Stabilization phenomena in Kac-Moody algebras and quiver varieties

Abstract

Let X be the Dynkin diagram of a symmetrizable Kac-Moody algebra, and X 0 a subgraph with all vertices of degree 1 or 2. Using the crystal structure on the components of quiver varieties for X, we show that if we expand X by extending X 0 , the branching multiplicities and tensor product multiplicities stabilize, provided the weights involved satisfy a condition which we call "depth" and are supported outside X 0 . This extends a theorem of Kleber and Viswanath.Furthermore, we show that the weight multiplicities of such representations are polynomial in the length of X 0 , generalizing the same result for A ℓ by Benkart, et al.

Dynkin Diagram Sequences and Stabilization Phenomena

“…The results and formulation in [9] and in section 2 of the present article overlap substantially. Webster's approach also proves a 'polynomiality of weight multiplicities' result for these kinds of Dynkin diagram sequences.…”

“…As mentioned in the introduction, Webster [9] has recently proved more general versions of the stabilization results of the present article using quiver varieties and their connections to representation theory. He has also proved the polynomiality of weight multiplicities for more general Z k 's, thereby extending the result of [2].…”

“…He has also proved the polynomiality of weight multiplicities for more general Z k 's, thereby extending the result of [2]. In fact [9] uses the result of [2] for the A k to prove it for all Z k 's. However the method originally used in [2] is very different in flavor than that in [9].…”

“…In fact [9] uses the result of [2] for the A k to prove it for all Z k 's. However the method originally used in [2] is very different in flavor than that in [9].…”

“…While this article was in preparation, Webster [9] has shown that a more general version of our stabilization result (Theorem 1) can be proved using quiver varieties and their connections with representations of Kac-Moody algebras. The results and formulation in [9] and in section 2 of the present article overlap substantially. Webster's approach also proves a 'polynomiality of weight multiplicities' result for these kinds of Dynkin diagram sequences.…”