Lusztig Polynomials Research Articles

Boolean elements in Kazhdan–Lusztig theory

Kazhdan–Lusztig polynomials have been proven to play an important role in different fields. Despite this, there are still few explicit formulae for them. Here we give closed product formulae for the Kazhdan–Lusztig polynomials indexed by Boolean elements in a class of Coxeter systems that we call linear. Boolean elements are elements smaller than a reflection that admits a reduced expression of the form s 1 … s n − 1 s n s n − 1 … s 1 ( s i ∈ S , s i ≠ s j if i ≠ j ). Then we provide several applications of this result concerning the combinatorial invariance of the Kazhdan–Lusztig polynomials, the classification of the pairs ( u , v ) with u ≺ v , the Kazhdan–Lusztig elements and the intersection homology Poincaré polynomials of the Schubert varieties.

Special matchings and Kazhdan–Lusztig polynomials

In 1979 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 , and which have proven to be of importance in several areas of mathematics. In this paper, we show that the combinatorial concept of a special matching plays a fundamental role in the computation of these polynomials. Our results also imply, and generalize, the recent one in [Adv. in Math. 180 (2003) 146–175] on the combinatorial invariance of Kazhdan–Lusztig polynomials.

Non-negativity properties of R-polynomials

An explicit formula expressing the coefficients of the R̃-polynomials as a linear combinations of those of the corresponding R-polynomials is given. Several systems of inequalities satisfied by the coefficients of the R-polynomials are shown and applied to obtain a lower bound for the coefficient of q2 in Kazhdan–Lusztig polynomials. Finally we present a basis of the space of polynomials of a given degree with respect to which both Kazhdan–Lusztig and R-polynomials have some nice (conjectural) non-negativity properties.

A two-sided q-analogue of the Coxeter complex

We define a complex of bimodules over the Iwahori–Hecke algebra associated to a finite Coxeter group, calculate its cohomology and show that it induces a derived equivalence over its module category extending the Morita equivalence given by a certain algebra automorphism. We show further that when tensored with the index representation this complex becomes isomorphic to the one-sided q-analogue of the Coxeter complex previously defined by V. Deodhar [On some geometric aspects of Bruhat orderings. II. The parabolic analogue of Kazhdan–Lusztig polynomials, J. Algebra 111 (2) (1987) 483–506] and A. Mathas [A q-analogue of the Coxeter complex, J. Algebra 164 (3) (1994) 831–848].

The leading coefficient of certain Kazhdan–Lusztig polynomials of the permutation group [formula omitted

In this paper we show that the leading coefficient μ ( y , w ) of certain Kazhdan–Lusztig polynomials P y , w of the permutation group S n of 1 , 2 , … , n are not greater than 1. More precisely, we show that the leading coefficients μ ( y , w ) are not greater than 1 whenever a ( y ) < a ( w ) , where a : S n → N is the function defined in [G. Lusztig, Cells in affine Weyl groups, in: Algebraic Groups and Related Topics, in: Adv. Stud. Pure Math., vol. 6, Kinokunia–North-Holland, 1985, pp. 255–287]. See 1.5 for a simple interpretation of the function a.

COMPOSITION FACTORS OF QUOTIENTS OF THE UNIVERSAL ENVELOPING ALGEBRA BY PRIMITIVE IDEALS

Graded versions of the principal series modules of the category O of a semisimple complex Lie algebra g are defined. Their combinatorial descriptions are given by some Kazhdan–Lusztig polynomials. A graded version of the Duflo–Zhelobenko four-term exact sequence is proved. This gives results about composition factors of quotients of the universal enveloping algebra of g by primitive ideals; in particular an upper bound is obtained for the multiplicities of such composition factors. Explicit descriptions are given of principal series modules for Lie algebras of rank 2. It can be seen that these graded versions of principal series representations are neither rigid nor Koszul modules.

The t-analogs of q-characters at roots of unity for quantum affine algebras and beyond

The q-characters were introduced by Frenkel and Reshetikhin [The q-characters of representations of quantum affine algebras and deformations of W-algebras, in: Recent Developments in Quantum Affine Algebras and Related Topics, in: Contemp. Math., vol. 248, Amer. Math. Soc., Providence, RI, 1999, pp. 163–205] to study finite dimensional representations of the untwisted quantum affine algebra U q( g ̂ ) for q generic. The ε-characters at roots of unity were constructed by Frenkel and Mukhin [The q-characters at roots of unity, Adv. Math. 171 (1) (2002) 139–167] to study finite dimensional representations of various specializations of U q( g ̂ ) at q s =1. In the finite simply laced case Nakajima [ t-analogue of the q-characters of finite dimensional representations of quantum affine algebras, in: Physics and Combinatorics, Proc. Nagoya 2000 Internat. Workshop, World Scientific, Singapore, 2001, pp. 181–212; Quiver varieties and t-analogs of q-characters of quantum affine algebras, Ann. of Math., in press; preprint arXiv: math.QA/0105173] defined deformations of q-characters called q, t-characters for q generic and also at roots of unity. The definition is combinatorial but the proof of the existence uses the geometric theory of quiver varieties which holds only in the simply laced case. In [Algebraic approach to q, t-characters, Adv. Math., in press, preprint arXiv: math.QA/0212257] we proposed an algebraic general (non-necessarily simply laced) new approach to q, t-characters for q generic. In this paper we propose two developments of this approach: we treat the root of unity case and the case of a larger class of generalized Cartan matrices (including finite and affine cases except A 1 (1), A 2 (2)). In particular, we generalize the construction of analogs of Kazhdan–Lusztig polynomials at roots of unity of [H. Nakajima, Quiver varieties and t-analogs of q-characters of quantum affine algebras, Ann. of Math., in press, preprint arXiv: math.QA/0105173] to those cases. We also study properties of various objects used in this article: deformed screening operators at roots of unity, t-deformed polynomial algebras, bicharacters arising from symmetrizable Cartan matrices, deformation of the Frenkel–Mukhin's algorithm.

The intersection cohomology of Schubert varieties is a combinatorial invariant

We give an explicit and entirely poset-theoretic way to compute, for any permutation v, all the Kazhdan–Lusztig polynomials P x, y for x, y≤ v, starting from the Bruhat interval [ e, v] as an abstract poset. This proves, in particular, that the intersection cohomology of Schubert varieties depends only on the inclusion relations between the closures of its Schubert cells.

Algebraic approach to q, t-characters

Frenkel and Reshetikhin (in: Recent Developments in Quantum Affine Algebras and Related Topics, Contemporary Mathematics, Vol. 248, 1999, pp. 163–205) introduced q-characters to study finite dimensional representations of the quantum affine algebra U q( g ̂ ) . In the simply laced case Nakajima (in: Physics and Combinatorics, Proceedings of the Nagoya 2000 International Workshop, World Scientific, Singapore, 2001, pp. 181–212; Preprint arXiv:math.QA/0105173) defined deformations of q-characters called q, t-characters. The definition is combinatorial but the proof of the existence uses the geometric theory of quiver varieties which holds only in the simply laced case. In this article we propose an algebraic general (non-necessarily simply laced) new approach to q, t-characters motivated by the deformed screening operators (Internat. Math. Res. Not. 2003 (8) (2003) 451). The t-deformations are naturally deduced from the structure of U q( g ̂ ) : the parameter t is analog to the central charge c∈ U q( g ̂ ) . The q, t-characters lead to the construction of a quantization of the Grothendieck ring and to general analogues of Kazhdan–Lusztig polynomials in the same spirit as Nakajima did for the simply laced case.

Explicit enumeration of 321, hexagon-avoiding permutations

The 321, hexagon-avoiding ( 321-hex) permutations were introduced and studied by Billey and Warrington (J. Alg. Comb. 13 (2001) 111–136). as a class of elements of S n whose Kazhdan–Lusztig polynomials and the singular loci of whose Schubert varieties have certain fairly simple and explicit descriptions. This paper provides a 7-term linear recurrence relation leading to an explicit enumeration of the 321-hex permutations. A complete description of the corresponding generating tree is obtained as a by-product of enumeration techniques used in the paper, including Schensted's 321-subsequences decomposition, a 5-parameter generating function and the symmetries of the octagonal patterns avoided by the 321-hex permutations.

A formula for certain inverse Kazhdan–Lusztig polynomials in Sn

Let w 0 denote the permutation [ n, n−1,…,2,1]. We give a new explicit formula for the Kazhdan–Lusztig polynomials P w 0 w, w 0 x in S n when x indexes an irreducible component of the singular locus of the Schubert variety X w . To do this, we utilize a standard identity that relates P x, w and P w 0 w, w 0 x .

On certain distinguished involutions in the Weyl group of type Dn

Let ( W, S) be a Weyl group. Let A= Z[u,u −1] be the Laurent polynomial ring in an indeterminate u. Kazhdan and Lusztig [Invent. Math. 33 (1979) 165–184] introduced two A -bases { T w } w∈ W and { C w } w∈ W for the Hecke algebra H associated to W. Let Y w =∑ y⩽ w u l( w)− l( y) T y . Then { Y w } w∈ W is also an A -basis for the Hecke algebra. In this paper we assume W of type D n and we express certain Kazhdan–Lusztig basis elements C w as A -linear combination of Y x 's. This in turn gives an explicit expression for certain Kazhdan–Lusztig basis elements C w as A -linear combination of T x 's. Thus we describe explicitly the Kazhdan–Lusztig polynomials for certain pairs of elements of W. We also study the joint relation among some elements in W. In particular, we find certain distinguished involutions in the two-sided cell Ω t of W with a-value 1 2 (n 2−n+4t 2−2t) for 1⩽2 t⩽ n and n even ( 1 2 (n 2−n+4t 2+2t) for n odd), where the two-sided cell Ω t does not contain the longest element ( w 0) J in subgroup W J of W for any J⊂ S.

Kazhdan–Lusztig polynomials of the central extension of the elliptic Hecke algebras

Kazhdan–Lusztig polynomials of the central extension of the elliptic Hecke algebras

Diagram algebras, Hecke algebras and decomposition numbers at roots of unity

We prove that the cell modules of the affine Temperley–Lieb algebra have the same composition factors, when regarded as modules for the affine Hecke algebra of type A, as certain standard modules which are defined homologically. En route, we relate these to the cell modules of the Temperley–Lieb algebra of type B, which provides a connection between Temperley–Lieb algebras on n and n−1 strings. Applications include the explicit determination of some decomposition numbers of standard modules at roots of unity, which in turn has implications for certain Kazhdan–Lusztig polynomials associated with nilpotent orbit closures. The methods involve the study of the relationships among several algebras defined by concatenation of braid-like diagrams and between these and Hecke algebras. Connections are made with earlier work of Bernstein–Zelevinsky on the “generic case” and of Jones on link invariants.

The distinguished involutions with a-value n2−3 n+3 in the Weyl group of type Dn

Let ( W, S) be a Weyl group and H its associated Hecke algebra. Let A= Z[u,u −1] be the Laurent polynomial ring. Kazhdan and Lusztig [Representation of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979) 165–184] introduced two A -bases { T w } w∈ W and { C w } w∈ W for the Hecke algebra H associated to W. Let Y w =∑ y⩽ w u l( w)− l( y) T y . Then { Y w } w∈ W is also an A -base for the Hecke algebra. In this paper we give an explicit expression for certain Kazhdan–Lusztig basis elements C w as A -linear combination of Y x 's in the Hecke algebra of type D n . In fact, this gives also an explicit expression for certain Kazhdan–Lusztig basis elements C w as A -linear combination of T x 's in the Hecke algebra of type D n . Thus we describe also explicitly the Kazhdan–Lusztig polynomials for certain elements of the Weyl group. We study also the joint relation among some elements in W and some distinguished involutions with a-value n 2−3 n+3 in the Weyl group of type D n .

Kazhdan–Lusztig polynomials and character formulae for the Lie superalgebra [formula omitted

We compute the characters of the finite dimensional irreducible representations of the Lie superalgebra q(n) and formulate a precise conjecture for the other irreducible characters in category O . The guiding principle is that the representation theory of q(n) should be described by Lusztig's canonical basis for “tensor space” over the quantum group of type b ∞ .

On the combinatorics of B× B-orbits on group compactifications

It is shown that there is an order isomorphism ϕ′ from the poset V of B× B-orbits on the wonderful compactification of a semi-simple adjoint group G with Weyl group W to an interval in reverse Chevalley–Bruhat order on a non-canonically associated Coxeter group W (in general neither finite nor affine). Moreover, ϕ′ preserves the corresponding Kazhdan–Lusztig polynomials. Springer's (partly conjectural) construction of Kazhdan–Lusztig polynomials for the analogues of V for general Coxeter groups W is completed by reducing it by a similar order isomorphism to known results involving a “twisted” Chevalley–Bruhat order on W .

Rigidity of Schubert closures and invariance of Kazhdan–Lusztig polynomials

Let [ e, w], [ e, w′] be intervals from the origin in two Coxeter groups W, W′. We prove that any poset isomorphism ϕ : [e,w]→[e,w′] preserves Kazhdan–Lusztig polynomials, in the sense that P ϕ( x), ϕ( y) = P x, y for any x⩽ y in [ e, w], in the case where one of the two groups W, W′ has the property that the Coxeter graph of each of its irreducible constituents is either a tree, or affine of type A ̃ n ; in particular, the result holds for all finite or affine Coxeter groups. We obtain this result as a consequence of an explicit description of all the poset isomorphisms from [ e, w] to [ e, w′].

P-kernels, IC bases and Kazhdan–Lusztig polynomials

In [J. Amer. Math. Soc. 5 (1992) 805–851] Stanley introduced the concept of a P-kernel for any locally finite partially ordered set P. In [Proc. Sympos. Pure Math., Vol. 56, AMS, 1994, pp. 135–148] Du introduced, for any set P, the concept of an IC basis. The purpose of this article is to show that, under some mild hypotheses, these two concepts are equivalent, and to characterize, for a given Coxeter group W, partially ordered by Bruhat order, the W-kernel corresponding to the Kazhdan–Lusztig basis of the Hecke algebra of W. Finally, we show that this W-kernel factorizes as a product of other W-kernels, and that these provide a solution to the Yang–Baxter equations for W.

Enumerative and Combinatorial Properties of Dyck Partitions

The purpose of this paper is to study the combinatorial and enumerative properties of a new class of (skew) integer partitions. This class is closely related to Dyck paths and plays a fundamental role in the computation of certain Kazhdan–Lusztig polynomials of the symmetric group related to Young's lattice. As a consequence of our results, we obtain some new identities for these polynomials.