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

TL;DR

This paper establishes new inequalities on the coefficients of R- and Kazhdan-Lusztig polynomials for crystallographic Coxeter groups, linking combinatorial properties with geometric singularities of Bruhat intervals.

Contribution

It introduces three novel inequalities on polynomial coefficients, connecting combinatorial, algebraic, and geometric aspects of Coxeter groups and Bruhat graphs.

Findings

Nonnegativity of (q-1)-coefficients of R-polynomials

A new criterion for rational singularities of Bruhat intervals

Existence of a strict coefficientwise inequality of Kazhdan-Lusztig polynomials

Abstract

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.