Kazhdan-Lusztig Polynomials Research Articles

Andersen Filtration and Hard Lefschetz

On the space of homomorphisms from a Verma module to an indecomposable tilting module of the BGG-category $${\mathcal{O}}$$ we define a natural filtration following Andersen [A] and establish a formula expressing the dimensions of the filtration steps in terms of coefficients of Kazhdan–Lusztig polynomials.

Embedded Factor Patterns for Deodhar Elements in Kazhdan-Lusztig Theory

The Kazhdan-Lusztig polynomials for nite Weyl groups arise in the geometry of Schubert varieties and representation theory. It was proved very soon after their introduction that they have nonnegative integer coefcients, but no simple all positive interpretation for them is known in general. Deodhar (16) has given a framework for computing the Kazhdan-Lusztig polynomials which generally involves recursion. We dene embedded factor pattern avoidance for general Coxeter groups and use it to characterize when Deodhar's algorithm yields a simple combinatorial formula for the Kazhdan-Lusztig polynomials of nite Weyl groups. Equivalently, if (W; S) is a Coxeter system for a nite Weyl group, we classify the elements w 2 W for which the Kazhdan-Lusztig basis element C 0 can be written as a monomial of C 0 where s 2 S. This work generalizes results of Billey-Warrington (8) that identied the Deodhar elements in type A as 321-hexagon-avoiding permutations, and Fan-Green (18) that identied the fully-tight Coxeter groups.

Enveloping algebras of Slodowy slices and the Joseph ideal

Let G be a simple algebraic group over an algebraically closed field \mathbb k of characteristic 0 , and \mathfrak g=\text{Lie}\,G . Let (e,h,f) be an \mathfrak{sl}_2 -triple in \mathfrak g with e being a long root vector in \mathfrak g . Let (\,\cdot\,,\,\cdot\,) be the G -invariant bilinear form on \mathfrak g with (e,f)=1 and let \chi\in\mathfrak g^* be such that \chi(x)=(e,x) for all x\in\mathfrak g . Let {\mathcal S} be the Slodowy slice at e through the adjoint orbit of e and let H be the enveloping algebra of {\mathcal S} ; see [31]. In this note we give an explicit presentation of H by generators and relations. As a consequence we deduce that H contains an ideal of codimension 1 which is unique if \mathfrak g is not of type \mathrm A . Applying Skryabin's equivalence of categories we then construct an explicit Whittaker model for the Joseph ideal of U(\mathfrak g) . Inspired by Joseph's Preparation Theorem we prove that there exists a homeomorphism between the primitive spectrum of H and the spectrum of all primitive ideals of infinite codimension in U(\mathfrak g) which respects Goldie rank and Gelfand–Kirillov dimension. We study highest weight modules for the algebra H and apply earlier results of Miličić–Soergel and Backelin to express the composition multiplicities of the Verma modules for H in terms of some inverse parabolic Kazhdan–Lusztig polynomials. Our results confirm in the minimal nilpotent case the de Vos–van Driel conjecture on composition multiplicities of Verma modules for finite {\mathcal W} -algebras. We also obtain some general results on the enveloping algebras of Slodowy slices and determine the associated varieties of related primitive ideals of U(\mathfrak g) . A sequel to this paper will treat modular aspects of this theory.

On the determination of Kazhdan–Lusztig cells for affine Weyl groups with unequal parameters

Let W be a Coxeter group and L be a weight function on W. Following Lusztig, we have a corresponding decomposition of W into left cells which have important applications in representation theory. We study the case where W is an affine Weyl group of type G ˜ 2 . Using explicit computation with COXETER and CHEVIE, we show that (1) there are only finitely many possible decompositions into left cells and (2) the number of left cells is finite in each case, thus confirming some of Lusztig's conjectures in this case. A key ingredient of the proof is a general result which shows that the Kazhdan–Lusztig polynomials of affine Weyl group are invariant under (large enough) translations.

TRIPLY-GRADED LINK HOMOLOGY AND HOCHSCHILD HOMOLOGY OF SOERGEL BIMODULES

We consider a class of bimodules over polynomial algebras which were originally introduced by Soergel in relation to the Kazhdan–Lusztig theory, and which describe a direct summand of the category of Harish–Chandra modules for sl(n). Rouquier used Soergel bimodules to construct a braid group action on the homotopy category of complexes of modules over a polynomial algebra. We apply Hochschild homology to Rouquier's complexes and produce triply-graded homology groups associated to a braid. These groups turn out to be isomorphic to the groups previously defined by Lev Rozansky and the author, which depend, up to isomorphism and overall shift, only on the closure of the braid. Consequently, our construction produces a homology theory for links.

Nilpotent orbits of linear and cyclic quivers and Kazhdan-Lusztig polynomials of type A

The intersection cohomologies of closures of nilpotent orbits of linear (respectively, cyclic) quivers are known to be described by Kazhdan-Lusztig polynomials for the symmetric group (respectively, the affine symmetric group). We explain how to simplify this description using a combinatorial cancellation procedure, and we derive some consequences for representation theory.

Completely compressible Bruhat intervals and Kazhdan–Lusztig polynomials

Our main result is that the recently proved combinatorial invariance property for Kazhdan–Lusztig polynomials on lower Bruhat intervals still holds for Bruhat intervals whose top element is critical. We conjecture that our theory also extends to completely compressible Bruhat intervals in type A,D,E, and we have checked this conjecture up to types A8,D7,E6 with a computer. Positive results in related special cases are also presented.

Generalized Jones traces and Kazhdan–Lusztig bases

We develop some applications of certain algebraic and combinatorial conditions on the elements of Coxeter groups, such as elementary proofs of the positivity of certain structure constants for the associated Kazhdan–Lusztig basis. We also explore some consequences of the existence of a Jones-type trace on the Hecke algebra of a Coxeter group, such as simple procedures for computing leading terms of certain Kazhdan–Lusztig polynomials.

Extensions, Levi subgroups and character formulas

This paper consists of three interconnected parts. Parts I, III study the relationship between the cohomology of a reductive group G and that of a Levi subgroup H. For example, we provide a sufficient condition, arising from Kazhdan–Lusztig theory, for a natural map Ext G • ( L , L ′ ) → Ext H • ( L H , L H ′ ) to be surjective, given irreducible G-modules L , L ′ and corresponding irreducible H-modules L H , L H ′ . In cohomological degree n = 1 , the map is always an isomorphism, under our hypothesis. These results were inspired by recent work of Hemmer [D. Hemmer, A row removal theorem for the Ext 1 quiver of symmetric groups and Schur algebras, Proc. Amer. Math. Soc. 133 (2) (2005) 403–414 (electronic)] for G = GL n , and both extend and improve upon the latter when our condition is met. Part II obtains results on Lusztig character formulas (LCFs) for reductive groups, obtaining new necessary and sufficient conditions for such formulas to hold. In the special case of G = GL n , these conditions can be recast in a striking way completely in terms of explicit representation theoretic properties of the symmetric group. This work on GL n improves upon [B. Parshall, L. Scott, Quantum Weyl reciprocity for cohomology, Proc. London Math. Soc. 90 (3) (2005) 655–688], which established only sufficient, rather than necessary and sufficient, conditions for the validity of the LCF in terms of the symmetric group.

Produit scalaire elliptique

Suppose that W is a Weyl group, let C ( W ) be a space of functions on W, with complex values, invariant under conjugation. We can define an “elliptic scalar product” on C ( W ) . It is a natural ingredient to the representation theory of p-adic reductive groups. Let G be a reductive group over the algebraic closure of a finite field. The generalized Springer correspondence gives a bijection between two sets: – the set of pairs ( U , E ) , where U is an unipotent orbit of G and E is a G-equivariant irreducible local system on U; – the disjoint union of the sets of irreducible representations of certain Weyl groups related to G. Using Kazhdan–Lusztig polynomials, we modify the generalized Springer correspondence. By the modified correspondence, a pair ( U , E ) as above maps to a representation of a certain Weyl group, and this representation is, in general, reducible. There is no simple formula that relates the elliptic scalar product and the generalized Springer correspondence. But a simple formula does exist, and we prove it, that relates the elliptic scalar product and the modified generalized Springer correspondence. Our result is, in fact, a corollary of a theorem of Lusztig on the restriction of character-sheaves to the unipotent variety.

Special matchings and permutations in Bruhat orders

For any permutation v, we show that the special matchings of v generate a Coxeter system. This gives new necessary conditions on an abstract poset to be isomorphic to a lower Bruhat interval of the symmetric group, and also answers in the affirmative, in the symmetric group case, a problem posed in [F. Brenti, F. Caselli, M. Marietti, Special matchings and Kazhdan–Lusztig polynomials, Adv. Math. 202 (2006) 555–601].

Character and dimension formulae for general linear superalgebra

The generalized Kazhdan–Lusztig polynomials for the finite dimensional irreducible representations of the general linear superalgebra are computed explicitly. The result is then applied to prove a conjectural character formula put forward by van der Jeugt et al. in the late 80s. We simplify this character formula to cast it into the Kac–Weyl form, and derive from it a closed formula for the dimension of any finite dimensional irreducible representation of the general linear superalgebra.

More on the combinatorial invariance of Kazhdan–Lusztig polynomials

We prove that the Kazhdan–Lusztig polynomials are combinatorial invariants for intervals up to length 8 in Coxeter groups of type A and up to length 6 in Coxeter groups of type B and D. As a consequence of our methods, we also obtain a complete classification, up to isomorphism, of Bruhat intervals of length 7 in type A and of length 5 in types B and D, which are not lattices.

An inversion formula for induced Kazhdan–Lusztig polynomials and duality ofW-graphs

Let \(\fancyscript{H}\) be the Hecke algebra associated with a Coxeter group W. Many interesting \(\fancyscript{H}\)-modules can be described using the concept of a W-graph, as introduced in the influential paper [Invent. Math. 53: 165–184,1979] of Kazhdan and Lusztig. In particular, Kazhdan and Lusztig showed that the regular representation of \(\fancyscript{H}\) has an associated W-graph. Let \(\fancyscript{H}_{J}\) be the Hecke algebra associated with WJ, a parabolic subgroup of W. In [Math Z 244:415–431, 2003], an algorithm was described for the construction of a so-called induced W-graph for an induced module \(\fancyscript{H} \bigotimes_{\fancyscript{H}_{J}}\)V, where V is an \(\fancyscript{H}_{J}\)-module derived from a WJ-graph. In this note, we continue to analyse the induced W-graphs and prove the following results: the induced Kazhdan–Lusztig polynomials for a pair of dual, induced W-graphs are related by an inversion formula. This result generalizes a result of Kazhdan and Lusztig [Invent Math 53:165–184, 1979. Theorem 3.1] that has already been generalized independently by Deodhar J Algebra 190:214–225,1997 and Matthew [Comm Algebra 18:371–387, 1990] the dual of an induced W-graph and the W-graph induced from the dual are associated to isomorphic \(\fancyscript{H}\)-modules(or simply, duality commutes with induction).

Combinatorial invariance of Kazhdan-Lusztig polynomials on intervals starting from the identity

We show that for Bruhat intervals starting from the identity in Coxeter groups the conjecture of Lusztig and Dyer holds, that is, the R-polynomials and the Kazhdan-Lusztig polynomials defined on [e,u] only depend on the isomorphism type of [e,u]. To achieve this we use the purely poset-theoretic notion of special matching. Our approach is essentially a synthesis of the explicit formula for special matchings discovered by Brenti and the general special matching machinery developed by Du Cloux.

Dual canonical bases and Kazhdan–Lusztig polynomials

We derive a formula for the entries of the (unitriangular) transition matrices between the standard monomial and dual canonical bases of the irreducible polynomial representations of U q ( gl n ) in terms of Kazhdan–Lusztig polynomials.

Formulas for multi-parameter classes of Kazhdan–Lusztig polynomials in [formula omitted

We give explicit formulas for several classes of Kazhdan–Lusztig polynomials of the symmetric group which are related to others already considered in the literature. In particular, we generalize two theorems of Brenti and Simion [Explicit formulae for some Kazhdan–Lusztig polynomials, J. Algebraic Combin. 11 (2000) 187–196], we prove an unpublished conjecture of Brenti, and we treat the case missing in Theorem 4.8 of F. Caselli [Proof of two conjectures of Brenti and Simion on Kazhdan–Lusztig polynomials, J. Algebraic Combin. 18 (2003) 171–187].

Lattice paths, lexicographic correspondence and Kazhdan–Lusztig polynomials

In this paper we give a new closed formula for the Kazhdan–Lusztig polynomials of finite Coxeter groups and affine Weyl groups. This formula is computationally more efficient than any existing one, and it conjecturally holds for all Coxeter groups.

The Combinatorics of Category O over symmetrizable Kac-Moody Algebras

We show that the structure of a block outside the critical hyperplanes of category O over a symmetrizable Kac-Moody algebra depends only on the corresponding integral Weyl group and its action on the parameters of the Verma modules. This is done by giving a combinatorial description of the projective objects in the block. As an application, we derive the Kazhdan-Lusztig conjecture for nonintegral blocks from the integral case for finite or affine Weyl groups. We also prove the uniqueness of Verma embeddings outside the critical hyperplanes.

On the combinatorial invariance of Kazhdan–Lusztig polynomials

In this paper, we solve the conjecture about the combinatorial invariance of Kazhdan–Lusztig polynomials for the first open cases, showing that it is true for intervals of length 5 and 6 in the symmetric group. We also obtain explicit formulas for the R-polynomials and for the Kazhdan–Lusztig polynomials associated with any interval of length 5 in any Coxeter group, showing in particular what they look like in the symmetric group.