Vertices in multiplicative eigenvalue problem for arbitrary groups

TL;DR

This paper characterizes the vertices of a specific polytope related to conjugacy classes in maximal compact subgroups of simple complex algebraic groups, extending previous work and introducing new geometric methods.

Contribution

It provides an inductive framework for determining polytope vertices and generalizes Fulton's conjecture to all groups using Kontsevich compactifications.

Findings

Vertices of the polytope P(s,K) are explicitly characterized.

A quantum generalization of Fulton's conjecture is established.

Kontsevich compactifications are used to replace quot schemes in type A.

Abstract

We determine, in an inductive framework, the vertices of the polytope $P(s,K)$ controlling the conjugacy classes of elements which product to one in the maximal compact subgroup $K$ of a simple complex algebraic group $G$. This extends earlier work of the authors in related contexts. One feature of this work is the use of Kontsevich compactifications of the moduli of $P$-bundles (replacing the use of quot schemes in type A) which are related to semi-infinite geometry. We also obtain a quantum generalization of Fulton's conjecture valid for all $G$.