Skew RSK and the switching on ballot tableau pairs

TL;DR

This paper explores the properties of skew RSK and ballot tableau pairs, revealing new internal insertion procedures and their connections to Littlewood-Richardson coefficients, crystal bases, and branching models in algebraic combinatorics.

Contribution

It introduces a novel internal insertion approach for skew RSK and ballot tableau pairs, linking various combinatorial models and solving existing conjectures.

Findings

Realization of switching involution as recursive internal insertion

Alternative construction of Gelfand-Tsetlin pairs without Schützenberger involution

Proof of the Lecouvey-Lenart conjecture on bijections between branching models

Abstract

In arXiv:1808.06095 we have introduced the Knuth class of the word recording a sequence of locations for repeated internal insertion operations in the Sagan-Stanley skew RSK correspondence, with no prescribed external insertion of new cells, to be a preserver for the $P$-tableau. As a consequence the Benkart-Sottile-Stroomer switching involution on ballot tableau pairs allows a realization as a recursive internal insertion procedure. This amounts to explain the various presentations of Littlewood-Richardson (LR) commuters and their coincidence predicted by Pak and Vallejo with contributions by Danilov and Koshevoi. In particular, the aforesaid presentation provides internal insertion as an alternative to Sch\"utzenberger- Lusztig involution (or evacuation) to constructing the Gelfand-Tsetlin pair in the Henriques-Kamnitzer $\mathfrak{gl}_n$-crystal commuter. In addition, the coincidence of LR commuters solves the Lecouvey-Lenart conjecture, recently further developed by Kumar-Torres, on bijections between the Kwon and Sundaram branching models.