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 ${\partial }_w(PQ)={\sum }_{v}v({\partial }_{w/v}P)\cdot {\partial }_{v}Q$ 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.

arXiv comment11 pages

References in corpus (1)

• Subalgebras of the Fomin-Kirillov algebra