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

2024/10/16 by Syu Kato, Kato, Syu · 1 citation
Computer Science · Mathematics · Physics and Astronomy · #Algebraic Geometry (math.AG) #Color Science and Applications #Combinatorics (math.CO) #Computer Graphics and Visualization Techniques #FOS: Mathematics #Mathematics and Applications #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2410.12231

openalex publication_date 2024/10/16 · openalex created_date 2024/10/20 · openalex updated_date 2026/07/28

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.

Citations

• A second proof of the Shareshian--Wachs conjecture, by way of a new Hopf algebra
• A geometric realization of Catalan functions

Cited by

• Toward Lower Bounds for Chromatic Symmetric Functions in the Elementary Basis

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