A simple proof of associativity and commutativity of LR-coefficients (or the hive ring)

TL;DR

This paper presents a straightforward bijective proof demonstrating the associativity and commutativity of Littlewood-Richardson coefficients, using polarized polymatroidal discretely concave functions on a tetrahedron.

Contribution

It introduces a simple, constructive proof of key algebraic properties of LR-coefficients via geometric functions on tetrahedra.

Findings

Proof of associativity and commutativity of LR-coefficients

Existence of polarized polymatroidal discretely concave functions on tetrahedra

Establishment of boundary value conditions for these functions

Abstract

We give a simple bijective proof of associativity and commutativity of the Littlewood-Richardson coefficients or the hive ring. Specifically, we establish existence a polarized polymatroidal discretely concave functions on the tetrahedron with given boundary values at two adjoint faces.