Perforated Tableaux as a Unified Combinatorial Model for Crystals on Words in $[n]^{\otimes k}$: Tensor Products, Commutators, and Lusztig Involutions
We present a combinatorial model, called perforated tableaux, to study $A_{n}$ crystals, unifying several combinatorial models "under one roof''. Nodes of the standard $A_{n}$ crystal graph are integers $[n] = \{ 1, 2, \ldots, n\}$. One can identify nodes in the crystal tensor product $[n]^{\otimes k}$ with length $k$ words in $[n]$. We replace words with perforated tableaux (ptableaux) and use them to simplify crystal operators and identify highest weights visually without computation (for all crystals directly, without reference to a canonical model $B_{\nu}$ of semistandard Young tableaux (SSYT) of shape $\nu$). We generalize 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 analyze commutators (the isomorphism $B_{\nu} \otimes B_{\mu} \rightarrow B_{\mu} \otimes B_{\nu}$) finding ptableaux algorithms relating prior results on tableaux switching and plactic equivalence. We relate evacuation (Lusztig involution) to products of ptableaux crystal operators, and find a natural bi-crystal structure on all crystals of a fixed isomorphism class.