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

TL;DR

This paper proves that skew divided difference operators in the Nichols algebra associated with a finite Coxeter group can be expressed as polynomials with nonnegative coefficients in simple divided difference operators, extending previous work and exploring algebraic and combinatorial implications.

Contribution

It generalizes Liu's results by expressing skew divided difference operators as polynomials with nonnegative coefficients in the Nichols algebra setting for finite Coxeter groups.

Findings

Skew divided difference operators can be written as polynomials with nonnegative coefficients.

The paper extends these results to the Nichols-Woronowicz algebra model.

It explores implications for Bruhat order chains and related combinatorics.

Abstract

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