Tutte Polynomials of arrangements of ideal type

理想类型排列的 Tutte 多项式

• 批准号: 286916001
• 负责人: Professor Dr. Gerhard Röhrle
• 金额: --
• 依托单位: Lehrstuhl Algebra/Zahlentheorie
• 依托单位国家: 德国
• 项目类别: Research Grants
• 财政年份: 2015
• 资助国家: 德国
• 起止时间: 2014-12-31 至 2020-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/286916001?language=en
• 关键词: Tutte; Polynomials; arrangements; ideal; type

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.

超平面排列的理论在过去的几十年里一直是数学的推动力。它自然位于代数，组合数学和代数几何的十字路口。这一建议反过来又是这些主题和代数李理论的核心。对安排的研究的大部分动机来自考克斯特安排。虽然后者是很好的研究，他们的子安排是相当少的理解。在这个研究计划中，我们想研究一类特殊的安排，这是与一个理想的Weyl群W的根系的正根的集合，所谓的安排的理想型。这些是由Sommers-Tymoczko在2006年定义和研究的。第一个这些progratures关注的乘法公式的Poincare多项式的子集的Weyl型的理想，推广了众所周知的因式分解的Poincare多项式的基础Weyl群。Sommers和Tymoczko证明了这一分解对A、B、C型根系和小秩例外型根系成立，对D、E7和E8型根系猜想仍然成立。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的猜想在设置的偏序集的区域的秩生成函数的基础安排和论证的秩W的归纳法，我们得到的情况下，不包含任何简单的根的理想的减少。我们的第二个研究方向是Sommers和Tymoczko的另一个猜想。这关系到理想类型的安排的自由性。Sommers和Tymoczko证明，如果Weyl群是经典的或小秩的例外类型，则理想类型的每个排列都是自由的。阿部、巴拉卡特、坎茨、霍格和寺尾只是在最近才以统一的方式解决了所有类型的一般案件。这推广了Shapiro-Steinberg-Kostant的一个开创性公式，该公式指出W的正根的高度分布的对偶分区是W的指数集。在这里，我们建议研究各种更强的自由度性质的安排的理想型。已知外尔群W的反射排列本身总是无诱导的。很可能，对于所有理想类型的安排，情况也是如此。在一般情况下，我们概述了一个归纳的方法来解决这个问题，通过归纳的秩W.除了产生新的结果超平面安排，这些结果安排的理想型反过来又提供了新的见解的几何形状的某些子品种的旗品种与一个复杂的还原群G与Weyl群W，所谓的Hessenberg品种。