The Agnihotri--Woodward--Belkale Polytope and Klyachko Cones

S. Yu. Orevkov ; Yu. P. Orevkov

Matematičeskie zametki, Tome 87 (2010) no. 1, pp. 101-107

Résumé

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

Détail

Keywords: unitary matrix, Weyl chamber, Horn problem, Gromov–Witten invariants, Littlewood–Richardson coefficients
Mots-clés : conjugacy class, Schubert calculus, Klyachko cone.