Littlewood-Richardson coefficients for Grothendieck polynomials from integrability

TL;DR

This paper connects Littlewood-Richardson coefficients of double Grothendieck polynomials to integrable vertex models, deriving new product rules and expressing coefficients via generalized puzzles, advancing understanding in algebraic geometry and combinatorics.

Contribution

It introduces a novel integrability-based approach to compute Littlewood-Richardson coefficients for double Grothendieck polynomials, generalizing existing puzzle methods.

Findings

Derived product rules using Yang-Baxter equation

Expressed coefficients in terms of gash-free puzzles

Extended puzzle models beyond previous frameworks

Abstract

We study the Littlewood-Richardson coefficients of double Grothendieck polynomials indexed by Grassmannian permutations. Geometrically, these are the structure constants of the equivariant $K$-theory ring of Grassmannians. Representing the double Grothendieck polynomials as partition functions of an integrable vertex model, we use its Yang-Baxter equation to derive a series of product rules for the former polynomials and their duals. The Littlewood-Richardson coefficients that arise can all be expressed in terms of puzzles without gashes, which generalize previous puzzles obtained by Knutson-Tao and Vakil.