Schur times Schubert via the Fomin-Kirillov algebra

TL;DR

This paper develops combinatorial formulas for multiplying Schubert polynomials by Schur polynomials, proving nonnegativity of expansion coefficients and Gromov-Witten invariants using the Fomin-Kirillov algebra.

Contribution

It provides explicit combinatorial expressions and proofs for nonnegativity of certain Schubert polynomial expansions and Gromov-Witten invariants, extending to quantum cohomology.

Findings

Explicit formulas for special partitions like hooks and 2x2 boxes

Proof of nonnegativity of Gromov-Witten invariants in specific cases

Evaluation of Schubert polynomials via Fomin-Kirillov algebra

Abstract

We study multiplication of any Schubert polynomial $\mathfrak{S}_w$ by a Schur polynomial $s_\lambda$ (the Schubert polynomial of a Grassmannian permutation) and the expansion of this product in the ring of Schubert polynomials. We derive explicit nonnegative combinatorial expressions for the expansion coefficients for certain special partitions $\lambda$, including hooks and the 2x2 box. We also prove combinatorially the existence of such nonnegative expansion when the Young diagram of $\lambda$ is a hook plus a box at the (2,2) corner. We achieve this by evaluating Schubert polynomials at the Dunkl elements of the Fomin-Kirillov algebra and proving special cases of the nonnegativity conjecture of Fomin and Kirillov. This approach works in the more general setup of the (small) quantum cohomology ring of the complex flag manifold and the corresponding (3-point) Gromov-Witten invariants. We provide an algebro-combinatorial proof of the nonnegativity of the Gromov-Witten invariants in these cases, and present combinatorial expressions for these coefficients.