[Paper Review] The shifted plactic monoid

Luís Serrano|ArXiv.org|Nov 13, 2008

Advanced Combinatorial Mathematics11 references4 citations

TL;DR

This paper introduces the shifted plactic monoid as a combinatorial structure analogous to the classical plactic monoid, defined via shifted Knuth relations or Haiman's mixed insertion. It establishes a shifted Littlewood-Richardson rule through plactic Schur $P$-functions, providing a noncommutative framework for symmetric function theory in the shifted setting.

ABSTRACT

We introduce a shifted analog of the plactic monoid of Lascoux and Schützenberger, 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-Schützenberger theory of noncommutative Schur functions in plactic variables; a characterization of shifted tableau words; and more.

Motivation & Objective

• To develop a shifted analog of the classical plactic monoid for semistandard shifted Young tableaux.
• To extend the Lascoux-Schützenberger theory of noncommutative Schur functions to the shifted setting.
• To provide a new combinatorial derivation of the shifted Littlewood-Richardson rule using plactic Schur $P$-functions.
• To characterize shifted tableau words via duality in the Sagan-Worley correspondence.
• To establish a canonical isomorphism between the shifted plactic algebra and the ring of Schur $P$-functions.

Proposed method

• Define the shifted plactic monoid $\mathbf{S}(X)$ as the quotient of the free monoid on $X$ by the shifted Knuth relations.
• Use Haiman's mixed insertion to define shifted plactic classes as sets of words with the same mixed insertion tableau.
• Establish that shifted plactic equivalence is preserved under concatenation, ensuring the monoid structure.
• Construct plactic Schur $P$-functions $\mathcal{P}_\lambda$ as sums of shifted plactic classes of shape $\lambda$.
• Prove that the ring generated by $\mathcal{P}_\lambda$ is isomorphic to the ring of ordinary Schur $P$-functions via the map $P_\lambda \mapsto \mathcal{P}_\lambda$.
• Use duality between mixed insertion and Sagan-Worley correspondence to characterize shifted tableau words via recording tableaux of inverses.

Experimental results

Research questions

• RQ1 How can the classical plactic monoid be generalized to the shifted setting using shifted Young tableaux?
• RQ2 What is the analog of the shifted Knuth relations that characterize equivalence classes of words under mixed insertion?
• RQ3 Can a shifted version of the Littlewood-Richardson rule be derived combinatorially using plactic Schur $P$-functions?
• RQ4 How do noncommutative Schur $P$-functions in the shifted plactic algebra relate to their commutative counterparts?
• RQ5 What characterizes words that yield semistandard shifted tableaux under mixed insertion?

Key findings

• The shifted plactic monoid $\mathbf{S}(X)$ is well-defined via the shifted Knuth relations and inherits a monoid structure from word concatenation.
• Two words are shifted Knuth-equivalent if and only if they have the same mixed insertion tableau.
• The shifted plactic Schur $P$-function $\mathcal{P}_\lambda$ is the sum of all shifted plactic classes of shape $\lambda$, and these functions span the shifted plactic algebra.
• The ring generated by the $\mathcal{P}_\lambda$ is canonically isomorphic to the ring of ordinary Schur $P$-functions, with $P_\lambda \mapsto \mathcal{P}_\lambda$.
• The shifted Littlewood-Richardson coefficient $b^\lambda_{\mu,\nu}$ equals the number of pairs $(T_\mu, T_\nu)$ of shifted plactic classes such that $T_\mu T_\nu = T_\lambda$ for a fixed $T_\lambda$ of shape $\lambda$.
• A word is a shifted tableau word if and only if the recording tableau of its inverse (as a biword) is a shifted standard tableau, via duality with the Sagan-Worley correspondence.