The ladder crystal

TL;DR

This paper introduces a new combinatorial model for the crystal $B( ext{Lambda}_0)$ of affine Lie algebra $ ext{sl}_ ext{hat}_ ext{ell}$, demonstrating its equivalence to the Misra-Miwa model through crystal isomorphism.

Contribution

It presents a novel description of the crystal $B( ext{Lambda}_0)$ and establishes its equivalence to the Misra-Miwa model via regularization as a crystal isomorphism.

Findings

New combinatorial model for $B( ext{Lambda}_0)$

Equivalence between the new model and Misra-Miwa model

Interpretation of regularization as a crystal isomorphism

Abstract

n this paper I introduce a new description of the crystal $B(\Lambda_0)$ of $\hat{\mathfrak{sl}_\ell}$. As in the Misra-Miwa model of $B(\Lambda_0)$, the nodes of this crystal are indexed by partitions and the $i$-arrows correspond to adding a box of residue $i$. I then show that the two models are equivalent by interpreting the operation of regularization introduced by James as a crystal isomorphism.