Rigged Configuration Descriptions of the Crystals B(infinity) and B(lambda) 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 $\bar{T}(\infty )$.

References in corpus (3)

• A rigged configuration model for $B(\infty )$
• Rigged configurations for all symmetrizable types
• Connecting marginally large tableaux and rigged configurations via crystals