色对称函数的牛顿多面体与洛伦兹性质
The Newton polytope and Lorentzian property of chromatic symmetric functions
AI总结：
本文研究Dyck路径无差别图的色对称函数，证明其允许着色权重构成置换多面体的格点并给出支配权公式，猜想并部分证明其洛伦兹性质，同时将牛顿多面体结果推广到(3+1)-自由偏序集。
摘要：
色对称函数是代数组合学中研究广泛的对称函数，它推广了色多项式，并与Hessenberg簇和对角调和函数相关。受Stanley–Stembridge猜想的启发，我们证明了Dyck路径的无差别图的允许着色权重是置换多面体$\mathcal{P}_λ$的格点，并给出了支配权$λ$的公式。此外，我们猜想这类色对称函数是洛伦兹的，这一性质由Brändén和Huh引入，作为离散凸分析与组合学中凹性性质之间的桥梁，并且我们证明了该猜想对阿贝尔Dyck路径成立。我们将关于牛顿多面体的结果推广到(3+1)-自由偏序集的无比较图，并给出了源于本工作的一系列猜想和结果，包括关于计算系数的复杂性以及与对角调和函数中$ζ$映射的关系的结果。
英文摘要：
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 $\mathcal{P}_λ$, and we give a formula for the dominant weight $λ$. 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 $ζ$ map from diagonal harmonics.
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