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A Note on Jing and Li's Type B Quasisymetric Schur Functions

Ezgi Kantarcı Oğuz

TL;DR

The paper addresses the positivity structure of Jing and Li's type B quasisymmetric Schur functions and proves their conjectured integral, unitriangular, positive expansion into peak functions. The authors introduce marked peak composition tableaux to realize explicit monomial, fundamental, and peak expansions, establishing that $\widehat{P}_{\alpha}(X)=\sum_{T\in\mathrm{MPCT}^*(\alpha)}M_{\mathrm{wt}(T)}(X)$ and $\widehat{Q}_{\alpha}(X)=\sum_{T\in\mathrm{MPCT}(\alpha)}M_{\mathrm{wt}(T)}(X)$, while also giving negative results showing lack of positivity in expansions to several other bases and non-positivity of certain products. These results illuminate the special role of peak functions in the type B setting and delineate the limits of positivity across related quasisymmetric bases.

Abstract

In 2015, Jing and Li defined type B quasisymmetric Schur functions and conjectured that these functions have a positive, integral and unitriangular expansion into peak functions. We prove this conjecture, and refine their combinatorial model to give explicit expansions in monomial, fundamental and peak bases. We also show that these functions are not quasisymmetric Schur, Young quasisymmetric Schur or dual immaculate positive, and do not have a positive multiplication rule.

A Note on Jing and Li's Type B Quasisymetric Schur Functions

TL;DR

The paper addresses the positivity structure of Jing and Li's type B quasisymmetric Schur functions and proves their conjectured integral, unitriangular, positive expansion into peak functions. The authors introduce marked peak composition tableaux to realize explicit monomial, fundamental, and peak expansions, establishing that and , while also giving negative results showing lack of positivity in expansions to several other bases and non-positivity of certain products. These results illuminate the special role of peak functions in the type B setting and delineate the limits of positivity across related quasisymmetric bases.

Abstract

In 2015, Jing and Li defined type B quasisymmetric Schur functions and conjectured that these functions have a positive, integral and unitriangular expansion into peak functions. We prove this conjecture, and refine their combinatorial model to give explicit expansions in monomial, fundamental and peak bases. We also show that these functions are not quasisymmetric Schur, Young quasisymmetric Schur or dual immaculate positive, and do not have a positive multiplication rule.

Paper Structure

This paper contains 6 sections, 11 theorems, 18 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Quasisymmetric Functions
  4. Peak Composition Tableaux and Quasisymmetric Schur $Q$-functions
  5. Marked Peak Composition Tableaux
  6. Further properties of quasisymmetric Schur $Q$-functions

Key Result

Theorem 2.3

For any strict partition $\lambda$ we have: where $\mathrm{sort}(\alpha)$ is the unique rearrangement of $\alpha$ into a partition, and $\sigma_{\alpha}$ is a corresponding permutation of minimal length.

Figures (4)

  • Figure 1: A peak composition tableau of shape (3,2,3,1) and weight (1,1,1,3,2,1)
  • Figure 2: The elements of $\mathrm{PCT}(2,1)$
  • Figure 3: The elements of $\mathrm{MPCT}^*(2,1)$
  • Figure 4: A marked standard peak composition tableau of shape $(3,2,3,1)$

Theorems & Definitions (25)

  • Remark 2.1
  • Definition 2.2: MR3366479
  • Theorem 2.3: MR3366479
  • Definition 3.1
  • Proposition 3.2
  • proof
  • Definition 3.3
  • Lemma 3.4
  • proof
  • Definition 3.5
  • ...and 15 more