A Molev-Sagan type formula for double Schubert polynomials

Abstract: We give a Molev-Sagan type formula for computing the product $\mathfrak{S}u(x;y)\mathfrak{S}_v(x;z)$ 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 $\mathfrak{S}_u(x;y)$ by a factorial elementary symmetric polynomial $E{p,k}(x;z)$. 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 $s_{\lambda}(x_1,\ldots,x_m;y)$ by a double Schubert polynomial $\mathfrak{S}_v(x_1,\ldots,x_p;y)$ such that $m\geq p$. 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.