leftcells as C = ` i ∈ I D i . Since [ C ] is constructible, the main result of [24] shows that[ C ] ∼ = [ e T ] as W n -modules where e T = { M T ( T, C ) | C ⊂ OC ∗ ( T ) } for some standarddomino tableau T of rank r . Let each D i be represented by T i = { M T ( T i , C ) | C ⊂ OC ∗ ( T i ) } and write T = ` i ∈ I T i . By Theorem 5.6, [ T ] = [ e T ] . This implies thatfor every i , the set of beginning and ending squares of non-core open cycles in T i is contained in the corresponding set in T . However, the size of this set is constantfor every partition in the set of possible shapes of tableaux in [ T ]. By Lemma 5.3,the only way this can occur is if | I | = 1, that is, C consists of a single combinatorialcell.Finally, we verify the last statement of the corollary. If s N , consider a left cell C represented by the tableau T . Then dim[ C ] = P | C ∩ C ′− | , the sum taken overall left cells C ′ in W n . But | C ∩ C ′− | = 1 iff the shape of the tableaux representing C and C ′ are the same; otherwise it is zero. Since each left cell is represented by aunique tableau, the above sum equals the number of tableaux of the same shape as T . This is the same as the number of elements of C with right tableau T . If s ∈ N ,consider left cells C and C ′ . For w ∈ C ∩ C ′− , [ shape T r ( w )] must be a componentof both [ C ] and [ C ′ ]. Furthermore, each w ∈ C ∩ C ′− must have the right tableau ofa unique shape, establishing a bijection between C ∩ C ′− and the set of irreduciblemodules common to [ C ] and [ C ′ ] . If we let C ′ vary over all left cells of W n , thestatement follows by Lemma 2.4. (cid:3) It should be remarked that the above statement classifying the module structureof left cells is not the strongest one could hope for. In the so-called “asymptotic” case when s is sufficiently large, M. Geck has shown that whenever the tableauxrepresenting [ C ] and [ C ′ ] equal, then not only are the underlying H -modules isomor-phic, but the underlying structure constants are the same. More precisely, there isa bijection C → C ′ sending x x ′ such that h w,x,y = h w,x ′ ,y ′ for all w ∈ W n and x, y ∈ C . It would be interesting to know under what circumstances this stronger statementholds for other values of s . References [1] D. Alvis. The left cells of the Coxeter group of type H . J. Algebra arXiv:math.RT/0806.0214 .[3] C. Bonnaf´e and L. Iancu. Left cells in type B n with unequal parameters. Represent. Theory ,7:587–609, 2003.[4] C. Bonnaf´e, M. Geck, L. Iancu, and T. Lam. On domino insertion and Kazhdan–Lusztig cellsin type B n , Progress in Math. (Lusztig Birthday Volume).
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E-mail address : [email protected]@bowdoin.edu