Positive expressions for skew divided difference operators

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.