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Quasisymmetric divided differences

Philippe Nadeau, Hunter Spink, Vasu Tewari

TL;DR

The paper constructs a comprehensive quasisymmetric analogue of Schubert calculus by introducing forest polynomials $\mathfrak{P}_F$ and trimming operators $\mathsf{T}_F$, organized through the Thompson monoid. It shows that forest polynomials form a basis dual to volume polynomials under a $D$-pairing, yields a basis for quasisymmetric coinvariants and harmonics, and provides positive expansions and Monk-type rules within this framework. Extending to $m$-colored quasisymmetric functions, the authors develop $m$-forest polynomials and $\mathsf{T}_F^{\underline{m}}$, along with $m$-ZigZag structures and a generalized Thompson monoid, broadening the scope to $m$-quasisymmetric polynomials and their coinvariants/harmonics. A central achievement is resolving the Aval--Bergeron--Li conjecture by showing that derivatives of top-degree harmonics span the whole quasisymmetric harmonic space, and the entire construction yields practical coefficient-extraction tools for fundamental expansions. The work also connects to geometric interpretations through forest polytopes and volume polynomials, enriching the interplay between combinatorics, algebraic geometry, and representation theory in the quasisymmetric setting.

Abstract

We develop a quasisymmetric analogue of the combinatorial theory of Schubert polynomials and the associated divided difference operators. Our counterparts are "forest polynomials", and a new family of linear operators, whose theory of compositions is governed by forests and the "Thompson monoid". Our approach extends naturally to $m$-colored quasisymmetric functions. We then give several applications of our theory to fundamental quasisymmetric functions, the study of quasisymmetric coinvariant rings and their associated harmonics, and positivity results for various expansions. In particular we resolve a conjecture of Aval-Bergeron-Li regarding quasisymmetric harmonics.

Quasisymmetric divided differences

TL;DR

The paper constructs a comprehensive quasisymmetric analogue of Schubert calculus by introducing forest polynomials and trimming operators , organized through the Thompson monoid. It shows that forest polynomials form a basis dual to volume polynomials under a -pairing, yields a basis for quasisymmetric coinvariants and harmonics, and provides positive expansions and Monk-type rules within this framework. Extending to -colored quasisymmetric functions, the authors develop -forest polynomials and , along with -ZigZag structures and a generalized Thompson monoid, broadening the scope to -quasisymmetric polynomials and their coinvariants/harmonics. A central achievement is resolving the Aval--Bergeron--Li conjecture by showing that derivatives of top-degree harmonics span the whole quasisymmetric harmonic space, and the entire construction yields practical coefficient-extraction tools for fundamental expansions. The work also connects to geometric interpretations through forest polytopes and volume polynomials, enriching the interplay between combinatorics, algebraic geometry, and representation theory in the quasisymmetric setting.

Abstract

We develop a quasisymmetric analogue of the combinatorial theory of Schubert polynomials and the associated divided difference operators. Our counterparts are "forest polynomials", and a new family of linear operators, whose theory of compositions is governed by forests and the "Thompson monoid". Our approach extends naturally to -colored quasisymmetric functions. We then give several applications of our theory to fundamental quasisymmetric functions, the study of quasisymmetric coinvariant rings and their associated harmonics, and positivity results for various expansions. In particular we resolve a conjecture of Aval-Bergeron-Li regarding quasisymmetric harmonics.
Paper Structure (45 sections, 75 theorems, 164 equations, 19 figures, 1 table)

This paper contains 45 sections, 75 theorems, 164 equations, 19 figures, 1 table.

Table of Contents

  1. Introduction
  2. Quasisymmetric polynomials
  3. Quasisymmetry via the Bergeron--Sottile map $\mathsf{R}_{i}$
  4. Quasisymmetric divided differences
  5. Indexed forests
  6. Binary trees and indexed forests
  7. The code $\mathsf{c}(F)$
  8. The left terminal set $\operatorname{LTer}_{}(F)$
  9. Left and right terminally supported forests
  10. Zigzag forests
  11. Trimming and blossoming
  12. Forests and the Thompson monoids
  13. A monoid structure on Forests
  14. The Thompson monoid
  15. A monoid factorization
  16. ...and 30 more sections

Key Result

Lemma 2.1

$f(x_1,\ldots,x_n)\in \operatorname{Pol}_n$ is quasisymmetric if and only if $f=\bm\sigma_1f=\cdots = \bm \sigma_{n-1}f$.

Figures (19)

  • Figure 1: An indexed forest in ${{\mathsf{For}_{14}}}$
  • Figure 2: A forest $F\in {\operatorname{\mathsf{For}}}$ in ${{\mathsf{ZigZag}_{5}}}$.
  • Figure 3: An $F\in {\operatorname{\mathsf{For}}}$ with $\mathsf{c}(F)=(2,1,0,1,0,0,1,0,\dots)$, and the corresponding $F/2$ and $F\cdot 4$.
  • Figure 4: The products $F\cdot G$ and $G\cdot F$ for $F,G\in {\operatorname{\mathsf{For}}}$, with both roots and leaves labeled
  • Figure 5: Example of the map $\Theta'_{n}$ for $n=13$. White and black vertices contribute to $G$ and $H$ respectively.
  • ...and 14 more figures

Theorems & Definitions (162)

  • Lemma 2.1: Hi00
  • Definition 2.2
  • Theorem 2.3
  • proof
  • Corollary 2.4
  • proof
  • Definition 2.5
  • Theorem 2.6
  • proof
  • Example 2.7
  • ...and 152 more