Reduction rules for Littlewood-Richardson coefficients

TL;DR

This paper introduces a geometric, type-independent method to derive reduction rules for Littlewood-Richardson coefficients, simplifying tensor product multiplicity problems for semisimple algebraic groups.

Contribution

It establishes a geometric approach linking regular faces of the Littlewood-Richardson cone to reduction rules applicable across all types, extending prior type-specific results.

Findings

Reduction rules correspond to regular faces of the Littlewood-Richardson cone.

The approach is geometric and independent of group type.

Simplifies tensor product multiplicity calculations.

Abstract

Let G be a semisimple algebraic group over an algebraically-closed field of characteristic zero. In this note we show that every regular face of the Littlewood-Richardson cone of G gives rise to a reduction rule: a rule which, given a problem "on that face" of computing the multiplicity of an irreducible component in a tensor product, reduces it to a similar problem on a group of smaller rank. In the type A case this result has already been proved by Derksen and Weyman using quivers, and by King, Tollu, and Tomazet using puzzles. The proof here is geometric and type-independent.