Fusion bases as facets of polytopes

TL;DR

This paper introduces a polytope-based method to construct fusion bases from elementary couplings, revealing symmetry properties and linear relations that influence the inequalities governing fusion rules in affine Lie algebras.

Contribution

It presents a novel polytope reinterpretation of fusion bases and analyzes the symmetry groups associated with different affine Lie algebras, uncovering new symmetries and relations.

Findings

Symmetry group order varies significantly among algebras.

Fusion basis inequalities can be generated from few elementary inequalities.

New symmetries of fusion coefficients are identified for b(3).

Abstract

A new way of constructing fusion bases (i.e., the set of inequalities governing fusion rules) out of fusion elementary couplings is presented. It relies on a polytope reinterpretation of the problem: the elementary couplings are associated to the vertices of the polytope while the inequalities defining the fusion basis are the facets. The symmetry group of the polytope associated to the lowest rank affine Lie algebras is found; it has order 24 for $\su(2)$, 432 for $\su(3)$ and quite surprisingly, it reduces to 36 for $\su(4)$, while it is only of order 4 for $\sp(4)$. This drastic reduction in the order of the symmetry group as the algebra gets more complicated is rooted in the presence of many linear relations between the elementary couplings that break most of the potential symmetries. For $\su(2)$ and $\su(3)$, it is shown that the fusion-basis defining inequalities can be generated from few (1 and 2 respectively) elementary ones. For $\su(3)$, new symmetries of the fusion coefficients are found.