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

Glenn D. Appleby, Tamsen Whitehead

Published 2020-07-23Version 1

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) with simpler crystal operators with which we can identify highest weights visually without computation (for all crystals directly, without reference to a canonical model of semistandard Young tableaux (SSYT)). 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 SSYT. We relate evacuation (Lusztig involution) to products of ptableaux crystal operators, and find a combinatorial algorithm to compute commutators of highest weight ptableaux.