On the quantum Horn problem

TL;DR

This paper refines the description of a convex polytope related to the multiplicative Horn problem for compact Lie groups, reducing the number of inequalities needed for its characterization.

Contribution

It provides a smaller, more efficient set of inequalities to describe the polytope associated with the multiplicative Horn problem, improving upon Teleman-Woodward's previous list.

Findings

Reduced the number of inequalities needed to describe the polytope.

Provided a more efficient characterization of the convex polytope.

Enhanced understanding of the multiplicative Horn problem for Lie groups.

Abstract

Let $K$ be a compact, connected, simply-connected simple Lie group. Given two conjugacy classes $\Orb_1$ and $\Orb_2$ in $K$, we consider the multiplicative Horn question: What conjugacy classes are contained in $\Orb_1\cdot\Orb_2$? It is known that answering this question remains to describe a convex polytope $\poly_K$. In 2003, Teleman-Woodward gave a complete list of inequalities for $\poly_K$. Their list contains redundant inequalities. In this paper, we describe $\poly_K$ by a smaller list of inequalities.