Stabilization phenomena in Kac-Moody algebras and quiver varieties

Details Stabilization phenomena in Kac-Moody algebras and quiver varieties Stabilization phenomena in Kac-Moody algebras and quiver varieties

2005-05-30

arxiv.org/abs/math/0505619v3

International Mathematics Research Notices, vol. 2006, Article ID 36856

Mathematics

Representation Theory

final version, to appear in International Math Research Notices. 17 pages, 4 figures

Scientific paper

10.1155/IMRN/2006/36856

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_\ell by Benkart, et al.

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