Schur times Schubert via the Fomin-Kirillov algebra

Karola Mészáros¹ ; Greta Panova² ; Alexander Postnikov³

¹Cornell University
²UCLA
³MIT

The electronic journal of combinatorics, Tome 21 (2014) no. 1

Résumé

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 $2\times 2$ 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.

Détail

DOI : 10.37236/3659

Classification : 05E05, 14N15
Mots-clés : Schubert polynomials, symmetric functions, Fomin-Kirillov algebra