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 $\alpha \in R$, $v,w\in W$, 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.

arXiv comment47 pages, 1 appendix