Faces of a hyperplane arrangement enumerated by ideal dimension, with application to plane, plaids, and Shi
type: Item
instance of: scholarly article
title: Faces of a hyperplane arrangement enumerated by ideal dimension, with application to plane, plaids, and Shi
zbMATH Open document ID: 1041.52021
DOI: 10.1023/A:1024029318990
author: Thomas Zaslavsky
published in: Geometriae Dedicata
publication date: 2003-08-06T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 52C35
Mathematics Subject Classification ID: 05A15
Mathematics Subject Classification ID: 05B35
Mathematics Subject Classification ID: 05C22
zbMATH DE Number: 1960057
zbMATH Keywords: additive real gain graph
zbMATH Keywords: affinographic arrangement
zbMATH Keywords: arrangement of hyperplanes
zbMATH Keywords: characteristic polynomial
zbMATH Keywords: extended Catalan arrangement
zbMATH Keywords: extended Linial arrangement
zbMATH Keywords: extended Shi arrangement
zbMATH Keywords: face count
zbMATH Keywords: geometric semilattice
zbMATH Keywords: graphic lift matroid
zbMATH Keywords: ideal dimension
zbMATH Keywords: plaid arrangement
zbMATH Keywords: Whitney-number polynomial
reviewed by: Kelly J. Pearson
MaRDI profile type: Publication
full work available at URL: a:1024029318990
OpenAlex ID: W221320767
description: scientific article; zbMATH DE number 1960057