Special isogenies and tensor product multiplicities

TL;DR

This paper explores how angle-preserving bijections between root systems induce inequalities in tensor product multiplicities for complex semisimple Lie groups, using combinatorial and geometric methods.

Contribution

It introduces a novel connection between root system bijections and tensor product multiplicities, expanding understanding of Lie group representations.

Findings

Inequalities relate tensor product multiplicities under angle-preserving root system bijections.

Uses Littelmann's Path Model to explain inequalities combinatorially.

Employs isogenies between algebraic groups over positive characteristic fields for geometric insights.

Abstract

We show that any bijection between two root systems that preserves angles (but not necessarily lengths) gives rise to inequalities relating tensor product multiplicities for the corresponding complex semisimple Lie groups (or Lie algebras). We explain the inequalities in two ways: combinatorially, using Littelmann's Path Model, and geometrically, using isogenies between algebaric groups defined over an algebraically closed field of positive characteristic.