su(N) tensor product multiplicities and virtual Berenstein-Zelevinsky triangles

TL;DR

This paper generalizes Berenstein-Zelevinsky triangles for su(N) tensor products by allowing negative entries, leading to a polytope-based formula for multiplicities and criteria for non-vanishing products.

Contribution

It introduces virtual triangles into BZ triangles, derives an explicit sum formula for tensor multiplicities, and provides inequalities to determine when these multiplicities are non-zero.

Findings

Explicit multiple sum formula for tensor product multiplicities

Polytope description of virtual BZ triangles

Criteria for non-vanishing tensor products

Abstract

Information on su(N) tensor product multiplicities is neatly encoded in Berenstein-Zelevinsky triangles. Here we study a generalisation of these triangles by allowing negative as well as non-negative integer entries. For a fixed triple product of weights, these generalised Berenstein-Zelevinsky triangles span a lattice in which one may move by adding integer linear combinations of so-called virtual triangles. Inequalities satisfied by the coefficients of the virtual triangles describe a polytope. The tensor product multiplicities may be computed as the number of integer points in this convex polytope. As our main result, we present an explicit formula for this discretised volume as a multiple sum. As an application, we also address the problem of determining when a tensor product multiplicity is non-vanishing. The solution is represented by a set of inequalities in the Dynkin labels. We also allude to the question of when a tensor product multiplicity is greater than a given non-negative integer.