Tutte Polynomials of arrangements of ideal type

几十年来，超平面排列理论一直是数学中的推动力。它自然位于代数、组合学和代数几何的十字路口。这一提议反过来又是这些主题和代数测谎理论的核心。研究安排的大部分动机来自于科克塞特安排。虽然后者得到了很好的研究，但对它们的次级安排却知之甚少。在这个研究方案中，我们想要研究Weyl群W的根系的正根集合中与理想相关的一类特殊的排列，即所谓的理想型排列。这些都是Sommers-Tymoczko在2006年定义和研究的。我们根据Sommers和Tymoczko的两个猜想提出了两个研究链。第一个猜想涉及理想的Weyl型子集的Poincare多项式的乘法公式，它推广了基础Weyl群的Poincare多项式的众所周知的因式分解。Sommers和Tymoczko证明了这种分解对A型、B型、C型和小秩例外类型的根系统成立，但对D型、E7和E8型的猜想仍然是开放的。我们提出了一个统一的方法来解决这个猜想。通过解释Sommers和Tymoczko的猜想，在基本排列的区域的偏序集的秩生成函数的设置下，并通过归纳法在W的秩上进行论证，我们得到了不含任何单根的理想的情形的约化。然后，我们通过归纳法进一步论证这种理想的基数。我们的第二个研究线索集中在Sommers和Tymoczko的另一个猜想上。这关系到理想型布局的自由性。Sommers和Tymoczko证明了当Weyl群是经典的或特殊的小秩型时，理想型的每一种排列都是自由的。直到最近，Abe、Barakat、Cuntz、Hoge和Terao才以统一的方式解决了一般案件。这推广了Shapiro-Steinberg-Kostant的一个开创性公式，即W的正根的高度分布的配分对偶是W的指数集。本文讨论了理想型排列的各种较强的自由性。众所周知，Weyl基团W本身的反射排列总是电感自由的。这很可能也是所有理想型安排的情况。一般说来，我们利用W的秩上的归纳法给出了这一问题的一种归纳法。除了得到超平面排列的新结果外，这些关于理想型排列的结果反过来又提供了新的见解，即与具有Weyl群W的复约群G相关的FLAG簇的某些子簇的几何，即所谓的Hessenberg簇。

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.