Perforated Tableaux in Type $A_{n-1}$ Crystal Graphs and the RSK Correspondence

TL;DR

This paper advances the combinatorial understanding of type A crystal graphs by extending perforated tableaux and exploring their connections to RSK correspondence and dual crystals, revealing new insights into insertion schemes and crystal structures.

Contribution

It introduces new results on perforated tableaux and dual crystals, deepening the combinatorial framework for type A crystal graphs and their relation to RSK theory.

Findings

New combinatorial models for crystal graphs

Enhanced understanding of RSK correspondence via dual crystals

Novel insertion scheme insights in crystal analysis

Abstract

We continue work begun in \cite{ptab} which introduced \emph{perforated tableaux} as a combinatorial model for crystals of type $A_{n-1}$, emphasizing connections to the classical Robinson-Schensted-Knuth (RSK) correspondence and Lusztig involutions, and, more generally, exploring the role of insertion schemes in the analysis of crystal graphs. An essential feature of our work is the role of \emph{dual} crystals (\cite{GerberLecouvey,vanLeeuwen}) from which we obtain new results within and beyond the classic RSK theory.