Littlewood--Richardson rules from quivers for two-step flag varieties
Linda Chen, Elana Kalashnikov, Appendix by Ellis Buckminster, Linda Chen, Elana Kalashnikov
TL;DR
This work develops a quiver-encoded, positivity-preserving basis framework for the bi-graded algebra $\\Lambda_1\otimes\\Lambda_2$ to study the cohomology of two-step flag varieties. By introducing the Remmel--Whitney quiver and its row-restricted subquivers, the authors obtain explicit, tableau-based Littlewood–Richardson rules for products involving Schubert and Schur polynomials, and they show these rules remain positive through infusion and diffusion operations. A central result is that there is a surjective map $\\epsilon_{Fl}$ from $\\Lambda_1\otimes\\Lambda_2$ to the cohomology $H^*(Fl(n;r_1,r_2))$ that identifies the quiver-based basis elements with Schubert classes, yielding a Remmel--Whitney style rule for a large family of two-step flag products. The framework also provides a Pieri-type corollary and connects to alternative generator sets via Chern-root Schur polynomials, enabling translation between combinatorial tableau data and Schubert calculus. Overall, the paper offers a positive, tableau-driven approach to a broad class of Littlewood–Richardson coefficients in two-step flag varieties, with potential ties to puzzle and pipe-dream formalisms and extensions to broader flag varieties.
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.