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Stabilization phenomena in Kac-Moody algebras and quiver varieties

Ben Webster

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_\ell by Benkart, et al.

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 . 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.

Paper Structure

This paper contains 13 sections, 14 theorems, 25 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Summary of results
  4. Deep weights
  5. The case of Viswanath
  6. Crystal structures on quiver varieties
  7. Quivers and their representations
  8. Quiver varieties as moduli spaces
  9. Connections with Kac-Moody algebras
  10. Stabilization of quiver varieties
  11. Notation
  12. Stabilization
  13. The polynomial behavior of weight multiplicities

Key Result

Theorem 1.1

(Kleber, Viswanath) Let $\lambda,\mu,\nu$ be a deep triple of weights (as defined in Section sec:deep-weights) on $X(m)$ supported on the head and end of tail as described above. Then the tensor product multiplicity $c^\lambda_{\mu,\nu}(m)$ for the Kac-Moody algebra $\mathfrak{g}(m)=\mathfrak{g}(X(m

Figures (4)

  • Figure 1: The effect of expansion on a Dynkin diagram
  • Figure 2: The Dynkin diagram $B_n$
  • Figure 3: The addition of an edge
  • Figure 4: The structure of a quiver representation on $X_0^s$.

Theorems & Definitions (19)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Proposition 2.1
  • Definition
  • Proposition 3.1
  • proof
  • Theorem 3.2
  • Proposition 3.3
  • Theorem 4.1
  • ...and 9 more