Kazhdan-Lusztig Polynomials Research Articles

Permutations with Kazhdan-Lusztig Polynomial $P_{id,w}(q)=1+q^h$

Using resolutions of singularities introduced by Cortez and a method for calculating Kazhdan-Lusztig polynomials due to Polo, we prove the conjecture of Billey and Braden characterizing permutations $w$ with Kazhdan-Lusztig polynomial $P_{id,w}(q)=1+q^h$ for some $h$. (Appendix by Sara Billey and Jonathan Weed)

ON THE MARKOV TRACE FOR TEMPERLEY–LIEB ALGEBRAS OF TYPE En

We show that there is a unique Markov trace on the tower of Temperley–Lieb type quotients of Hecke algebras of Coxeter type En(for all n ≥ 6). We explain in detail how this trace may be computed easily using tom Dieck's calculus of diagrams. As applications, we show how to use the trace to show that the diagram representation is faithful, and to compute leading coefficients of certain Kazhdan–Lusztig polynomials.

Permutations with Kazhdan-Lusztig polynomial $ P_id,w(q)=1+q^h$

Using resolutions of singularities introduced by Cortez and a method for calculating Kazhdan-Lusztig polynomials due to Polo, we prove the conjecture of Billey and Braden characterizing permutations w with Kazhdan-Lusztig polynomial$ P_id,w(q)=1+q^h$ for some $h$. On démontre la conjecture de Billey et Braden sur les permutations w pour lesquelles le polynôme de Kazhdan-Lusztig $P_id,w(q)=1+q^h$ pour un entier $h$. On emploie une résolution des singularités présentées par Cortez et une méthode de Polo pour calculer ces polynômes.

Weights of mixed tilting sheaves and geometric Ringel duality

We describe several general methods for calculating weights of mixed tilting sheaves. We introduce a notion called "non-cancellation property" which implies a strong uniqueness of mixed tilting sheaves and enables one to calculate their weights effectively. When we have a certain Radon transform, we prove a geometric analogue of Ringel duality which sends tilting objects to projective objects. We apply these methods to (partial) flag varieties and affine (partial) flag varieties and show that the weight polynomials of mixed tilting sheaves on flag and affine flag varieties are essentially given by Kazhdan-Lusztig polynomials. This verifies a mixed geometric analogue of a conjecture by W.Soergel in \cite{Sg1}.

An immanant formulation of the dual canonical basis of the quantum polynomial ring

We show that dual canonical basis elements of the quantum polynomial ring in $n^2$ variables can be expressed as specializations of dual canonical basis elements of $0$-weight spaces of other quantum polynomial rings. Our results rely upon the natural appearance in the quantum polynomial ring of Kazhdan-Lusztig polynomials, $R$-polynomials, and certain single and double parabolic generalizations of these. Nous démontrons que des éléments de la base canonique duale de l'anneau quantique des polynômes en $n^2$ variables peuvent s'exprimer en termes des spécialisations d'éléments de la base canonique duale des espaces de poids $0$ d'autres anneaux quantiques. Nos résultats dépendent fortement de l'apparition naturelle des polynômes de Kazhdan-Lusztig, des $R$-polynômes, et de certaines généralisations simplement et doublement paraboliques de ces polynômes dans l'anneau quantique.

Type B plactic relations for $r$-domino tableaux

The recent work of Bonnafé et al. (2007) shows through two conjectures that $r$-domino tableaux have an important role in Kazhdan-Lusztig theory of type $B$ with unequal parameters. In this paper we provide plactic relations on signed permutations which determine whether given two signed permutations have the same insertion $r$-domino tableaux in Garfinkle's algorithm (1990). Moreover, we show that a particular extension of these relations can describe Garfinkle's equivalence relation on $r$-domino tableaux which is given through the notion of open cycles. With these results we enunciate the conjectures of Bonnafé et al. and provide necessary tool for their proofs.

Brundan–Kazhdan–Lusztig and Super Duality Conjectures

We formulate a general super duality conjecture on connections between parabolic categories \mathcal O of modules over Lie superalgebras and Lie algebras of type A , based on a Fock space formalism of their Kazhdan–Lusztig theories which was initiated by Brundan. We show that the Brundan–Kazhdan–Lusztig (BKL) polynomials for \mathfrak{gl}(m|n) in our parabolic setup can be identified with the usual parabolic Kazhdan–Lusztig polynomials. We establish some special cases of the BKL conjecture on the parabolic category \mathcal O of \mathfrak{gl}(m|n) -modules and additional results which support the BKL conjecture and super duality conjecture.

Parabolic Kazhdan-Lusztig polynomials for Hermitian symmetric pairs

We study the parabolic Kazhdan-Lusztig polynomials for Hermitian symmetric pairs. In particular, we show that these polynomials are always either zero or a monic power of $q$, and that they are combinatorial invariants.

Leading coefficients of Kazhdan–Lusztig polynomials and fully commutative elements

Let W be a Coxeter group of type $\widetilde{A}_{n-1}$ . We show that the leading coefficient, μ(x,w), of the Kazhdan–Lusztig polynomial P x,w is always equal to 0 or 1 if x is fully commutative (and w is arbitrary).

Multiplicity matrices for the affine graded Hecke algebra

In this paper we are looking at the problem of determining the composition factors for the affine graded Hecke algebra via the computation of Kazhdan–Lusztig type polynomials. We review the algorithms of [G. Lusztig, Study of perverse sheaves arising from graded Lie algebras, Adv. Math. 112 (1995) 147–217; G. Lusztig, Graded Lie algebras and intersection cohomology, preprint], and use them in particular to compute, at every real central character which admits tempered modules, the geometric parameterization, the Kazhdan–Lusztig polynomials, the composition series, and the Iwahori–Matsumoto involution for the representations with Iwahori fixed vectors of the split p-adic groups of type G 2 and F 4 (and by the nature of the algorithms, for their Levi subgroups).

Super duality and Kazhdan-Lusztig polynomials

We establish a direct connection between the representation theories of Lie algebras and Lie superalgebras (of type A) via Fock space reformulations of their Kazhdan-Lusztig theories. As a consequence, the characters of finite-dimensional irreducible modules of the general linear Lie superalgebra are computed by the usual parabolic Kazhdan-Lusztig polynomials of type A. In addition, we establish closed formulas for canonical and dual canonical bases for the tensor product of any two fundamental representations of type A.

SOME Z/2-GRADED REPRESENTATION THEORY

In representation theory, the existence of a Z + -grading on a related finite dimensional algebra often plays an important role. For example, such a grading arises from the Koszul structure of the finite dimensional algebra representing the principal block of the BGG category O associated to a complex semisimple Lie algebra. But Koszul gradings in positive characteristic have proved elusive. For example, except for small values of a positive integer n, it is not known if the Schur algebra S(n, n) has such a Koszul grading, assuming the characteristic p of the base field satisfies p ≥ n, though this grading would suffice to establish Lusztig’s character formula for these algebras. (And even though the character formula is known for p sufficiently large [H. Andersen, J. Jantzen and W. Soergel, Representations of Quantum Groups at a pth Root of Unity and of Semisimple Groups in Characteristic p, Asterique, Vol. 220, 1994], it is not known if the Schur algebra is Koszul for p sufficiently large.) This paper introduces Z/2-gradings on quasi-hereditary algebras, and shows that these gradings are almost as useful as a full Z + -grading, while being possibly much easier to find. We define the notion of a Z/2-based Kazhdan–Lusztig theory, which appears to be more flexible than, and generalizes, the notion of a Kazhdan–Lusztig theory (as first defined in [E. Cline, B. Parshall and L. Scott, Abstract Kazhdan–Lusztig theories, Tohoku Math. J. 45 (1993), 511–534]). However, its existence suffices, as was the case with the original notion, to establish character formulas in the standard settings, determine Ext n -groups, and show that homological duals behave well. Finally, we present some suggestive symmetric group examples involving Schur algebras which were an outgrowth of this work.

Cellular structures on Hecke algebras of type B

The aim of this paper is to gather and (try to) unify several approaches for the modular representation theory of Hecke algebras of type B. We attempt to explain the connections between Geck's cellular structures (coming from Kazhdan–Lusztig theory with unequal parameters) and Ariki's Theorem on the canonical basis of the Fock spaces.

Parabolic Kazhdan–Lusztig polynomials, plethysm and generalized Hall–Littlewood functions for classical types

We use power sums plethysm operators to introduce H functions which interpolate between the Weyl characters and the Hall–Littlewood functions Q ′ corresponding to classical Lie groups. The coefficients of these functions on the basis of Weyl characters are parabolic Kazhdan–Lusztig polynomials and thus, by works of Kashiwara and Tanisaki, are nonnegative. We prove that they can be regarded as quantizations of branching coefficients obtained by restriction to certain subgroups of Levi type. The H functions associated to linear groups coincide with the polynomials introduced by Lascoux, Leclerc and Thibon in [A. Lascoux, B. Leclerc, J.Y. Thibon, Ribbon tableaux, Hall Littelwood functions, quantum affine algebras, J. Math. Phys. 38 (1996) 1041–1068].

Signatures of Invariant Hermitian Forms on Irreducible Highest-Weight Modules

Perhaps the most important problem in representation theory in the 1970s and early 1980s was the determination of the multiplicity of com- position factors in a Verma module. This problem was settled by the proof of the Kazhdan-Lusztig Conjecture which states that the multiplicities may be computed via Kazhdan-Lusztig polynomials. In this paper, we introduce signed Kazhdan-Lusztig polynomials, a variation of Kazhdan-Lusztig polyno- mials which encodes signature information in addition to composition factor multiplicities and Jantzen filtration level. Careful consideration of Gabber and Joseph's proof of Kazhdan and Lusztig's recursive formula for computing Kazhdan-Lusztig polynomials and an application of Jantzen's determinant for- mula lead to a recursive formula for the signed Kazhdan-Lusztig polynomials. We use these polynomials to compute the signature of an invariant Hermitian form on an irreducible highest weight module. Such a formula has applications to unitarity testing. 1.1. The Unitary Dual Problem. In the 1930s, I.M. Gelfand introduced a broad programme in abstract harmonic analysis which would permit the transfer of dif- ficult problems in areas as distinct from analysis as topology to more tractable problems in algebra. Fourier analysis is just one incarnation of this programme. An unresolved component in Gelfand's programme is the classification of the irre- ducible unitary representations of a group, known as the unitary dual problem. In the case of a real reductive Lie group, the problem is equivalent to identifying all irreducible Harish-Chandra modules which admit a positive definite invariant Hermitian form. As Harish-Chandra modules may be constructed via an algebraic method introduced by Zuckerman in 1978 known as cohomological induction, it is of interest to study signatures of invariant Hermitian forms on cohomologically induced modules and to understand how positivity can fail. Cohomological induction is a two-step process in which we compose an induction functor with a Zuckerman functor i . The intermediate module in cohomological induction is a generalized Verma module which admits an invariant Hermitian form if the module to which induction was applied admits an invariant Hermitian form. Formulas for the signatures of invariant Hermitian forms on these intermediate modules may be used to compute signatures of forms on corresponding cohomo- logically induced modules (eg. (Wal84)). This motivates the study of invariant Hermitian forms on Verma modules (see (Wal84), (Yee05)) and irreducible highest weight modules.

The combinatorics of Coxeter categories

We present an alternative construction of Soergel's category of bimodules associated to a reflection faithful representation of a Coxeter system. We show that its objects can be viewed as sheaves on the associated moment graph. We introduce an exact structure and show that Soergel's special bimodules are the projective objects. Then we construct the indecomposable projectives by both a global and a local method, discuss a version of the Kazhdan-Lusztig conjecture and prove it for universal Coxeter systems.

Leading coefficients of Kazhdan–Lusztig polynomials for Deodhar elements

We show that the leading coefficient of the Kazhdan---Lusztig polynomial P x,w (q) known as μ(x,w) is always either 0 or 1 when w is a Deodhar element of a finite Weyl group. The Deodhar elements have previously been characterized using pattern avoidance in Billey and Warrington (J. Algebraic Combin. 13(2):111---136, [2001]) and Billey and Jones (Ann. Comb. [2008], to appear). In type A, these elements are precisely the 321-hexagon avoiding permutations. Using Deodhar's algorithm (Deodhar in Geom. Dedicata 63(1):95---119, [1990]), we provide some combinatorial criteria to determine when μ(x,w)=1 for such permutations w.

Governing singularities of Schubert varieties

We present a combinatorial and computational commutative algebra methodology for studying singularities of Schubert varieties of flag manifolds. We define the combinatorial notion of interval pattern avoidance. For “reasonable” invariants P of singularities, we geometrically prove that this governs (1) the P -locus of a Schubert variety, and (2) which Schubert varieties are globally not P . The prototypical case is P = “singular”; classical pattern avoidance applies admirably for this choice [V. Lakshmibai, B. Sandhya, Criterion for smoothness of Schubert varieties in SL ( n ) / B , Proc. Indian Acad. Sci. Math. Sci. 100 (1) (1990) 45–52, MR 91c:14061], but is insufficient in general. Our approach is analyzed for some common invariants, including Kazhdan–Lusztig polynomials, multiplicity, factoriality, and Gorensteinness, extending [A. Woo, A. Yong, When is a Schubert variety Gorenstein?, Adv. Math. 207 (1) (2006) 205–220, MR 2264071]; the description of the singular locus (which was independently proved by [S. Billey, G. Warrington, Maximal singular loci of Schubert varieties in SL ( n ) / B , Trans. Amer. Math. Soc. 335 (2003) 3915–3945, MR 2004f:14071; A. Cortez, Singularités génériques et quasi-résolutions des variétés de Schubert pour le groupe linéaire, Adv. Math. 178 (2003) 396–445, MR 2004i:14056; C. Kassel, A. Lascoux, C. Reutenauer, The singular locus of a Schubert variety, J. Algebra 269 (2003) 74–108, MR 2005f:14096; L. Manivel, Le lieu singulier des variétés de Schubert, Int. Math. Res. Not. 16 (2001) 849–871, MR 2002i:14045]) is also thus reinterpreted. Our methods are amenable to computer experimentation, based on computing with Kazhdan–Lusztig ideals (a class of generalized determinantal ideals) using Macaulay 2. This feature is supplemented by a collection of open problems and conjectures.

Deodhar Elements in Kazhdan-Lusztig Theory

The Kazhdan-Lusztig polynomials for finite Weyl groups arise in representation theory as well as the geometry of Schubert varieties. It was proved very soon after their introduction that they have nonnegative integer coefficients, but no simple all positive interpretation for them is known in general. Deodhar has given a framework, which generally involves recursion, to express the Kazhdan-Lusztig polynomials in a very attractive form. We use a new kind of pattern-avoidance that can be defined for general Coxeter groups to characterize when Deodhar's algorithm yields a non-recursive combinatorial formula for Kazhdan-Lusztig polynomials $P_{x,w}(q)$ of finite Weyl groups. This generalizes results of Billey-Warrington which identified the $321$-hexagon-avoiding permutations, and Fan-Green which identified the fully-tight Coxeter groups. We also show that the leading coefficient known as $\mu (x,w)$ for these Kazhdan―Lusztig polynomials is always either $0$ or $1$. Finally, we generalize the simple combinatorial formula for the Kazhdan―Lusztig polynomials of the $321$-hexagon-avoiding permutations to the case when $w$ is hexagon avoiding and maximally clustered. Les polynômes de Kazhdan-Lusztig $P_{x,w}(q)$ des groupes de Weyl finis apparaissent en théorie des représentations, ainsi qu’en géométrie des variétés de Schubert. Il a été démontré peu après leur introduction qu’ils avaient des coefficients entiers positifs, mais on ne connaît toujours pas d’interprétation combinatoire simple de cette propriété dans le cas général. Deodhar a proposé un cadre donnant un algorithme, en général récursif, calculant des formules attractives pour les polynômes de Kazhdan-Lusztig. Billey-Warrington ont démontré que cet algorithme est non récursif lorsque$w$ évite les hexagones et les $321$ et qu’il donne des formules combinatoires simples. Nous introduisons une notion d’évitement de schémas dansles groupes de Coxeter quelconques nous permettant de généraliser les résultats de Billey-Warrington à tout groupe de Weyl fini. Nous montrons que le coefficient de tête $\mu (x,w)$ de ces polynômes de Kazhdan-Lusztig est toujours $0$ ou $1$. Cela généralise aussi des résultats de Fan-Greenqui identifient les groupes de Coxeter complètement serrés. Enfin, en type $A$, nous obtenons une classe plus large de permutations évitant la récursion.

Cyclic Sieving, Promotion, and Representation Theory

We prove a collection of conjectures due to Abuzzahab-Korson-Li-Meyer, Reiner, and White regarding the cyclic sieving phenomenon 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 $\mathbb{C}[x_{11}, \ldots , x_{nn}]$ due to Skandera. 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, handshake patterns, and noncrossing partitions.