Agnihotri-Woodward-Belkale polytope and the intersection of Klyachko cones

TL;DR

This paper investigates the Agnihotri-Woodward-Belkale polytope and Klyachko cones related to Horn's problem, revealing their intersection properties for matrices of size up to 14 and identifying a divergence at size 15.

Contribution

It provides computational evidence on the intersection of Klyachko cones under group actions, advancing understanding of the multiplicative Horn problem.

Findings

$ riangle$ equals the intersection of $gK$ for $g eq 15$

Discrepancy observed at $n=15$

Computational approach to Horn's problem

Abstract

Agnihotri-Woodward-Belkale polytope $\Delta$ (resp. Klyachko cone $K$) is the set of solutions of the multiplicative (resp. additive) Horn's problem, i.e., the set of triples of spectra of special unitary (resp. traceless Hermitian) $n\times n$ matrices satisfying $AB=C$ (resp. $A+B=C$). $K$ is the tangent cone of $\Delta$ at the origin. The group $G=\Bbb Z_n \oplus \Bbb Z_n$ acts naturally on $\Delta$. In this note, we report on a computer calculation which shows that $\Delta$ coincides with the intersection of $gK$, $g\in G$, for $n\le 14$ but does not coincide for $n=15$. Our motivation was an attempt to understand how to solve the multiplicative Horn problem in practice for given conjugacy classes in SU(n).