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 $s_λ(x_1,\ldots,x_m;y)$ 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.

Theorems & Definitions (71)

• Conjecture 1.1
• Definition 2.1: The symmetric group
• Definition 2.2: Double Schubert polynomials, $\partial^w$, $\partial_u^w$
• Definition 2.3: Factorial Schur polynomials, Grassmannian permutations
• Definition 2.4: The code $\mathfrak{c}(v)$, dominant permutations, the dominant approximation $\mu_{v}$, and the partition $\lambda(v)$
• Example 2.1
• Definition 2.5: $\xrightarrow{k}$, $P_k(u,w)$, $\mathrm{Path}_\lambda(u,w)$, $\mathrm{weight}_{P,\lambda}(y;z)$
• Example 2.2
• Definition 3.1: $d_{u,\lambda}^w(y;z)$, $e_{uv}^w(y;z)$
• Theorem 3.1