Tutte Polynomials of arrangements of ideal type

几十年来，超平面排列理论一直是数学的推动力。它自然地处于代数、组合学和代数几何的交叉点。这个提议反过来又是这些主题和代数李理论的核心。研究编曲的动机大多来自于考克斯特编曲。虽然后者得到了很好的研究，但它们的子结构却没有得到很好的理解。在这个研究计划中，我们想研究一类特殊的与Weyl群的根的正根集中的一个理想相关联的排列，即理想型排列。Sommers-Tymoczko在2006年定义并研究了这些。根据Sommers和Tymoczko的两个猜想，我们提出了两个研究方向。第一个猜想是关于一个理想的Weyl型子集的庞加莱多项式的乘法公式，它推广了下面Weyl群的庞加莱多项式的众所周知的因式分解。Sommers和Tymoczko证明这种分解对A、B、C型和小阶例外型的根系都成立。在D、E7和E8类型中，推测仍然是开放的。我们提出一个统一的方法来解决这个猜想。通过解释Sommers和Tymoczko在基本排列的区域偏序集的秩生成函数设置中的猜想，并通过归纳论证W的秩，我们得到了不包含任何单根的理想情况的约化。然后我们通过归纳进一步论证这些理想的基数性。我们的第二个研究方向集中在Sommers和Tymoczko的另一个猜想上。这关系到理想类型安排的自由。Sommers和Tymoczko证明，当Weyl群是经典的或者是特殊类型的小秩时，每种理想类型的排列都是自由的。一般案件直到最近才由Abe、Barakat、Cuntz、Hoge和Terao以统一的方式解决。这推广了Shapiro-Steinberg-Kostant的一个重要公式，该公式指出W的正根的高度分布的对偶分割是W的指数集。在这里，我们提出研究理想型排列的各种更强的自由性质。已知Weyl群W本身的反射排列总是归纳自由的。这很可能也是所有理想型排列的情况。总的来说，我们通过对W的秩进行归纳，概述了对这个问题的归纳方法。除了对超平面排列得到新的结果外，这些关于理想型排列的结果反过来又为与Weyl群W的复约群G相关的flag变体的某些子变体的几何提供了新的见解，即所谓的Hessenberg变体。

The theory of hyperplane arrangements has been a driving force in mathematics over many decades. It naturally lies at the crossroads of algebra, combinatorics and algebraic geometry. This proposal in turn lies at the very heart of these subject matters and algebraic Lie theory. Much of the motivation for the study of arrangements comes from Coxeter arrangements. While the latter are well studied, their subarrangements are considerably less well understood. In this research proposal we want to investigate a particular class of arrangements which are associated with an ideal in the set of positive roots of the root system of a Weyl group W, so called arrangements of ideal type. These were defined and investigated by Sommers-Tymoczko in 2006.We propose two research strands stemming from two conjectures due to Sommers and Tymoczko. The first of these conjectures concerns a multiplicative formula for the Poincare polynomial of the subsets of Weyl type of an ideal which generalizes the well known factorization of the Poincare polynomial of the underlying Weyl group. Sommers and Tymoczko showed that this factorization holds for root systems of types A, B, C and small rank exceptional types.The conjecture is still open in types D, E7 and E8. We propose a uniform approach to resolve this conjecture. By interpreting Sommers and Tymoczko's conjecture in the setting of rank-generating functions of the poset of regions for the underlying arrangements and arguing by induction on the rank of W, we obtain a reduction to the case of ideals which do not contain any simple roots. Then we argue further by induction on the cardinality of such ideals.Our second research strand focuses on another conjecture by Sommers and Tymoczko. This concerns the freeness of the arrangements of ideal type. It was shown by Sommers and Tymoczko in case the Weyl group is classical or of exceptional type of small rank that each arrangement of ideal type is free. The general case was settled only very recently in a uniform manner for all types by Abe, Barakat,Cuntz, Hoge and Terao. This generalizes a seminal formula of Shapiro-Steinberg-Kostant which states that the partition dual to the height distribution of the positive roots of W is the set of exponents of W. Here we propose to investigate various stronger freeness properties for the arrangements of ideal type. It is known that the reflection arrangement of a Weyl group W itself is always inductively free. It is very likely that this is also the case for all the arrangements of ideal type. In general, we outline an inductive approach to this question by means of induction on the rank of W.Apart from yielding new results for hyperplane arrangements, these results on arrangements of ideal type in turn provide new insight into the geometry of certain subvarieties of the flag variety associated with a complex reductive group G with Weyl group W, so called Hessenberg varieties.