Schubert puzzles and integrability III: separated descents

TL;DR

This paper introduces positive combinatorial formulas for new classes of Schubert polynomial products involving separated and almost separated descents, extending to K-theory and equivariant K-theory, with connections to quantum integrability.

Contribution

It provides the first positive formulas for Schubert products with separated and almost separated descents, generalizing previous cases and linking to quantum algebra.

Findings

Formulas extend to K-theory and equivariant K-theory.

Fusion of minuscule quantum loop algebra representations underpins the formulas.

New classes of Schubert product problems are solved with positive combinatorial formulas.

Abstract

In paper I of this series we gave positive formulae for expanding the product $\mathfrak S^\pi \mathfrak S^\rho$ of two Schubert polynomials, in the case that both $\pi,\rho$ had shared descent set of size $\leq 3$. Here we introduce and give positive formulae for two new classes of Schubert product problems: separated descent in which $\pi$'s last descent occurs at (or before) $\rho$'s first, and almost separated descent in which $\pi$'s last two descents occur at (or before) $\rho$'s first two respectively. In both cases our puzzle formulae extend to $K$-theory (multiplying Grothendieck polynomials), and in the separated descent case, to equivariant $K$-theory. The two formulae arise (via quantum integrability) from fusion of minuscule quantized loop algebra representations in types $A$, $D$ respectively.