Graded decompositions of fusion products in rank two

TL;DR

This paper explicitly determines the graded decompositions of fusion products for rank two Lie algebras, providing generators, relations, and confirming the Schur positivity conjecture in this setting.

Contribution

It offers a complete description of graded decompositions, generators, relations, and confirms Schur positivity for rank two Lie algebra fusion products.

Findings

Graded decompositions are parametrized by lattice points in convex polytopes.

Explicit hyperplane descriptions for these decompositions are provided.

Schur positivity conjecture holds for rank two cases.

Abstract

We determine the graded decompositions of fusion products of finite-dimensional irreducible representations for simple Lie algebras of rank two. Moreover, we give generators and relations for these representations and obtain as a consequence that the Schur positivity conjecture holds in this case. The graded Littlewood-Richardson coefficients in the decomposition are parametrized by lattice points in convex polytopes and an explicit hyperplane description is given in the various types.