Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The geometry of quasisymmetric coinvariants

Philippe Nadeau, Hunter Spink, Vasu Tewari

Abstract

We develop a quasisymmetric analogue of the theory of Schubert cycles, building off of our previous work on a quasisymmetric analogue of Schubert polynomials and divided differences. Our constructions result in a natural geometric interpretation for the ring of quasisymmetric coinvariants.

The geometry of quasisymmetric coinvariants

Abstract

We develop a quasisymmetric analogue of the theory of Schubert cycles, building off of our previous work on a quasisymmetric analogue of Schubert polynomials and divided differences. Our constructions result in a natural geometric interpretation for the ring of quasisymmetric coinvariants.

Paper Structure

This paper contains 39 sections, 66 theorems, 183 equations, 11 figures.

Table of Contents

  1. Introduction
  2. The geometry of quasisymmetric coinvariants
  3. Applications
  4. Results
  5. The quasisymmetric operations $\mathsf{R}_{i}$, and ${{\mathsf{T}_{i}}}$
  6. A quasisymmetric Schubert cycle
  7. Duality with forest polynomials
  8. The $\Omega$-flag variety
  9. Applications
  10. Generalized Littlewood--Richardson coefficients
  11. Divided symmetrization
  12. A formula of Gessel
  13. Composites of R and T via nested forests
  14. Composites of ${{\mathsf{T}_{}}}$ via indexed forests, and forest polynomials
  15. Composites of $\mathsf{R}_{}$ and ${{\mathsf{T}_{}}}$ via marked nested forests
  16. ...and 24 more sections

Key Result

Theorem 2.2

If $\Omega=\Omega'\mathsf{x}_{}\in \mathrm{RTSeq}_{n}$ and $f\in H^\bullet(\mathrm{Fl}_{n})$ then the following are true. In particular, we have $\deg_{X(\Omega)}f=\Pi_{\Omega}^n f.$

Figures (11)

  • Figure 1: Unit cube subdivision for $n=3$ where we have indicated the face labelings by $\mathrm{RTSeq}_3$ on the right, and nested forests on the left.
  • Figure 2: A nested forest.
  • Figure 3: A marked nested forest.
  • Figure 4: Trimming diagram for $\mathsf{r}_{1}^4\mathsf{t}_{2}\mathsf{r}_{3}\mathsf{r}_{4}\mathsf{t}_{6}\mathsf{t}_{2}\mathsf{t}_{7}\mathsf{r}_{1}\mathsf{t}_{6}\mathsf{r}_{5}\in \mathrm{RTSeq}_{13}$ and the associated nested forest.
  • Figure 5: Recovering $\widehat{F}(\Omega)$ from $M(\Omega)$
  • ...and 6 more figures

Theorems & Definitions (164)

  • Example 2.1
  • Theorem 2.2: \ref{['thm:mainP1degree']}
  • Example 2.3
  • Theorem A
  • Theorem B
  • Theorem 2.4: \ref{['thm:LR']}
  • Theorem 2.5
  • Conjecture 2.6: \ref{['conj:HL_positivity']}
  • Definition 3.1
  • Definition 3.2
  • ...and 154 more