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

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.