Monomial Crystals and Partition Crystals

TL;DR

This paper explores various combinatorial models of the fundamental crystal for affine sl(n), establishing isomorphisms between different realizations including partitions, ladder crystals, and Nakajima's monomial crystal.

Contribution

It demonstrates that a specific case of Fayers' partition-based crystal realization is naturally isomorphic to Nakajima's monomial crystal, unifying different models.

Findings

Special case of Fayers' realization matches Nakajima's monomial crystal

Connections between Misra-Miwa, ladder, and monomial crystal models established

Unification of multiple crystal realizations for affine sl(n)

Abstract

Recently Fayers introduced a large family of combinatorial realizations of the fundamental crystal for affine sl(n), 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.