论文标题
牛顿多层和洛伦兹的色度对称函数的特性
The Newton polytope and Lorentzian property of chromatic symmetric functions
论文作者
论文摘要
色度对称函数是概括化色多项式的代数组合学中良好的对称函数，与Hessenberg品种和对角谐波有关。由Stanley-Stembridge猜想的激励，我们表明，染色器路径无差异图的允许着色权重是Permutahedron $ \ Mathcal {p}_λ$的晶格点，我们给出了一个主要权重$λ$的公式。此外，我们猜想这种色度对称函数是洛伦兹（Lorentzian），这是Brändén和Huh引入的属性，作为Compinatorics中离散凸分析和凹陷性能之间的桥梁，我们证明了Abelian Dyck路径的这种猜想。我们将结果扩展到牛顿多层人士上，并将无效Posets的无与伦比图表扩展到无与伦比的图形，我们给出了许多猜想和来自我们工作的结果，包括有关计算系数的复杂性以及与对角谐波的$ζ$ MAP的复杂性。
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.
