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

الأصل · EN

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.