Kazhdan Lusztig Research Articles

Pircon kernels and up-down symmetry

We show that a symmetry property that we call the up-down symmetry implies that the Kazhdan–Lusztig Rx-polynomials of a pircon P are a P-kernel, and we show that this property holds in the classical cases. Then, we enhance and extend to this context a duality of Deodhar in parabolic Kazhdan–Lusztig theory.

On a conjecture about cellular characters for the complex reflection group G(d,1,n)

We propose a conjecture relating two different sets of characters for the complex reflection group G(d,1,n). From one side, the characters are afforded by Calogero–Moser cells, a conjectural generalisation of Kazhdan–Lusztig cells for a complex reflection group. From the other side, the characters arise from a level d irreducible integrable representations of 𝒰 q (𝔰𝔩 ∞ ). We prove this conjecture in some cases: in full generality for G(d,1,2) and for generic parameters for G(d,1,n).

On ordered k-paths and rims for certain families of Kazhdan–Lusztig cells of Sn

For a composition [Formula: see text] of [Formula: see text] we consider the Kazhdan–Lusztig cell in the symmetric group [Formula: see text] containing the longest element of the standard parabolic subgroup of [Formula: see text] associated to [Formula: see text]. In this paper, we extend some of the ideas and results in [Beiträge zur Algebra und Geometrie, 59(3) (2018) 523–547]. In particular, by introducing the notion of an ordered [Formula: see text]-path, we are able to obtain alternative explicit descriptions for some additional families of cells associated to compositions. This is achieved by first determining the rim of the cell, from which reduced forms for all the elements of the cell are easily obtained.

The Berenstein–Kirillov group and cactus groups

Berenstein and Kirillov have studied the action of Bender–Knuth moves on semistandard tableaux. Losev has studied a cactus group action in Kazhdan–Lusztig theory; in type A this action can also be identified in the work of Henriques and Kamnitzer. We establish the relationship between the two actions. We show that the Berenstein–Kirillov group is a quotient of the cactus group. We use this to derive previously unknown relations in the Berenstein–Kirillov group.We also determine precise implications between subsets of relations in the two groups, which yields a presentation for cactus groups in terms of Bender–Knuth generators.

Blob algebra approach to modular representation theory

Two decades ago P. Martin and D. Woodcock made a surprising and prophetic link between statistical mechanics and representation theory. They observed that the decomposition numbers of the blob algebra (that appeared in the context of transfer matrix algebras) are Kazhdan-Lusztig polynomials in type $\tilde{A}_1$. In this paper we take that observation far beyond its original scope. We conjecture that for $\tilde{A}_n$ there is an equivalence of categories between the characteristic $p$ diagrammatic Hecke category and a "blob category" that we introduce (using certain quotients of KLR algebras called \emph{generalized blob algebras}). Using alcove geometry we prove the "graded degree" part of this equivalence for all $n$ and all prime numbers $p$. If our conjecture was verified, it would imply that the graded decomposition numbers of the generalized blob algebras in characteristic $p$ give the $p$-Kazhdan Lusztig polynomials in type $\tilde{A}_n$. We prove this for $\tilde{A}_1$, the only case where the $p$-Kazhdan Lusztig polynomials are known.

The ABC of p-cells

Parallel to the very rich theory of Kazhdan–Lusztig cells in characteristic 0, we try to build a similar theory in positive characteristic. We study cells with respect to the p-canonical basis of the Hecke algebra of a crystallographic Coxeter system (see Jensen and Williamson in Categorification and higher representation theory, American Mathematical Society, Providence, 2017). Our main technical tools are the star-operations introduced by Kazhdan–Lusztig (Invent. Math. 53(2):65–184, 1979) which have interesting numerical consequences for the p-canonical basis. As an application, we explicitly describe p-cells in finite type A (i.e. for symmetric groups) using the Robinson–Schensted correspondence. Moreover, we show that Kazhdan–Lusztig cells in finite types B and C decompose into p-cells for $$p > 2$$ p > 2 .

A note on specializations of Grothendieck polynomials

Buch and Rimányi proved a formula for a specialization of double Grothendieck polynomials based on the Yang–Baxter equation related to the degenerate Hecke algebra. A geometric proof was found by Yong and Woo by constructing a Gröbner basis for the Kazhdan–Lusztig ideals. In this note, we give an elementary proof for this formula by using only divided difference operators.

Kazhdan-Lusztig Polynomials of Matroids Under Deletion

We present a formula which relates the Kazhdan–Lusztig polynomial of a matroid $M$, as defined by Elias, Proudfoot and Wakefield, to the Kazhdan–Lusztig polynomials of the matroid obtained by deleting an element, and various contractions and localizations of $M$. We give a number of applications of our formula to Kazhdan–Lusztig polynomials of graphic matroids, including a simple formula for the Kazhdan–Lusztig polynomial of a parallel connection graph.

A Kazhdan-Lusztig Algorithm for Whittaker Modules

We study a category of Whittaker modules over a complex semisimple Lie algebra by realizing it as a category of twisted $\mathcal {D}$ -modules on the associated flag variety using Beilinson–Bernstein localization. The main result of this paper is the development of a geometric algorithm for computing the composition multiplicities of standard Whittaker modules. This algorithm establishes that these multiplicities are determined by a collection of polynomials we refer to as Whittaker Kazhdan–Lusztig polynomials. In the case of trivial nilpotent character, this algorithm specializes to the usual algorithm for computing multiplicities of composition factors of Verma modules using Kazhdan–Lusztig polynomials.

Type $A$ admissible cells are Kazhdan–Lusztig

Admissible W-graphs were defined and combinatorially characterised by Stembridge in reference [12]. The theory of admissible W-graphs was motivated by the need to construct W-graphs for Kazhdan-Lusztig cells, which play an important role in the representation theory of Hecke algebras, without computing Kazhdan-Lusztig polynomials. In this paper, we shall show that type A-admissible W-cells are Kazhdan-Lusztig as conjectured by Stembridge in his original paper.

Kazhdan–Lusztig representations and Whittaker space of some genuine representations

We prove a formula for the dimension of Whittaker functionals of irreducible constituents of a regular unramified genuine principal series for covering groups. The formula explicitly relates such dimension to the Kazhdan-Lusztig representations associated with certain right cells of the Weyl group. We also state a refined version of the formula, which is proved under some natural assumption. The refined formula is also verified unconditionally in several important cases.

A Combinatorial Formula for the Coefficient of q in Kazhdan–Lusztig Polynomials

Abstract We propose a combinatorial interpretation of the coefficient of $q$ in Kazhdan–Lusztig polynomials and we prove it for finite simply-laced Weyl groups.

Right-angled Coxeter groups, universal graphs, and Eulerian polynomials

We define a class of representations for any right-angled Coxeter group R and its Hecke algebra H(R). The group R and the algebra H(R) act on any Bruhat interval of any Coxeter system (W,S), once given a suitable function from the set of Coxeter generators of R to the power set of S. The existence of such a function is related to the problem of universality of a graph G2 constructed from the unlabeled Coxeter graph G of (W,S). When G=Pn is a path with n vertices, we conjecture that G2 is n-universal; this property is equivalent to the existence of an action of R and H(R) on the Bruhat intervals of the symmetric group Sn+1, for all right-angled groups with n generators. We prove that (Pn)2 is n-universal for forests. Eulerian polynomials arise as characters of our representations, when the Coxeter graph of R is a path. We also give a formula for the toric h-polynomial of any lower Bruhat interval in a universal Coxeter group Un, using results on the Kazhdan–Lusztig basis of H(Un).

Functorial invariants of trees and their cones

We study the category whose objects are trees (with or without roots) and whose morphisms are contractions. We show that the corresponding contravariant module categories are locally Noetherian, and we study two natural families of modules over these categories. The first takes a tree to a graded piece of the homology of its unordered configuration space, or to the homology of the unordered configuration space of its cone. The second takes a tree to a graded piece of the intersection homology of the reciprocal plane of its cone, which is a vector space whose dimension is given by a Kazhdan–Lusztig coefficient. We prove finite generation results for each of these modules, which allow us to draw conclusions about the growth of Betti numbers of configuration spaces and of Kazhdan–Lusztig coefficients of graphical matroids.

Kazhdan–Lusztig R-polynomials for pircons

The purpose of this work is to provide a common combinatorial framework for some of the analogues and generalizations of Kazhdan–Lusztig R-polynomials that have appeared since the introduction of these remarkable polynomials (e.g., parabolic Kazhdan–Lusztig R-polynomials, Kazhdan–Lusztig R-polynomials of zircons, and Kazhdan–Lusztig–Vogan polynomials for fixed point free involutions).

The normal closure of a power of a half-twist has infinite index in the mapping class group of a punctured sphere

In this paper we show that the normal closure of the [Formula: see text]th power of a half-twist has infinite index in the mapping class group of a punctured sphere if [Formula: see text] is at least five. Furthermore, in some cases we prove that the quotient of the mapping class group of the punctured sphere by the normal closure of a power of a half-twist contains a free abelian subgroup. As a corollary we prove that the quotient of the hyperelliptic mapping class group of a surface of genus at least two by the normal closure of the [Formula: see text]th power of a Dehn twist has infinite order, and for some integers [Formula: see text] the quotient contains a free group. As a second corollary we recover a result of Coxeter: the normal closure of the [Formula: see text]th power of a half-twist in the braid group of at least four strands has infinite index. Our method is to reformulate the Jones representation of the mapping class group of a punctured sphere, using the action of Hecke algebras on [Formula: see text]-graphs, as introduced by Kazhdan–Lusztig.

AN IDENTITY OF PARABOLIC KAZHDAN–LUSZTIG POLYNOMIALS ARISING FROM SQUARE-IRREDUCIBLE MODULES

AbstractWe show a precise formula, in the form of a monomial, for certain families of parabolic Kazhdan–Lusztig polynomials of the symmetric group. The proof stems from results of Lapid–Mínguez on irreducibility of products in the Bernstein–Zelevinski ring. By quantizing those results into a statement on quantum groups and their canonical bases, we obtain identities of coefficients of certain transition matrices that relate Kazhdan–Lusztig polynomials to their parabolic analogues. This affirms some basic cases of conjectures raised recently by Lapid.

Diagrammatics for Kazhdan–Lusztig [formula omitted]-polynomials

Let (W,S) be an arbitrary Coxeter system. We introduce a family of polynomials, {R̃u,v̲(t)}, indexed by pairs (u,v̲) formed by an element u∈W and a (non-necessarily reduced) word v̲ in the alphabet S. The polynomial R̃u,v̲(t) is obtained by considering a certain subset of Libedinsky’s light leaves associated to the pair (u,v̲). Given a reduced expression v̲ of an element v∈W, we show that R̃u,v̲(t) coincides with the Kazhdan–Lusztig R̃-polynomial R̃u,v(t). Using the diagrammatic approach, we obtain some closed formulas for R̃-polynomials.

Blob algebra and two-color Soergel calculus

In 2003, Martin and Woodcock noticed a connection between the representation theory of the blob algebra and the Kazhdan–Lusztig polynomials associated with the infinite dihedral group. However, no conceptual explanation for this coincidence has yet been provided. In this study, a possible explanation of this phenomenon is suggested by enunciating a conjecture that relates the endomorphism algebra of Bott–Samelson bimodules to certain subalgebras of the blob algebra obtained by idempotent truncation. Evidence supporting this conjecture is provided.

PARABOLIC KAZHDAN–LUSZTIG BASIS, SCHUBERT CLASSES, AND EQUIVARIANT ORIENTED COHOMOLOGY

We study the equivariant oriented cohomology ring $\mathtt{h}_{T}(G/P)$ of partial flag varieties using the moment map approach. We define the right Hecke action on this cohomology ring, and then prove that the respective Bott–Samelson classes in $\mathtt{h}_{T}(G/P)$ can be obtained by applying this action to the fundamental class of the identity point, hence generalizing previously known results of Chow groups by Brion, Knutson, Peterson, Tymoczko and others. Our main result concerns the equivariant oriented cohomology theory $\mathfrak{h}$ corresponding to the 2-parameter Todd genus. We give a new interpretation of Deodhar’s parabolic Kazhdan–Lusztig basis, i.e., we realize it as some cohomology classes (the parabolic Kazhdan–Lusztig (KL) Schubert classes) in $\mathfrak{h}_{T}(G/P)$. We make a positivity conjecture, and a conjecture about the relationship of such classes with smoothness of Schubert varieties. We also prove the latter in several special cases.