Perforated Tableaux: A Combinatorial Model for Crystal Graphs in Type $A_{n-1}$

TL;DR

This paper introduces perforated tableaux as a new combinatorial model for type A crystal graphs, unifying existing models and simplifying the understanding of crystal operators, highest weights, and tensor product rules.

Contribution

It develops perforated tableaux to model $A_{n-1}$ crystals, extending combinatorial descriptions to all tensor products and relating evacuation to crystal operators.

Findings

Simplified crystal operator definitions in ptableaux

Visual identification of highest weights

Generalization of Littlewood-Richardson tensor products

Abstract

We present a combinatorial model, called \emph{perforated tableaux}, to study $A_{n-1}$ crystals, unifying several previously studied combinatorial models. We identify nodes in the $k$-fold tensor product of the standard crystal with length $k$ words in $[n]= \{ 1, \ldots n\}$. We model this crystal with perforated tableaux (ptableaux), extending this identification isomorphically to biwords, RSK $(P,Q)$ tableaux pairs, and matrix models. In the ptableaux setting, crystal operators are more simply defined and we can identify highest weights visually without computation. We generalize the tensor products in the Littlewood-Richardson rule to all of $[n]^{\otimes k}$, and not just the irreducible crystals whose reading words come from semistandard Young tableaux. We relate evacuation (Lusztig involution) to products of ptableaux crystal operators, and find a combinatorial algorithm to compute commutators of highest weight ptableaux.