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

Abstract

The rigged configuration realization RC(infinity) of the crystal B(infinity) 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(infinity) explicitly for the A(n)-type case. Two separate descriptions of RC(infinity) are obtained. These lead naturally to isomorphisms RC(infinity) congruent to T(infinity) and RC(infinity) congruent to (T) over bar(infinity), i.e., those with the marginally large tableau and marginally large reverse tableau realizations of B(infinity), 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(infinity), the two mentioned isomorphisms, and two existing realizations of B(lambda) that were based on T(infinity) and (T) over bar(infinity).