The shifted plactic monoid

TL;DR

This paper introduces the shifted plactic monoid, a new algebraic structure that extends classical plactic theory with applications to shifted tableaux, Schur functions, and combinatorial rules.

Contribution

It defines the shifted plactic monoid via shifted Knuth relations and Haiman's mixed insertion, providing new combinatorial and algebraic tools.

Findings

Derived a shifted Littlewood-Richardson rule

Established results for Schur P-function coefficients

Characterized shifted tableau words

Abstract

We introduce a shifted analog of the plactic monoid of Lascoux and Sch\"utzenberger, the \emph{shifted plactic monoid}. It can be defined in two different ways: via the \emph{shifted Knuth relations}, or using Haiman's mixed insertion. Applications include: a new combinatorial derivation (and a new version of) the shifted Littlewood-Richardson Rule; similar results for the coefficients in the Schur expansion of a Schur $P$-function; a shifted counterpart of the Lascoux-Sch\"utzenberger theory of noncommutative Schur functions in plactic variables; a characterization of shifted tableau words; and more.