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A geometric realization of the chromatic symmetric function of a unit interval graph

Syu Kato

Abstract

Shareshian-Wachs, Brosnan-Chow, and Guay-Pacquet [Adv. Math. ${\bf 295}$ (2016), ${\bf 329}$ (2018), arXiv:1601.05498] realized the chromatic (quasi-)symmetric function of a unit interval graph in terms of Hessenberg varieties. Here we exhibit another realization of these chromatic (quasi-)symmetric functions in terms of the Betti cohomology of the variety $\mathscr X_Ψ$ defined in [arXiv:2301.00862]. This yields a new inductive combinatorial expression of these chromatic symmetric functions. Based on this, we propose a geometric refinement of the Stanley-Stembridge conjecture, whose validity would imply the Shareshian-Wachs conjecture.

A geometric realization of the chromatic symmetric function of a unit interval graph

Abstract

Shareshian-Wachs, Brosnan-Chow, and Guay-Pacquet [Adv. Math. (2016), (2018), arXiv:1601.05498] realized the chromatic (quasi-)symmetric function of a unit interval graph in terms of Hessenberg varieties. Here we exhibit another realization of these chromatic (quasi-)symmetric functions in terms of the Betti cohomology of the variety defined in [arXiv:2301.00862]. This yields a new inductive combinatorial expression of these chromatic symmetric functions. Based on this, we propose a geometric refinement of the Stanley-Stembridge conjecture, whose validity would imply the Shareshian-Wachs conjecture.

Paper Structure

This paper contains 10 sections, 16 theorems, 72 equations.

Table of Contents

  1. Preliminaries
  2. Algebraic Groups
  3. Combinatorial setup
  4. Chromatic symmetric functions and the modular law
  5. Geometry of $\mathscr X_\Psi$
  6. Inductions of character sheaves
  7. Main Results
  8. Statements
  9. Proof of Theorem \ref{['thm:main']}
  10. Proof of Theorem \ref{['thm:alg']}

Key Result

Theorem A

We have $( \mathsf{m}_\Gamma )_*{\mathbb C}_{\mathscr X_\Gamma} \cong \bigoplus _{\lambda \in \mathtt{P}^+} V^{\lambda} (\mathscr X_\Gamma) \boxtimes \mathsf{IC}_\lambda$, where $V^\lambda (\mathscr X_\Gamma) \in D^b(\mathrm{pt})$ is a $($graded$)$ vector space. Using this, we have

Theorems & Definitions (30)

  • Theorem A: $\doteq$ Theorem \ref{['thm:main']}
  • Theorem B: $\doteq$ Theorem \ref{['thm:alg']}
  • Corollary C: $\doteq$ Corollary \ref{['cor:EP']}, see e.g. DK22
  • Conjecture D: Geometric Stanley-Stembridge conjecture
  • Definition 1.1: Hessenberg function
  • Definition 1.2: Unit interval graph
  • Definition 1.3: Root ideals
  • Lemma 1.4: Cellini Cel00 § 3
  • Proposition 1.5: De Mari-Procesi-Shayman MPS92 Lemma 1
  • Theorem 1.6: Abreu-Nigro AB21 Theorem 1.1, cf. AS20 (12)
  • ...and 20 more