Mixed jeu de taquin and a problem of Soojin Cho
Santiago Estupiñán-Salamanca, Oliver Pechenik
TL;DR
The paper addresses the need for a type B analogue of the plactic/RSK framework by constructing a new skew plactic Schur $P$-function that aligns with shifted combinatorics and by introducing a mixed jeu de taquin that computes Haiman's mixed insertion. The authors define a mixed rectification process ${\rm rect}_{\rm mix}$ using bullets and diagonal slides, establishing that the rectification sequence encodes the mixed insertion steps and preserves semistandard structure. They then define skew plactic Schur $P$-functions $\mathcal{P}_{\nu/\mu}$ via a modified Sagan–Worley rectification and a diagonal-raising operator, proving these functions lie in the ring generated by $\mathcal{P}_{\lambda}$ and have expansion coefficients matching the classical skew Schur $P$-function expansion with LR-like numbers $b_{\lambda,\mu}^{\nu}$. Consequently, the work provides an algebraic framework that resolves Cho's open problem and connects mixed insertion, shifted plactic theory, and skew Schur $P$-functions in type B combinatorics.
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.