Littlewood--Richardson coefficients and integrable tilings

TL;DR

This paper offers direct combinatorial proofs for product and coproduct formulas of Schur functions using puzzle counting, leveraging quantum integrability of tiling models to unify and extend previous results.

Contribution

It introduces a new proof approach for Littlewood--Richardson coefficients via integrable tilings, connecting combinatorics with quantum integrability.

Findings

Provided direct proofs for product and coproduct formulas of Schur functions.

Extended formulas to include a second alphabet, recovering known results for factorial Schur functions.

Linked combinatorial tiling models with quantum integrability principles.

Abstract

We provide direct proofs of product and coproduct formulae for Schur functions where the coefficients (Littlewood--Richardson coefficients) are defined as counting puzzles. The product formula includes a second alphabet for the Schur functions, allowing in particular to recover formulae of [Molev--Sagan '99] and [Knutson--Tao '03] for factorial Schur functions. The method is based on the quantum integrability of the underlying tiling model.