Kazhdan-Lusztig Polynomials Research Articles

Characteristic Cycles in Hermitian Symmetric Spaces

AbstractWe give explicit combinatorial expresssions for the characteristic cycles associated to certain canonical sheaves on Schubert varieties X in the classical Hermitian symmetric spaces: namely the intersection homology sheaves IHX and the constant sheaves ℂX. The three main cases of interest are the Hermitian symmetric spaces for groups of type An (the standard Grassmannian), Cn (the Lagrangian Grassmannian) and Dn. In particular we find that CC(IHX) is irreducible for all Schubert varieties X if and only if the associated Dynkin diagramis simply laced. The result for Schubert varieties in the standard Grassmannian had been established earlier by Bressler, Finkelberg and Lunts, while the computations in the Cn and Dn cases are new.Our approach is to compute CC(ℂX) by a direct geometric method, then to use the combinatorics of the Kazhdan-Lusztig polynomials (simplified for Hermitian symmetric spaces) to compute CC(IHX). The geometric method is based on the fundamental formula where the Xr ↓ X constitute a family of tubes around the variety X. This formula leads at once to an expression for the coefficients of CC(ℂX) as the degrees of certain singular maps between spheres.

Kazhdan-Lusztig polynomials and a combinatoric for tilting modules

This article gives a self-contained treatment of the theory of Kazhdan-Lusztig polynomials with special emphasis on affine reflection groups. There are only a few new results but several new proofs. We close with a conjectural character formula for tilting modules, which formed the starting point of these investigations.

Kazhdan-Lusztig-Polynome und eine Kombinatorik für Kipp-Moduln

This article gives a selfcontained treatment of the theory of Kazhdan-Lusztig polynomials with special emphasis on affine reflection groups. There are only a few new results but several new proofs. We close with a conjectural character formula for tilting modules, which formed the starting point of these investigations.

Homological Duality and Quasi-Heredity

AbstractThis paper represents a general study of the (Yoneda) Ext-algebra A* of a finite dimensional K-algebra A. Our motivation lies in the problem of establishing conditions under which (i) the species of A* coincides with the dual species of A and (ii) the quasi-heredity of A (or A*) yields the quasi-heredity of A* (or A, respectively). These questions are closely related to the Kazhdan—Lusztig Theory as presented by [CPS2]. The main results include introducing the concept of a solid algebra and the relevant Theorem 4.5 as well as a rather complete description of the situation in the case of monomial algebras in Section 5.

Kazhdan-Lusztig polynomials and character formula for the Lie superalgebragI(m/n)

We find the character formula for irreducible finite-dimensionalgl(m/n)-modules. Also multiplicities of the composition factors in a Kac module are calculated.

Probing integrable perturbations of conformal theories using singular vectors

It has been known for some time that the (1,3) perturbations of the (2 k + 1,2) Virasoro minimal models have conserved currents which are also singular vectors of the Virasoro algebra. This also turns out to hold for the (1,2) perturbation of the (3 k ± 1,3) models. In this paper we investigate the requirement that a perturbation of an extended conformal field theory has conserved currents which are also singular vectors. We consider conformal field theories with W 3 and (bosonic) WBC 2 = W(2,4) extended symmetries. Our analysis relies heavily on the general conjecture of de Vos and van Driel relating the multiplicities of W-algebra irreducible modules to the Kazhdan-Lusztig polynomials of a certain double coset. Granting this conjecture, the singular-vector argument provides a direct way of recovering all known integrable perturbations. However, W models bring a slight complication in that the conserved densities of some (1, 2)-type perturbations are actually subsingular vectors, that is, they become singular vectors only in a quotient module.

The Kazhdan–Lusztig conjecture for 𝒲 algebras

The main result in this paper is the character formula for arbitrary irreducible highest weight modules of 𝒲 algebras. The key ingredient is the functor provided by quantum Hamiltonian reduction, which constructs the 𝒲 algebras from affine Kac–Moody (KM) algebras and in a similar fashion 𝒲 modules from KM modules. Assuming certain properties of this functor, the 𝒲 characters are subsequently derived from the Kazhdan–Lusztig conjecture for KM algebras. The result can be formulated in terms of a double coset of the Weyl group of the KM algebra: the Hasse diagrams give the embedding diagrams of the Verma modules and the Kazhdan–Lusztig polynomials give the multiplicities in the characters.

Topology of intersections of Schubert cells and Hecke algebra

We consider intersections of Schubert cells Bα · B and σBσ − β · B in the space of complete flags F = SL/ B, where B denotes the Borel subgroup of upper triangular matrices, while α, β and σ belong to the Weyl group W (coinciding with the symmetric group). We obtain a special decomposition of F which subdivides all Bα · B ∩ σBσ − β · B into strata of a simple form. It enables us to establish a new geometrical interpretation of the structure constants for the corresponding Hecke algebra and in particular of the so-called R-polynomials used in Kazhdan-Lusztig theory. Structure constants of the Hecke algebra appear to be the alternating sums of the Hodge numbers for the mixed Hodge structure in the cohomology with compact supports of the above intersections. We derive a new efficient combinatorial algorithm calculating the R-polynomials and structure constants in general.

Proof of the Kazhdan–Lusztig Conjecture for Kac–Moody Algebras (The CharacterschLωρ−ρ)

Proof of the Kazhdan–Lusztig Conjecture for Kac–Moody Algebras (The CharacterschLωρ−ρ)

The state of the art in the computation of Kazhdan-Lusztig polynomials

We present the different approaches available today for the computation of Kazhdan-Lusztig polynomials by computer, and describe the scope of the existing programs in this area. We also outline some possible directions for future developments.

The State of the Art in the Computation of Kazhdan-Lusztig Polynomials

The State of the Art in the Computation of Kazhdan-Lusztig Polynomials

The Kazhdan-Lusztig conjecture for finiteW-algebras

We study the representation theory of finiteW-algebras. After introducing parabolic subalgebras to describe the structure ofW-algebras, we define the Verma modules and give a conjecture for the Kac determinant. This allows us to find the completely degenerate representations of the finiteW-algebras. To extract the irreducible representations we analyse the structure of singular and subsingular vectors and find that, forW-algebras, in general the maximal submodule of a Verma module is not generated by singular vectors only. Surprisingly, the role of the (sub)singular vectors can be encapsulated in terms of a ‘dual’ analogue of the Kazhdan-Lusztig theorem for simple Lie algebras. These involve dual relative Kazhdan-Lusztig polynomials. We support our conjectures with some examples, and briefly discuss applications and the generalisation to infiniteW-algebras.

A Symmetric Function Generalization of the Chromatic Polynomial of a Graph

For a finite graph G with d vertices we define a homogeneous symmetric function XG of degree d in the variables x1, x2, ... . If we set x1 = ... = xn= 1 and all other xi = 0, then we obtain χG(n), the chromatic polynomial of G evaluated at n. We consider the expansion of XG in terms of various symmetric function bases. The coefficients in these expansions are related to partitions of the vertices into stable subsets, the Möbius function of the lattice of contractions of G, and the structure of the acyclic orientations of G. The coefficients which arise when XG is expanded in terms of elementary symmetric functions are particularly interesting, and for certain graphs are related to the theory of Hecke algebras and Kazhdan-Lusztig polynomials.

KOSZUL ALGEBRAS AND THE FROBENIUS AUTOMORPHISM

LET %’ be a highest weight category [CPSl] with finitely many simple objects and such that every object has finite length. Then Ce = mod S for a finite dimensional quasi-hereditary algebra S. In a series of papers, [CPS4,5,6,7], we have studied various Kazhdan-Lusztig theories for %. This concept arises naturally in, for example, the modular representation theory of semisimple algebraic groups. As shown in [CPS4; § 41, the existence of a Kazhdan-Lusztig theory in this case is equivalent to the validity of the Lusztig conjecture (and leads to a considerable simplification of that conjecture). Also, the theory permits a combinatorial calculation of the Yoneda extension groups for irreducible representations with regular high weight in the Jantzen region. More generally, as shown in [CPS7; 5 21, if (e has a Kazhdan-Lusztig theory, its homological dual is also a highest weight category. When the algebra S is (positively) graded, the associated category ?& of finitely generated, graded S-modules has the structure of a graded highest weight category. (For a discussion of this notion, see [CPS7; $ 11.) By analogy with the ungraded theory, one can consider the notion of a graded Kazhdan-Lusztig theory. This concept has been developed in detail in [CPS4,7]. The property of having a graded Kazhdan-Lusztig theory is closely related to another property of graded algebras: a positively graded algebra S is called Koszul provided each irreducible S-module L, when regarded as a graded S-module concentrated in degree 0, has a projective resolution * * * --+ P-‘-+ PO+ L + 0 in %& in which each P-” is generated by its term in grade n. (see [P], [BGS].) A main result in [CPS’I; (3.9)] establishes that if %& has a graded KazhdanLusztig theory, the associated algebra S is Koszul. The converse also holds in the presence of an (ungraded) Kazhdan-Lusztig theory on (e. We begin in Q 1 by formulating a new criterion guaranteeing that a (finite dimensional) k-algebra S is Koszul. Our result states essentially that, given an algebra S and an automorphism (7, if the Ext-algebra associated to S can be given its natural graded structure by means of the

Kazhdan-Lusztig conjecture for generalized Kac-Moody algebras. II. Proof of the conjecture

Generalized Kac-Moody algebras were introduced by Borcherds in the study of Conway and Norton’s moonshine conjectures for the Monster sporadic simple group. In this paper, we prove the Kazhdan-Lusztig conjecture for generalized Kac-Moody algebras under a certain mild condition, by using a generalization (to the case of generalized Kac-Moody algebras) of Jantzen’s character sum formula. Our (main) formula generalizes the celebrated result for the case of Kac-Moody algebras, and describes the characters of irreducible highest weight modules over generalized Kac-Moody algebras in terms of the "extended" Kazhdan-Lusztig polynomials.

𝐷-sets and BG-functors in Kazhdan-Lusztig theory

By using Deodhar’s combinatorial setting and Bernstein-Gelfand projective functors, this paper provides some necessary and sufficient conditions for a highest weight category to have a Kazhdan-Lusztig theory. A consequence of these conditions is that in the semisimple Lie algebra case, the Kazhdan-Lusztig conjecture on the multiplicities of a Verma module implies the nonnegativity conjecture on the coefficients of Kazhdan-Lusztig polynomials.

A combinatorial formula for Kazhdan-Lusztig polynomials

In their fundamental paper [14] Kazhdan and Lusztig defined, for every Coxeter group W, a family of polynomials, indexed by pairs of elements of W, which have become known as the Kazhdan-Lusztig polynomials of W (see, e.g., [12], Chap. 7). These polynomials are intimately related to the Bruhat order of W and to the algebraic geometry of Schubert varieties, and have proven to be of fundamental importance in representation theory. Our aim in this work is to give a simple, nonrecursive, combinatorial formula for any Kazhdan-Lusztig polynomial of any Coxeter group (Theorem 4.1), and to study some consequences of it. The main idea involved in the proof and statement of this formula is that of extending the concept of the Rpolynomial (see, e.g., [12], w to any (finite) multichain of W (so that, for multichains of length 1, one obtains, apart from sign, the usual R-polynomials). Once this has been done, then the Kazhdan-Lusztig polynomial of a pair u, v turns out to be just the sum, over all multichains from u to v, of the corresponding (generalized) R-polynomials. The R-polynomial of a multichain can be readily defined, and computed from the ordinary R-polynomials (see (9), (10), and (11)). Since several combinatorial formulas and interpretations are known for these polynomials (see, e.g., [5, 10], and, for the case of symmetric groups, [3]) and simple recurrences exist for them, we feel that this formula is a significant step forward in the understanding of the KazhdanLusztig polynomials. Though combinatorial formulas for Kazhdan-Lusztig polynomials have appeared before in the literature (see, e.g., [16, 21, 8, 6]), none of them hold in complete generality.

The decomposition numbers of Hecke algebras of typeF 4 with unequal parameters

The purpose of this paper is to calculate the decomposition numbers for Hecke algebras of typeF4 (andC3) with unequal parameters. The problem is reduced to specifying decomposition maps of the generic Hecke algebras. The concept of Kazhdan-Lusztig polynomials and left cells serves to determine their irreducible representations. We also prove a result about the minimal ring over which the irreducible representations of the generic algebras can be realized.

Homological characterization of lean algebras

Certain classes of lean quasi-hereditary algebras play a central role in the representation theory of semisimple complex Lie algebras and algebraic groups. The concept of a lean semiprimary ring, introduced recently in [1] is given here a homological characterization in terms of the surjectivity of certain induced maps between Ext1-groups. A stronger condition requiring the surjectivity of the induced maps between Extk-groups for allk≥1, which appears in the recent work of Cline, Parshall and Scott on Kazhdan-Lusztig theory, is shown to hold for a large class of lean quasi-hereditary algebras.

Sandwich Matrices, Solomon Algebras, and Kazhdan-Lusztig Polynomials

Sandwich matrices have proved to be of importance in semigroup theory for the last 50 years. The work of the author on algebraic monoids leads to sandwich matrices in group theory. In this paper, we find some connections between sandwich matrices and the Hecke algebras (for monoids) introduced recently by Louis Solomon. At the local level we then obtain an explicit isomorphism between Solomon’s Hecke algebra and the complex monoid algebra of the Renner monoid. In the simplest case of monoids associated with a Borel subgroup, we find that the entries of the inverse of the sandwich matrix, as well as those of the related structure matrix of Solomon’s Hecke algebra are ’almost’ the polynomials ${R_{x,y}}$ associated with the Kazhdan-Lusztig polynomials.