Faces of a hyperplane arrangement enumerated by ideal dimension, with application to plane, plaids, and Shi
An arrangement of hyperplanes \(\mathcal H\) in real affine space divides the affine space into open cells referred to as the faces of \(\mathcal H\). Well-known formulae express the number of faces of each dimension. When considering the complex complement arrangement, these numbers have a cohomological interpretation. In this paper, the author considers a refinement of the problem of enumerating faces as proposed by Richard Ehrenborg. He considers the ideal dimension of each face \(F\) which is the dimension of the topological closure of \(F\) as it intersects the infinite hyperplane. The author compares this definition of ideal dimension to the definition of ideal dimension given by the level of a set as introduced by Ehrenborg. From the author's abstract: ``We obtain a formula for the number of faces of a real hyperplane arrangement having given dimension and ideal dimension. We apply this formula to the plane, the plaids, which are arrangements of parallel families in general position, and to affinographic arrangements.'' The author concludes by considering the complex complement complex and asking whether there is a cohomological interpretation.
scientific article; zbMATH DE number 1960057