Some results on the crystal commutor and affine sl(n) crystals

TL;DR

This paper explores the crystal commutor's relationship with the R-matrix and introduces new models for affine sl(n) crystals, leading to insights into their structure and connections to combinatorics and representation theory.

Contribution

It provides a new definition of the crystal commutor applicable to symmetrizable Kac-Moody algebras and develops combinatorial models for affine sl(n) crystals, including a basis for irreducible representations.

Findings

New definition of crystal commutor for symmetrizable Kac-Moody algebras

Combinatorial models for affine sl(n) crystals using partitions and abacus configurations

Partition function calculation for cylindric plane partitions

Abstract

There are two parts to this work, which are largely independent. The first consists of a series of results concerning the crystal commutor of Henriques and Kamnitzer. We first describe the relationship between the crystal commutor and Drinfeld's unitarized R-matrix. We then give a new definition for the crystal commutor, which makes sense for any symmetrizable Kac-Moody algebra. We show that this new definition agrees with A. Henriques and J. Kamnitzer's definition in the finite type case, but we cannot prove our commutor remains a coboundary structure in other cases. Next, we extend these ideas to give a new formula for the standard R-matrix. In the second part, we define three combinatorial models for affine sl(n) crystals. These are parameterized by partitions, configurations of beads on an "abacus", and cylindric plane partitions, respectively. Our models are reducible, but we can identify an irreducible subcrystal corresponding to any dominant integral highest weight. Cylindric plane partitions actually parameterize a basis for an irreducible representation of affine gl(n). This allows us to calculate the partition function for a system of random cylindric plane partitions first studied by A. Borodin. We also observe a form of rank-level duality, originally due to I. Frenkel.