Littlewood--Richardson rules from quivers for two-step flag varieties

TL;DR

This paper introduces quiver-based bases for symmetric function algebras to derive positive tableau formulas for Littlewood--Richardson coefficients in two-step flag varieties, extending classical rules like Pieri's.

Contribution

It provides a novel quiver-based approach to compute Littlewood--Richardson coefficients for two-step flag varieties, generalizing existing combinatorial rules.

Findings

Derived positive tableau formulas for Littlewood--Richardson coefficients

Extended classical Pieri rule to two-step flag varieties

Connected quiver representations with symmetric function bases

Abstract

Let $\bigwedge_1$ and $\bigwedge_2$ be two symmetric function algebras in independent sets of variables. We define vector space bases of $\bigwedge_1 \otimes_\mathbb{Z} \bigwedge_2$ coming from certain quivers, with vertex sets indexed by pairs of partitions. We use these vector space bases to give a positive tableau formula for Littlewood--Richardson coefficients for the product of Schubert polynomials with certain Schur polynomials in two-step flag varieties, in the spirit of the Remmel-Whitney rule for the product of two Schur polynomials in Grassmannians. This in particular covers the cases considered by the Pieri rule.