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

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

Bruhat intervals as rooks on skew Ferrers boards

We characterise the permutations pi such that the elements in the closed lower Bruhat interval [id,pi] of the symmetric group correspond to non-taking rook configurations on a skew Ferrers board. It turns out that these are exactly the permutations pi such that [id,pi] corresponds to a flag manifold defined by inclusions, studied by Gasharov and Reiner. Our characterisation connects the Poincare polynomials (rank-generating function) of Bruhat intervals with q-rook polynomials, and we are able to compute the Poincare polynomial of some particularly interesting intervals in the finite Weyl groups A_n and B_n. The expressions involve q-Stirling numbers of the second kind. As a by-product of our method, we present a new Stirling number identity connected to both Bruhat intervals and the poly-Bernoulli numbers defined by Kaneko.Comment: 16 pages, 9 figure

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

The Number of Convex Permutominoes

Permutominoes are polyominoes defined by suitable pairs of permutations. In this paper we provide a formula to count the number of convex permutominoes of given perimeter. To this aim we define the transform of a generic pair of permutations, we characterize the transform of any pair defining a convex permutomino, and we solve the counting problem in the transformed space

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

Flipclasses and Combinatorial Invariance for Kazhdan--Lusztig polynomials

In this work, we investigate a novel approach to the Combinatorial Invariance Conjecture of Kazhdan--Lusztig polynomials for the symmetric group. Using the new concept of flipclasses, we introduce some combinatorial invariants of intervals in the symmetric group whose analysis leads us to a recipe to compute the coefficients of $q^h$ of the Kazhdan--Lusztig $\widetilde{R}$-polynomials, for $h\leq 6$. This recipe depends only on the isomorphism class (as a poset) of the interval indexing the polynomial and thus provides new evidence for the Combinatorial Invariance Conjecture.Comment: Comments are welcom

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