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Symmetric quasisymmetric Schur-like functions

Maria Esipova, Jinting Liang, Stephanie van Willigenburg

Abstract

In this paper we classify when (row-strict) dual immaculate functions and (row-strict) extended Schur functions, as well as their skew generalizations, are symmetric. We also classify when their natural variants, termed advanced functions, are symmetric. In every case our classification recovers classical skew Schur functions.

Symmetric quasisymmetric Schur-like functions

Abstract

In this paper we classify when (row-strict) dual immaculate functions and (row-strict) extended Schur functions, as well as their skew generalizations, are symmetric. We also classify when their natural variants, termed advanced functions, are symmetric. In every case our classification recovers classical skew Schur functions.

Paper Structure

This paper contains 16 sections, 18 theorems, 85 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Compositions and partitions
  4. Symmetric and quasisymmetric functions
  5. Symmetric (row-strict) dual immaculate functions
  6. Dual immaculate functions
  7. Row-strict dual immaculate functions
  8. Symmetric (row-strict) extended Schur functions
  9. Extended Schur functions
  10. Row-strict extended Schur functions
  11. Symmetric advanced functions
  12. Strictly advanced functions
  13. Weakly advanced functions
  14. Strictly advanced extended functions
  15. Weakly advanced extended functions
  16. ...and 1 more sections

Key Result

Lemma 2.8

Suppose $f=g\cdot h$ for some $f,g,h\in \mathbb{Q}[[x_1,x_2,\ldots]]$. Then $f$ is symmetric if and only if $g$ and $h$ are both symmetric. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (63)

  • Example 2.1
  • Example 2.2
  • Example 2.3
  • Definition 2.4
  • Example 2.5
  • Example 2.6
  • Example 2.7
  • Lemma 2.8: gps
  • Definition 3.1
  • Example 3.2
  • ...and 53 more