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

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.