Rigged configuration descriptions of the crystals B(∞) and B(λ) for special linear Lie algebras

Abstract

The rigged configuration realization $RC(\infty)$ of the crystal $B(\infty)$ was originally presented as a certain connected component within a larger crystal. In this work, we make the realization more concrete by identifying the elements of $RC(\infty)$ explicitly for the $A_n$-type case. Two separate descriptions of $RC(\infty)$ are obtained. These lead naturally to isomorphisms $RC(\infty)\cong T(\infty)$ and $RC(\infty)\cong \bar{T}(\infty)$, i.e., those with the marginally large tableau and marginally large reverse tableau realizations of $B(\infty)$, that may be computed explicitly. We also present two descriptions of the irreducible highest weight crystal $B(\lambda)$ in terms of rigged configurations. These are obtained by combining our two descriptions of $RC(\infty)$, the two mentioned isomorphisms, and two existing realizations of $B(\lambda)$ that were based on $T(\infty)$ and $\bar{T}(\infty)$.