Twisted Schubert polynomials

Abstract

We prove that twisted versions of Schubert polynomials defined by $$\widetilde{\mathfrak S}_{w_0} = x_1^{n-1}x_2^{n-2} \cdots x_{n-1}$$ and $$\widetilde{\mathfrak S}_{ws_i} = (s_i+\partial _i)\widetilde{\mathfrak S}_w$$ 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 $$\widetilde{\mathfrak S}_w$$ as well as their localizations.