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 Deodharʼs inequality in a connection with a sum of R-polynomials and edges of Bruhat graphs.