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Shifted insertion algorithms for primed words

Eric Marberg

TL;DR

New insertion algorithms that associate pairs of shifted tableaux to finite integer sequences in which certain terms may be primed, namely, a shifted form of Edelman-Greene insertion, Sagan-Worley insertion, and Haiman's shifted mixed insertion are studied.

Abstract

This article studies some new insertion algorithms that associate pairs of shifted tableaux to finite integer sequences in which certain terms may be primed. When primes are ignored in the input word these algorithms reduce to known correspondences, namely, a shifted form of Edelman-Greene insertion, Sagan-Worley insertion, and Haiman's shifted mixed insertion. These maps have the property that when the input word varies such that one output tableau is fixed, the other output tableau ranges over all (semi)standard tableaux of a given shape with no primed diagonal entries. Our algorithms have the same feature, but now with primes allowed on the main diagonal. One application of this is to give another Littlewood-Richardson rule for products of Schur $Q$-functions. It is hoped that there will exist set-valued generalizations of our bijections that can be used to understand products of $K$-theoretic Schur $Q$-functions.

Shifted insertion algorithms for primed words

TL;DR

New insertion algorithms that associate pairs of shifted tableaux to finite integer sequences in which certain terms may be primed, namely, a shifted form of Edelman-Greene insertion, Sagan-Worley insertion, and Haiman's shifted mixed insertion are studied.

Abstract

This article studies some new insertion algorithms that associate pairs of shifted tableaux to finite integer sequences in which certain terms may be primed. When primes are ignored in the input word these algorithms reduce to known correspondences, namely, a shifted form of Edelman-Greene insertion, Sagan-Worley insertion, and Haiman's shifted mixed insertion. These maps have the property that when the input word varies such that one output tableau is fixed, the other output tableau ranges over all (semi)standard tableaux of a given shape with no primed diagonal entries. Our algorithms have the same feature, but now with primes allowed on the main diagonal. One application of this is to give another Littlewood-Richardson rule for products of Schur -functions. It is hoped that there will exist set-valued generalizations of our bijections that can be used to understand products of -theoretic Schur -functions.

Paper Structure

This paper contains 28 sections, 51 theorems, 134 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Outline
  3. Motivation
  4. Preliminaries
  5. Involution words
  6. Primed words
  7. Tableaux
  8. Shifted Edelman-Greene insertion
  9. Definitions for the standard case
  10. Extension to the semistandard case
  11. Application to multiplying Schur $Q$-functions
  12. Orthogonal Coxeter-Knuth equivalence
  13. Dual equivalence operators for shifted tableaux
  14. Properties of marked cycles
  15. Proofs of the two main theorems
  16. ...and 13 more sections

Key Result

Theorem 1.1

The map $a \mapsto (P_{\textsf{EG}}^\mathsf{O}(a),Q_{\textsf{EG}}^\mathsf{O}(a))$ is a bijection from $\mathcal{R}_{\mathsf{inv}}^+(z)$ (respectively, $\mathsf{Incr}_\infty(\mathcal{R}_{\mathsf{inv}}^+(z))$) to the set of pairs $(P,Q)$ where $P$ is a shifted tableau with increasing rows and columns,

Figures (3)

  • Figure 1: Examples for the proof of Proposition \ref{['o-lem2']}.
  • Figure 2: Example for the proof of Lemma \ref{['gamma-lem']}; compare with Figure \ref{['running-fig1']}(a).
  • Figure 3: Example for the proof of Lemma \ref{['gamma-lem']}; compare with Figure \ref{['running-fig1']}(b).

Theorems & Definitions (130)

  • Theorem 1.1: See Theorems \ref{['o-thm1']} and \ref{['o-summary-thm']}
  • Example 1.2
  • Theorem 1.3: See Theorem \ref{['ck-fkd-thm']} and Corollary \ref{['o-cor2']}
  • Theorem 1.4: See Marberg2019a
  • Lemma 2.1
  • proof
  • Proposition 2.2: HuZhang1
  • proof
  • Remark 2.3
  • Proposition 2.4: Marberg2021
  • ...and 120 more