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# Journal of Combinatorics

## Volume 9 (2018)

### Number 2

### Characterization of $\mathcal{B} (\infty)$ using marginally large tableaux and rigged configurations in the $A_n$ case via integer sequences

Pages: 345 – 399

DOI: https://dx.doi.org/10.4310/JOC.2018.v9.n2.a7

#### Author

#### Abstract

Marginally large tableaux are semi-standard Young tableaux of special form that give a combinatorial realization of the crystals $\mathcal{B} (\infty)$. Rigged configurations are combinatorial objects prominent in the study of solvable lattice models, and give combinatorial realizations of the crystals $\mathcal{B} (\lambda)$ and $\mathcal{B} (\infty)$ in simply-laced and affine Kacs–Moody types. However, $\mathcal{B} (\infty)$ rigged configurations have not yet been characterized explicitly at the time of this writing.

We introduce certain nice integer sequences, called cascading sequences, to characterize marginally large tableaux. Then we use cascading sequences and a known non-explicit crystal isomorphism between marginally large tableaux and rigged configurations to give an explicit characterization of the latter set in the $A_n$ case, revealing interesting structural properties of rigged configurations along the way, and then to give an explicit bijection between the two sets.

#### Keywords

crystal, marginally large tableau, rigged configuration, Kashiwara operator

The author was partially supported by NSF grants DMS-1001256, OCI-1147247, DMS-1500050.

Received 17 February 2017

Published 22 January 2018