论文标题
后尼科夫 - 史丹利的外线布置猜想
Postnikov-Stanley Linial arrangement conjecture
论文作者
论文摘要
特征多项式在超平面排列领域是重要的不变。对于任何不可约根系的外线排列，尼科夫和斯坦利的后者都认为，特征多项式的所有根都具有相同的实际部分。关于这一猜想，Yoshinaga获得了特征性的准多物种和Ehrhart Quasi-Polynomial之间的明确关系，用于基本的壁co。在本文中，我们通过将ehrhart准级 - 多项式分解计算为Yoshinaga的显式公式，并将改良的移位运算符分解为几个准偏移运算符，并为特征性的Quasi-PolyNomial获得新的公式。特别是，当线性排列的参数与Ehrhart Quasi-Polynomial的周期相对较好时，我们证明了Nikov-Stanley后线排列的猜想。这概括了尼科夫 - 史丹利和阿萨纳西迪斯证明的经典类型的根系的一些结果。对于其他情况，我们使用计算方法为特殊根系验证了此猜想。
A characteristic polynomial is an important invariant in the field of hyperplane arrangement. For the Linial arrangement of any irreducible root system, Postnikov and Stanley conjectured that all roots of the characteristic polynomial have the same real part. In relation to this conjecture, Yoshinaga obtained an explicit relationship between the characteristic quasi-polynomial and the Ehrhart quasi-polynomial for the fundamental alcove. In this paper, we calculate Yoshinaga's explicit formula through the decomposition of the Ehrhart quasi-polynomial into several quasi-polynomials and a modified shift operator, and obtain new formulas for the characteristic quasi-polynomial of the Linial arrangement. In particular, when the parameter of the Linial arrangement is relatively prime to the period of the Ehrhart quasi-polynomial, we prove the Postnikov-Stanley Linial arrangement conjecture. This generalizes some of the results for the root systems of classical types that have been proved by Postnikov-Stanley and Athanasiadis. For other cases, we verify this conjecture for exceptional root systems using a computational approach.
