The combinatorial invariance conjecture for parabolic Kazhdan-Lusztig polynomials of lower intervals

The aim of this work is to prove a conjecture related to the Combinatorial Invariance Conjecture of Kazhdan-Lusztig polynomials, in the parabolic setting, for lower intervals in every arbitrary Coxeter group. This result improves and generalizes, among other results, the main results of [Advances in Math. {202} (2006), 555-601], [Trans. Amer. Math. Soc. {368} (2016), no. 7, 5247--5269].Comment: to appear in Advances in Mathematic

A simple characterization of special matchings in lower Bruhat intervals

We give a simple characterization of special matchings in lower Bruhat intervals (that is, intervals starting from the identity element) of a Coxeter group. As a byproduct, we obtain some results on the action of special matchings.Comment: accepted for publication on Discrete Mathematic

A simple characterization of special matchings in lower Bruhat intervals

We give a simple characterization of special matchings in lower Bruhat intervals (that is, intervals starting from the identity element) of a Coxeter group. As a byproduct, we obtain some results on the action of special matchings.Comment: accepted for publication on Discrete Mathematic

Inequalities on Bruhat graphs, R- and Kazhdan-Lusztig polynomials

From a combinatorial perspective, we establish three inequalities on coefficients of $R$- and Kazhdan-Lusztig polynomials for crystallographic Coxeter groups: (1) Nonnegativity of $(q-1)$-coefficients of $R$-polynomials, (2) a new criterion of rational singularities of Bruhat intervals by sum of quadratic coefficients of $R$-polynomials, (3) existence of a certain strict inequality (coefficientwise) of Kazhdan-Lusztig polynomials. Our main idea is to understand Deodhar's inequality in a connection with a sum of $R$-polynomials and edges of Bruhat graphs.Comment: 16 page

On the combinatorial invariance of Kazhdan–Lusztig polynomials

AbstractIn 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

Towards combinatorial invariance for Kazhdan-Lusztig polynomials

Kazhdan-Lusztig polynomials are important and mysterious objects in representation theory. Here we present a new formula for their computation for symmetric groups based on the Bruhat graph. Our approach suggests a solution to the combinatorial invariance conjecture for symmetric groups, a well-known conjecture formulated by Lusztig and Dyer in the 1980s.Comment: 47 pages, comments welcom

Path representation of maximal parabolic Kazhdan-Lusztig polynomials

We provide simple rules for the computation of Kazhdan--Lusztig polynomials in the maximal parabolic case. They are obtained by filling regions delimited by paths with "Dyck strips" obeying certain rules. We compare our results with those of Lascoux and Sch\"utzenberger.Comment: v3: fixed proof of lemma

Combinatorial invariance for elementary intervals

We adapt the hypercube decompositions introduced by Blundell-Buesing-Davies-Veli\v{c}kovi\'{c}-Williamson to prove the Combinatorial Invariance Conjecture for Kazhdan-Lusztig polynomials in the case of elementary intervals in $S_n$. This significantly generalizes the main previously-known case of the conjecture, that of lower intervals.Comment: 15 pages, comments welcom