论文标题
通过定向的矩形的简单超平面布置的一致性正态性
Congruence Normality of Simplicial Hyperplane Arrangements via Oriented Matroids
论文作者
论文摘要
1971年，格伦鲍姆（Grünbaum）首先给出了简单的超平面布置目录。这些布置自然概括了有限的coxeter排列和通过区域poset的弱顺序。对于简单的安排，实际上是晶格的区域。我们更新Grünbaum的目录，为所有已知的零星简单安排提供正常和不变，最多37行。已知弱顺序是一致的正常，并且可以使用称为碎片的多面体锥确定简单排列的一致性正态性。 在本文中，我们通过确定始终/有时/永远不会导致区域的一致性正常晶格来为简单超平面排列的目录提供其他结构。为此，我们使用定向的矩阵将碎片重新铸造为共vector，以确定大型超平面布置的一致性正态性。由于这种方法的结果，我们特别得出了哪些零星简单排列区域的晶格始终是正常的。我们还表明，从任何等级的有限Weyl群的区域的晶格都是一致的。
A catalogue of simplicial hyperplane arrangements was first given by Grünbaum in 1971. These arrangements naturally generalize finite Coxeter arrangements and the weak order through the poset of regions. For simplicial arrangements, posets of regions are in fact lattices. We update Grünbaum's catalogue, providing normals and invariants for all known sporadic simplicial arrangements with up to 37 lines. The weak order is known to be congruence normal, and congruence normality for simplicial arrangements can be determined using polyhedral cones called shards. In this article, we provide additional structure to the catalogue of simplicial hyperplane arrangements by determining which arrangements always/sometimes/never lead to congruence normal lattices of regions. To this end, we use oriented matroids to recast shards as covectors to determine congruence normality of large hyperplane arrangements. As a consequence of this approach we derive in particular which lattices of regions of sporadic simplicial arrangements of rank 3 are always congruence normal. We also show that lattices of regions from finite Weyl groupoids of any rank are congruence normal.
