Twisted Schubert polynomials.

We prove that twisted versions of Schubert polynomials defined by S ~ w 0 = x 1 n - 1 x 2 n - 2 ⋯ x n - 1 and S ~ w s i = (s i + ∂ i) 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 S ~ w as well as their localizations. [ABSTRACT FROM AUTHOR]