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 limit n → ∞. Motivated by the classical theory of A n = sl n+1 C, 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.

Dynkin Diagram Sequences and Stabilization Phenomena

“…These were notable exceptions to the extensibility condition of [4]. So, while nice stabilization results hold for the A n , nothing much could be said about these other classical types.…”

“…It was also shown in [4] that one could use the stable values of the c ν λµ (k) to define an operation * , which mimics the limit as k → ∞ of the tensor product. A very surprising fact discovered there was the associativity of * .…”

“…where X 1 and X 2 are fixed Dynkin diagrams and the string of intermediate nodes has length k. The article [4] considered the Dynkin diagram sequences Z k arising in the special case when X 2 = A 1 (the diagram with just a single node).…”

“…First, we recall that the goal of the previous article [4] was slightly different; it sought to find conditions on the diagram X 1 (X 2 = A 1 there) which would ensure stabilization of c ν λµ (k) for all weights λ, µ, ν. The condition on X 1 which made this work was called extensibility.…”

“…In [4], the primary object of interest was the multiplicity c ν λµ (k) of the irreducible highest weight representation L(ν (k) ) of…”