Skew divided difference operators in the Nichols algebra associated to a finite Coxeter group

Abstract

Let (W,S) be a finite Coxeter system with root system R and with set of positive roots R+. For α∈R, v,w∈W, we denote by ∂α, ∂w and ∂w/v the divided difference operators and skew divided difference operators acting on the coinvariant algebra of W. Generalizing the work of Liu [15], we prove that ∂w/v can be written as a polynomial with nonnegative coefficients in ∂α where α∈R+. In fact, we prove the stronger and analogous statement in the Nichols–Woronowicz algebra model for Schubert calculus on W after Bazlov [4]. We draw consequences of this theorem on saturated chains in the Bruhat order, and partially treat the question when ∂w/v can be written as a monomial in ∂α where α∈R+. In an appendix, we study related combinatorics on shuffle elements and Bruhat intervals of length two.