Combinatorial realizations of crystals via torus actions on quiver varieties

TL;DR

This paper develops a geometric framework using torus actions on quiver varieties to derive combinatorial models of crystals associated with Kac-Moody algebras, generalizing existing results and providing new insights.

Contribution

It introduces a novel method to extract combinatorial realizations of crystals from geometric quiver varieties via torus actions and Morse theory, extending known models.

Findings

Constructed torus actions on quiver varieties to index crystal components.

Reproduced and generalized Fayers' combinatorial realizations for fundamental representations.

Established connections with Nakajima's monomial crystal.

Abstract

Consider Kashiwara's crystal associated to a highest weight representation of a symmetric Kac-Moody algebra. There is a geometric realization of this object using Nakajima's quiver varieties, but in many particular cases it can also be realized by elementary combinatorial methods. Here we propose a framework for extracting combinatorial realizations from the geometric picture: We construct certain torus actions on the quiver varieties and use Morse theory to index the irreducible components by connected components of the subvariety of torus fixed points. We then discuss the case of affine sl(n). There the fixed point components are just points, and are naturally indexed by multi-partitions. There is some choice in our construction, leading to a family of combinatorial models for each highest weight crystal. Applying this construction to the crystal of the fundamental representation recovers a family of combinatorial realizations recently constructed by Fayers. This gives a more conceptual proof of Fayers' result as well as a generalization to higher level. We also discuss a relationship with Nakajima's monomial crystal.