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

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).