Higher su(N) tensor products

TL;DR

This paper generalizes the calculation of tensor product multiplicities in su(N) to higher-point couplings, providing explicit formulas and inequalities for their existence, extending previous work on three-point couplings.

Contribution

It introduces a framework for computing higher su(N) tensor product multiplicities, including explicit sum formulas and criteria for non-vanishing couplings.

Findings

Explicit multiple sum formulas for higher-point multiplicities

Inequalities determining when higher-point couplings exist

Extension of convex polytope volume methods to higher couplings

Abstract

We extend our recent results on ordinary su(N) tensor product multiplicities to higher su(N) tensor products. Particular emphasis is put on four-point couplings where the tensor product of four highest weight modules is considered. The number of times the singlet occurs in the decomposition is the associated multiplicity. In this framework, ordinary tensor products correspond to three-point couplings. As in that case, the four-point multiplicity may be expressed explicitly as a multiple sum measuring the discretised volume of a convex polytope. This description extends to higher-point couplings as well. We also address the problem of determining when a higher-point coupling exists, i.e., when the associated multiplicity is non-vanishing. The solution is a set of inequalities in the Dynkin labels.