The ladder crystal

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.