Positive expressions for skew divided difference operators

TL;DR

This paper proves that skew divided difference operators for permutations have positive expressions in terms of basic divided difference operators, confirming a conjecture and extending results to the Fomin-Kirillov algebra.

Contribution

It establishes positivity of skew divided difference operators in terms of elementary operators and confirms a conjecture in the Fomin-Kirillov algebra setting.

Findings

Proved positivity of skew divided difference operators.

Extended positivity results to the Fomin-Kirillov algebra.

Settled a conjecture of Kirillov.

Abstract

For permutations $v,w \in \mathfrak S_n$, Macdonald defines the skew divided difference operators $\partial_{w/v}$ as the unique linear operators satisfying $\partial_w(PQ) = \sum_v v(\partial_{w/v}P) \cdot \partial_vQ$ for all polynomials $P$ and $Q$. We prove that $\partial_{w/v}$ has a positive expression in terms of divided difference operators $\partial_{ij}$ for $i<j$. In fact, we prove that the analogous result holds in the Fomin-Kirillov algebra $\mathcal E_n$, which settles a conjecture of Kirillov.