Tensor product stabilization in Kac-Moody algebras

Abstract

We consider a large class of series of symmetrizable Kac-Moody algebras (generically denoted X_n). This includes the classical series A_n as well as others like E_n whose members are of Indefinite type. The focus is to analyze the behavior of representations in the large n limit. Motivated by the classical theory of A_n, we consider tensor product decompositions of irreducible highest weight representations of X_n and study how these vary with n. The notion of ``double headed'' dominant weights is introduced. For such weights, we show that tensor product decompositions in X_n do stabilize, generalizing the classical results for A_n. The main tool used is Littelmann's celebrated path model. One can also use the stable multiplicities as structure constants to define a multiplication operation on a suitable space. We define this so called "stable representation ring'' and show that the multiplication operation is associative.