Kazhdan-Lusztig Polynomials Research Articles

Cohomological growth rates and Kazhdan-Lusztig polynomials

In a previous work, the authors established various bounds for the dimensions of degree n cohomology and Ext-groups, for irreducible modules of semisimple algebraic groups G (in positive characteristic p) and (Lusztig) quantum groups U ζ (at roots of unity ζ). These bounds depend only on the root system, and not on the characteristic p or the size of the root of unity ζ. This paper investigates the rate of growth of these bounds. Both in the quantum and algebraic group situation, these rates of growth represent new and fundamental invariants attached to the root system ϕ. For quantum groups U ζ with a fixed ϕ, we show the sequence {max L irred dim Hn(U ζ , L)} n has polynomial growth independent of ζ. In fact, we provide upper and lower bounds for the polynomial growth rate. Applications of these and related results for \(Ext_{U_\zeta }^n \) are given to Kazhdan-Lusztig polynomials. Polynomial growth in the algebraic group case remains an open question, though it is proved that {log max L irred dim Hn(G,L)} has polynomial growth ≤ 3 for any fixed prime p (and ≤ 4 if p is allowed to vary with n). We indicate the relevance of these issues to (additional structure for) the constants proposed in the theory of higher cohomology groups for finite simple groups with irreducible coefficients by Guralnick, Kantor, Kassabov and Lubotzky [13].

Casselman’s Basis of Iwahori Vectors and the Bruhat Order

AbstractW. Casselman defined a basis fu of Iwahori fixed vectors of a spherical representation of a split semisimple p-adic group G over a nonarchimedean local field F by the condition that it be dual to the intertwining operators, indexed by elements u of the Weyl group W. On the other hand, there is a natural basis , and one seeks to find the transition matrices between the two bases. Thus, let and . Using the Iwahori–Hecke algebra we prove that if a combinatorial condition is satisfied, then , where z are the Langlands parameters for the representation and α runs through the set S(u, v) of positive coroots (the dual root systemof G) such that with rα the reflection corresponding to α. The condition is conjecturally always satisfied if G is simply-laced and the Kazhdan–Lusztig polynomial Pw0v,w0u = 1 with w0 the long Weyl group element. There is a similar formula for conjecturally satisfied if Pu,v = 1. This leads to various combinatorial conjectures.

Equivalence Classes for the μ-Coefficient of Kazhdan–Lusztig Polynomials in Sn

We study equivalence classes relating to the Kazhdan–Lusztig μ(x, w) coefficients in order to help explain the scarcity of distinct values. Each class is conjectured to contain a “crosshatch” pair. We also compute the values attained by μ(x, w) for the permutation groups S 10 and S 11.

Quasisymmetric functions and Kazhdan-Lusztig polynomials

We associate a quasisymmetric function to any Bruhat interval in a general Coxeter group. This association can be seen to be a morphism of Hopf algebras to the subalgebra of all peak functions, leading to an extension of the cd-index of convex polytopes. We show how the Kazhdan-Lusztig polynomial of the Bruhat interval can be expressed in terms of this complete cd-index and otherwise explicit combinatorially defined polynomials. In particular, we obtain the simplest closed formula for the Kazhdan-Lusztig polynomials that holds in complete generality.

Diagrammatic Kazhdan–Lusztig theory for the (walled) Brauer algebra

We determine the decomposition numbers for the Brauer and walled Brauer algebras in characteristic zero in terms of certain polynomials associated to cap and curl diagrams (recovering a result of Martin in the Brauer case). We consider a second family of polynomials associated to such diagrams, and use these to determine projective resolutions of the standard modules. We then relate these two families of polynomials to Kazhdan–Lusztig theory via the work of Lascoux–Schützenberger and Boe, inspired by work of Brundan and Stroppel in the cap diagram case.

Kazhdan–Lusztig polynomials, tight quotients and Dyck superpartitions

We give an explicit combinatorial product formula for the parabolic Kazhdan–Lusztig polynomials of the tight quotients of the symmetric group. This formula shows that these polynomials are always either zero or a monic power of q and implies the main result in [F. Brenti, Kazhdan–Lusztig and R-polynomials, Youngʼs lattice, and Dyck partitions, Pacific J. Math. 207 (2002) 257–286] on the parabolic Kazhdan–Lusztig polynomials of the maximal quotients. The formula depends on a new class of superpartitions.

Kazhdan–Lusztig coefficients for an affine Weyl group of type [formula omitted

In this paper we compute the leading coefficients μ ( u , w ) of the Kazhdan–Lusztig polynomials P u , w for an affine Weyl group of type B ˜ 2 . By using the a-function of a Coxeter group defined by Lusztig, we compute most μ ( u , w ) explicitly. With part of these values μ ( u , w ) , we show that a conjecture of Lusztig on distinguished involutions is true for an affine Weyl group of type B ˜ 2 . We also show that the conjectural formula concerning some semilinear equations needs a modification.

Generic Hecke algebras for monomial groups

In this paper we define a two-variable, generic Hecke algebra,H\mathcal H, for each complex reflection groupG(b,1,n)G(b,1,n). The algebraH\mathcal Hspecializes to the group algebra ofG(b,1,n)G(b,1,n)and also to an endomorphism algebra of a representation ofGLn⁡(Fq)\operatorname {GL}_n(\mathbb F_q)induced from a solvable subgroup. We construct Kazhdan-Lusztig “RR-polynomials” forH\mathcal {H}and show that they may be used to define a partial order onG(b,1,n)G(b,1,n). Using a generalization of Deodhar’s notion of distinguished subexpressions we give a closed formula for theRR-polynomials. After passing to a one-variable quotient of the ring of scalars, we construct Kazhdan-Lusztig polynomials forH\mathcal Hthat reduce to the usual Kazhdan-Lusztig polynomials for the symmetric group whenb=1b=1.

Bounding Ext for modules for algebraic groups, finite groups and quantum groups

Given a finite root system Φ , we show that there is an integer c = c ( Φ ) such that dim Ext G 1 ( L , L ′ ) < c , for any reductive algebraic group G with root system Φ and any irreducible rational G -modules L , L ′ . There also is such a bound in the case of finite groups of Lie type, depending only on the root system and not on the underlying field. For quantum groups, a similar result holds for Ext n , for any integer n ⩾ 0 , using a constant depending only on n and the root system. When L is the trivial module, the same result is proved in the algebraic group case, thus giving similar bounded properties, independent of characteristic, for algebraic and generic cohomology. (A similar result holds for any choice of L = L ( λ ) , even allowing λ to vary, provided the p -adic expansion of lambda is limited to a fixed number of terms.) In particular, because of the interpretation of generic cohomology as a limit for underlying families of finite groups, the same boundedness properties hold asymptotically for finite groups of Lie type. The results both use, and have consequences for, Kazhdan–Lusztig polynomials. Appendix A proves a stable version, needed for small prime arguments, of Donkin's tilting module conjecture.

Super duality and irreducible characters of ortho-symplectic Lie superalgebras

We formulate and establish a super duality which connects parabolic categories O for the ortho-symplectic Lie superalgebras and classical Lie algebras of BCD types. This provides a complete and conceptual solution of the irreducible character problem for the ortho-symplectic Lie superalgebras in a parabolic category O, which includes all finite dimensional irreducible modules, in terms of classical Kazhdan-Lusztig polynomials.

Some non-trivial Kazhdan-Lusztig coefficients of an affine Weyl group of type à n

In this paper we show that the leading coefficient µ(y,w) of some Kazhdan-Lusztig polynomials P y,w with y,w in an affine Weyl group of type à n is n + 2. This fact has some consequences on the dimension of first extension groups of finite groups of Lie type with irreducible coefficients.

Irreducible Characters of General Linear Superalgebra and Super Duality

We develop a new method to solve the irreducible character problem for a wide class of modules over the general linear superalgebra, including all the finite-dimensional modules, by directly relating the problem to the classical Kazhdan-Lusztig theory. Furthermore, we prove that certain parabolic BGG categories over the general linear algebra and over the general linear superalgebra are equivalent. We also verify a parabolic version of a conjecture of Brundan on the irreducible characters in the BGG category of the general linear superalgebra.

Alcove geometry and a translation principle for the Brauer algebra

There are similarities between algebraic Lie theory and a geometric description of the blocks of the Brauer algebra. Motivated by this, we study the alcove geometry of a certain reflection group action. We provide analogues of translation functors for a tower of recollement, and use these to construct Morita equivalences between blocks containing weights in the same facet. Moreover, we show that the determination of decomposition numbers for the Brauer algebra can be reduced to a study of the block containing the weight 0. We define parabolic Kazhdan–Lusztig polynomials for the Brauer algebra and show in certain low rank examples that they determine standard module decomposition numbers and filtrations.

Kazhdan–Lusztig polynomials for maximally-clustered hexagon-avoiding permutations

We provide a nonrecursive description for the bounded admissible sets of masks used by Deodhar's algorithm to calculate the Kazhdan–Lusztig polynomials P x , w ( q ) of type A, in the case when w is hexagon avoiding and maximally clustered. This yields a combinatorial description of the Kazhdan–Lusztig basis elements of the Hecke algebra associated to such permutations w. The maximally-clustered hexagon-avoiding elements are characterized by avoiding the seven classical permutation patterns { 3421 , 4312 , 4321 , 46718235 , 46781235 , 56718234 , 56781234 } . We also briefly discuss the application of heaps to permutation pattern characterization.

The degenerate analogue of Ariki’s categorification theorem

We explain how to deduce the degenerate analogue of Ariki’s categorification theorem over the ground field \({\mathbb{C}}\) as an application of Schur–Weyl duality for higher levels and the Kazhdan–Lusztig conjecture in finite type A. We also discuss some supplementary topics, including Young modules, tensoring with sign, tilting modules and Ringel duality.

Relations between the Clausen and Kazhdan–Lusztig representations of the symmetric group

We use Kazhdan–Lusztig polynomials and subspaces of the polynomial ring C [ x 1 , 1 , … , x n , n ] to give a new construction of the Kazhdan–Lusztig representations of S n . This construction produces exactly the same modules as those which Clausen constructed using a different basis in [M. Clausen, Multivariate polynomials, standard tableaux, and representations of symmetric groups, J. Symbolic Comput. (11), 5-6 (1991) 483–522. Invariant-theoretic algorithms in geometry (Minneapolis, MN, 1987)], and does not employ the Kazhdan–Lusztig preorders. We show that the two resulting matrix representations are related by a unitriangular transition matrix. This provides a C [ x 1 , 1 , … , x n , n ] -analog of results due to Garsia and McLarnan, and McDonough and Pallikaros, who related the Kazhdan–Lusztig representations to Young’s natural representations.

Enumerative properties of shifted-Dyck partitions

We study the enumerative properties of a new class of (skew) shifted partitions. This class arises in the computation of certain parabolic Kazhdan–Lusztig polynomials and is closely related to ballot sequences. As consequences of our results, we obtain new identities for the parabolic Kazhdan–Lusztig polynomials of Hermitian symmetric pairs and for the ordinary Kazhdan–Lusztig polynomials of certain Weyl groups.

An Inversion Formula for -Relative Kazhdan–Lusztig Polynomials

The article proves a relative version of one of the results from the influential article [4] of Kazhdan and Lusztig which introduced the Kazhdan–Lusztig polynomials. Given a Coxeter group W and a set S of simple reflections, let ℋ denote the corresponding Hecke algebra, it has a “standard” basis T w and another basis C w with many remarkable properties. The Kazhdan–Lusztig polynomials p x, w give the transition matrix between these bases. One of their results proved by Kazhdan and Lusztig is an inversion formula, which states that if W is finite with longest element w 0, then for all x ≤ w in W. The main result of this article generalizes this result to the following setting: for any subset J of S, we define elements η J , and , and consider the two “dual” ideals and , where their standard basis and a Kazhdan–Lusztig basis are, respectively, indexed by subsets E J and of W.

Parabolic Kazhdan–Lusztig and [formula omitted]-polynomials for Boolean elements in the symmetric group

The parabolic analogue of the Kazhdan–Lusztig and R -polynomials has been introduced by Deodhar [V. Deodhar, On some geometric aspects of Bruhat orderings. II. The parabolic analogue of Kazhdan–Lusztig polynomials, J. Algebra 111 (1987) 483–506]. Answering a question of Brenti, we give closed combinatorial product formulae for the parabolic R -polynomials and for the parabolic Kazhdan–Lusztig polynomials of type q in the case where the indexing permutations are smaller than the top transposition in Bruhat order. These formulae are valid in complete generality on the parabolic subgroup W J .

Cyclic sieving, promotion, and representation theory

We prove a collection of conjectures of White [D. White, personal communication, 2007], as well as some related conjectures of Abuzzahab, Korson, Li, and Meyer [O. Abuzzahab, M. Korson, M. Li, S. Meyer, Cyclic and dihedral sieving for plane partitions, U. Minnesota REU Report, 2005] and of Reiner and White [V. Reiner, personal communication, 2007; D. White, personal communication, 2007], regarding the cyclic sieving phenomenon of Reiner, Stanton and White [V. Reiner, D. Stanton, D. White, The cyclic sieving phenomenon, J. Combin. Theory Ser. A 108 (2004)] as it applies to jeu-de-taquin promotion on rectangular tableaux. To do this, we use Kazhdan–Lusztig theory and a characterization of the dual canonical basis of C [ x 11 , … , x n n ] due to Skandera [M. Skandera, On the dual canonical and Kazhdan–Lusztig bases and 3412, 4231-avoiding permutations, 2006, submitted for publication]. Afterwards, we extend our results to analyzing the fixed points of a dihedral action on rectangular tableaux generated by promotion and evacuation, suggesting a possible sieving phenomenon for dihedral groups. Finally, we give applications of this theory to cyclic sieving phenomena involving reduced words for the long elements of hyperoctohedral groups and noncrossing partitions.