On the faces of the tensor cone of symmetrizable Kac-Moody lie algebras

TL;DR

This paper investigates the structure of the tensor cone of symmetrizable Kac-Moody Lie algebras, proving that certain inequalities correspond to its faces, extending known results from finite-dimensional cases.

Contribution

It establishes that specific inequalities conjectured to describe the tensor cone indeed correspond to its codimension one faces in the Kac-Moody setting.

Findings

Identifies inequalities as faces of the tensor cone.

Extends finite-dimensional results to Kac-Moody algebras.

Provides a proof linking inequalities to cone faces.

Abstract

In this paper, we are interested in the decomposition of the tensor product of two representations ofa symmetrizable Kac-Moody Lie algebra ${\mathfrak g}$, or more precisely in the tensor cone of~${\mathfrak g}$.As usual, we parametrize the integrable, highest weight (irreducible) representations of~${\mathfrak g}$ by their highest weights. Then, the triples of such representations such that the last one is contained in the tensor product of the first two is a semigroup.This semigroup generates a rational convex cone $ \Gamma({\mathfrak g})$ called tensor cone.If ${\mathfrak g}$~is finite-dimensional, $\Gamma({\mathfrak g})$~is a polyhedral convex cone. In 2006, Belkale and the first author described this cone by an explicit finite list of inequalities.In 2010, this list of inequalities was proved to be irredundant by the second author:each such inequality corresponds to a codimension one face.In general, $\Gamma({\mathfrak g})$~is neither polyhedral, nor closed.Brown and the first author obtained a list of inequalities that describe $\Gamma({\mathfrak g})$ conjecturally. Here, we prove that each of these inequalities corresponds to a codimension one face of~$\Gamma({\mathfrak g})$.