We prove the relative hard Lefschetz theorem for Soergel bimodules. It follows that structure constants of Kazhdan–Lusztig basis are unimodal. explain why implies tensor category associated by Lusztig to any two-sided cell in a Coxeter group is rigid and pivotal.

These are notes for a talk on Kazhdan-Lusztig Cells for Hecke Algebras. In this talk, we construct the Kazhdan-Lusztig basis for the Hecke algebra associated to an arbitrary Coxeter group, in full multiparameter generality. We then use this basis to construct a partition of the Coxeter group into the Kazhdan-Lusztig cells and describe the corresponding cell representations. Finally, we speciali...

for a composition $lambda$ of $n$ our aim is to obtain reduced forms ‎for all the elements in the kazhdan-lusztig (right) cell containing ‎$w_{j(lambda)}$‎, ‎the longest element of the standard parabolic‎ ‎subgroup of $s_n$ corresponding to $lambda$‎. ‎we investigate how far this is possible to achieve by looking at‎ ‎elements of the form $w_{j(lambda)}d$‎, ‎where $d$ is a prefix of ‎an element...

We provide a combinatorial proof for the coincidence of Knuth equivalence classes, Kazhdan–Lusztig left cells and Vogan classes for the symmetric group, involving only Robinson-Schensted algorithm and the combinatorial part of the Kazhdan–Lusztig cell theory. The determination of Kazhdan–Lusztig cells for the symmetric group is given in the proof of [4, Thm1.4]. The argument is largely combinat...

Abstract The affine matrix-ball construction (abbreviated AMBC) was developed by Chmutov, Lewis, Pylyavskyy, and Yudovina as an generalization of the Robinson–Schensted correspondence. We show that AMBC gives a simple way to compute distinguished involution in each Kazhdan–Lusztig cell symmetric group. then use give 1st known canonical presentation for asymptotic Hecke algebras extended groups....

We study the nonnegative part G>0 of the De Concini-Procesi compactification of a semisimple algebraic group G, as defined by Lusztig. Using positivity properties of the canonical basis and parametrization of flag varieties, we will give an explicit description of G>0. This answers the question of Lusztig in Total positivity and canonical bases, Algebraic groups and Lie groups (ed. G.I. Lehrer)...

In their famous paper [6], Kazhdan and Lusztig introduced the concept of equivalence classes such as left cell, right cell and two-sided cell in a Coxeter group W . We inherit the notations 6 L , 6 R , 6 LR , ∼ L , ∼ R and ∼ LR in [6]. Thus w ∼ LR y (resp. w ∼ L y, resp. w ∼ R y) means that the elements w, y ∈ W are in the same two-sided cell (resp. left cell, resp. right cell) of W , etc. Conc...

We show that the totally nonnegative part of a partial flag variety $G/P$ (in sense Lusztig) is regular CW complex, confirming conjecture Williams. In particular, closure each positroid cell inside Grassmannian homeomorphic to ball, Postnikov.

In this paper we compute the leading coefficients μ(y,w) of the Kazhdan-Lusztig polynomials Py,w for an affineWeyl group of type B̃2. When a(y) ≤ a(w) or a(y) = 2 and a(w) = 1, we compute all μ(y,w) clearly, where a(y) is the a-function of a Coxeter group defined by Lusztig (see [L1]). With these values μ(y,w), we are able to show that a conjecture of Lusztig on distinguished involutions is true...

We give a criterion which determines when a union of one-dimensional Deligne-Lusztig varieties has a connected closure. We obtain a new, short proof of the connectedness criterion for Deligne-Lusztig varieties due to Lusztig.