A cohomology free description of eigencones in type A, B and C

TL;DR

This paper provides a new cohomology-free method to describe the eigencones of certain Lie groups, simplifying the characterization of these geometric objects in types A, B, and C.

Contribution

It introduces an inductive, cohomology-free approach to determine the inequalities defining eigencones for simple Lie groups of types A, B, and C.

Findings

Provides minimal linear inequalities for eigencones

Simplifies previous cohomology-based descriptions

Applicable to types A, B, and C Lie groups

Abstract

Let $K$ be a connected compact Lie group. The triples $(O_1,\,O_2,\,O_3)$ of adjoint $K$-orbits such that $O_1+O_2+O_3$ contains $0$ are parametrized by a closed convex polyhedral cone called the eigencone of $K$. For $K$ simple of type $A$, $B$ or $C$ we give an inductive cohomology free description of the minimal set of linear inequalities which characterizes the eigencone of $K$.