The Newton polytope and Lorentzian property of chromatic symmetric functions

Selecta Mathematica Pub Date : 2024-04-03 DOI:10.1007/s00029-024-00928-4

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Abstract

Chromatic symmetric functions are well-studied symmetric functions in algebraic combinatorics that generalize the chromatic polynomial and are related to Hessenberg varieties and diagonal harmonics. Motivated by the Stanley–Stembridge conjecture, we show that the allowable coloring weights for indifference graphs of Dyck paths are the lattice points of a permutahedron $\mathcal {P}_\lambda $ , and we give a formula for the dominant weight $\lambda $ . Furthermore, we conjecture that such chromatic symmetric functions are Lorentzian, a property introduced by Brändén and Huh as a bridge between discrete convex analysis and concavity properties in combinatorics, and we prove this conjecture for abelian Dyck paths. We extend our results on the Newton polytope to incomparability graphs of $(3+1)$ -free posets, and we give a number of conjectures and results stemming from our work, including results on the complexity of computing the coefficients and relations with the $\zeta $ map from diagonal harmonics.

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牛顿多面体和色度对称函数的洛伦兹性质

摘要色度对称函数是代数组合学中研究得很透彻的对称函数，它概括了色度多项式，并与海森堡变体和对角谐波有关。在斯坦利-斯坦布里奇猜想的激励下，我们证明了戴克路径的无差别图的允许着色权重是 permutahedron $\mathcal {P}_\lambda $ 的晶格点，并给出了主导权重 $\lambda $ 的公式。此外，我们猜想这种色度对称函数是洛伦兹性的，这是布兰登（Brändén）和胡（Huh）作为离散凸分析和组合学中凹性性质之间的桥梁而引入的性质，我们证明了非等边戴克路径的这一猜想。我们把关于牛顿多面体的结果扩展到了$(3+1)$的不可比性图。-我们给出了一系列源自我们工作的猜想和结果，包括计算系数的复杂性以及与来自对角谐波的（\zeta \）映射的关系的结果。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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