Mixed jeu de taquin and a problem of Soojin Cho

TL;DR

This paper introduces a new definition of skew shifted plactic Schur functions that corrects previous issues and develops a new jeu de taquin theory for mixed insertion, advancing algebraic combinatorics related to Young tableaux.

Contribution

It provides a corrected definition of skew shifted plactic Schur functions and a new jeu de taquin theory for mixed insertion, addressing prior conjectures and problems.

Findings

New definition of skew shifted plactic Schur functions that behaves as desired.

A novel jeu de taquin theory for mixed insertion.

Resolution of previous conjecture by Cho.

Abstract

Serrano (2010) introduced the shifted plactic monoid, governing Haiman's (1989) mixed insertion algorithm, as a type B analogue of the classical plactic monoid that connects jeu de taquin of Young tableaux with the Robinson-Schensted-Knuth insertion algorithm. Serrano proposed a corresponding definition of skew shifted plactic Schur functions. Cho (2013) disproved Serrano's conjecture regarding this definition, by showing that the functions do not live in the desired ring and hence cannot provide an algebraic interpretation of tableau rectification or of the corresponding structure coefficients. Cho asked for a new definition with particular properties. We introduce such a definition and prove that it behaves as desired. We also introduce a new jeu de taquin theory that computes mixed insertion.