Perforated Tableaux as a Unified Combinatorial Model for Crystals on Words in [n]⊗k: Tensor Products, Commutators, and Lusztig Involutions
We present a combinatorial model, called perforated tableaux, to study An crystals, unifying several combinatorial models "under one roof''. Nodes of the standard An crystal graph are integers [n]={1,2,…,n}. One can identify nodes in the crystal tensor product [n]⊗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 Bν of semistandard Young tableaux (SSYT) of shape ν). We generalize tensor products in the Littlewood-Richardson rule to all of [n]⊗k, and not just the irreducible crystals whose reading words come from SSYT. We analyze commutators (the isomorphism Bν⊗Bμ→Bμ⊗Bν) 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.