Twisted Schubert polynomials

Abstract

We prove that twisted versions of Schubert polynomials defined by ${\tilde{\mathfrak{S}}}_{w_0}=x_1^{n-1}{x}_2^{n-2}\cdots x_{n-1}$ and ${\tilde{\mathfrak{S}}}_{w{s}_i}=(s_i+{\partial }_i){\tilde{\mathfrak{S}}}_w$ are monomial positive and give a combinatorial formula for their coefficients. In doing so, we reprove and extend a previous result about positivity of skew divided difference operators and show how it implies the Pieri rule for Schubert polynomials. We also give positive formulas for double versions of the ${\tilde{\mathfrak{S}}}_w$ as well as their localizations.

19 pages