Rigged Configuration Descriptions of the Crystals B(infinity) and B(lambda) for Special Linear Lie Algebras

The rigged configuration realization $RC(\mathrm{\infty })$ of the crystal $B(\mathrm{\infty })$ 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 $RC(\mathrm{\infty })$ explicitly for the $A_n$-type case. Two separate descriptions of $RC(\mathrm{\infty })$ are obtained. These lead naturally to isomorphisms $RC(\mathrm{\infty })\cong T(\mathrm{\infty })$ and $RC(\mathrm{\infty })\cong \bar{T}(\mathrm{\infty })$, i.e., those with the marginally large tableau and marginally large reverse tableau realizations of $B(\mathrm{\infty })$, 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 $RC(\mathrm{\infty })$, the two mentioned isomorphisms, and two existing realizations of $B(\lambda )$ that were based on $T(\mathrm{\infty })$ and $\bar{T}(\mathrm{\infty })$.