Monomial Crystals and Partition Crystals

Recently Fayers introduced a large family of combinatorial realizations of the fundamental crystal $B(\Lambda_0)$ for $\widehat{\mathfrak{sl}}_\ell$, where the vertices are indexed by certain partitions. He showed that special cases of this construction agree with the Misra–Miwa realization and with Berg's ladder crystal. Here we show that another special case is naturally isomorphic to a realization using Nakajima's monomial crystal.