A Molev-Sagan type formula for double Schubert polynomials

Abstract

We give a Molev-Sagan type formula for computing the product Su(x;y)Sv(x;z) of two double Schubert polynomials in different sets of coefficient variables where the descents of u and v satisfy certain conditions that encompass Molev and Sagan's original case and conjecture positivity in the general case. Additionally, we provide a Pieri formula for multiplying an arbitrary double Schubert polynomial Su(x;y) by a factorial elementary symmetric polynomial Ep,k(x;z). Both formulas remain positive in terms of the negative roots when we set y=z, so in particular this gives a new equivariant Littlewood-Richardson rule for the Grassmannian, and more generally a positive formula for multiplying a factorial Schur polynomial sλ(x1,…,xm;y) by a double Schubert polynomial Sv(x1,…,xp;y) such that m≥p. An additional new result we present is a combinatorial proof of a conjecture of Kirillov of nonnegativity of the coefficients of skew Schubert polynomials, and we conjecture a weight-preserving bijection between a modification of certain diagrams used in our formulas and RC-graphs/pipe dreams arising in formulas for double Schubert polynomials.