About the Bohr-Sommerfeld polytope and the multiplicative $\operatorname{SU}(2)$-eigenvalue problem for the trinion

TL;DR

This paper explores the geometric structure of the Bohr-Sommerfeld polytope and its relation to the SU(2)-multiplicative eigenvalue problem for three conjugacy classes, building on Jeffrey and Weitsman's foundational work.

Contribution

It provides a detailed description of how inequalities defining the Bohr-Sommerfeld polytope lead to solutions of the SU(2) eigenvalue problem for three conjugacy classes.

Findings

Characterization of the Bohr-Sommerfeld polytope inequalities.

Connection between the polytope and the eigenvalue problem.

Extension of Jeffrey and Weitsman's results to specific cases.

Abstract

We study in detail the work of Jeffrey and Weitsman, especially \cite{Jeffrey:Weitsman:1992} and \cite{Jeffrey:Weitsman:1994}. In particular, we describe the manner in which the inequalities that cut out the Bohr-Sommerfeld moment polytope give rise to a solution of the $\operatorname{SU}(2)$-multiplicative eigenvalue problem for the case of three conjugacy classes.