Reduction Rules for Littlewood–Richardson Coefficients

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 $\bar G$ of smaller rank.