A class of hypergraph arrangements with shellable intersection lattice
type: Item
instance of: scholarly article
title: A class of hypergraph arrangements with shellable intersection lattice
zbMATH Open document ID: 0924.05049
author: Dmitry N. Kozlov
published in: Journal of Combinatorial Theory. Series A
publication date: 1999-11-03T00:00:00Z
review text: A hypergraph \(H\) on the ground set \([n] := \{ 1, \ldots, n\}\) is given by a set of incomparable subsets of \([n]\). To each hypergraph \(H\) an arrangement \(A_H\) of linear subspaces in real \(n\)-space is given by the linear subspaces \(U_S\) for \(S \in H\) where \(U_S\) is the set of all \(n\)-tuples of real numbers that coincide on the coordinates in \(S\). This construction was first suggested by \textit{A. Björner} [In: Joseph, A. (ed.) et al., First European congress of mathematics (ECM), Paris, France, July 6-10, 1992. Volume I: Invited lectures (Part 1). Basel: Birkhäuser. Prog. Math. 119, 321-370 (1994; Zbl 0844.52008)]. It generalizes various types of arrangements that had been considered before. Notably, the arrangement of reflecting hyperplanes of the symmetric group on \(n\) letters is the hypergraph arrangement associated to the complete graph on \(n\) vertices (i.e. the set of all two element subsets of \([n]\)). In the paper the author studies a class of hypergraph arrangements that generalize all previously introduced ones. He shows that the intersection lattice of an arrangement of this class is EL-shellable. Since all the intersection lattices can be regarded as join-sublattices of the lattice of set partitions of \([n]\) the result in particular provides a large class of shellable subposets of the partition lattice.
Mathematics Subject Classification ID: 05C65
Mathematics Subject Classification ID: 06A11
Mathematics Subject Classification ID: 52C35
zbMATH DE Number: 1278832
zbMATH Keywords: arrangement of linear subspaces
zbMATH Keywords: EL-shellability
zbMATH Keywords: partition lattice
zbMATH Keywords: hypergraph
reviewed by: Volkmar Welker
MaRDI profile type: Publication
full work available at URL: jcta.1998.2937
OpenAlex ID: W2092859683
cites work: Shellable and Cohen-Macaulay Partially Ordered Sets
cites work: Q4871782
cites work: Q4324358
cites work: Bruhat order of Coxeter groups and shellability
cites work: On Lexicographically Shellable Posets
cites work: The homology of ``\(k\)-equal'' manifolds and related partition lattices
cites work: General lexicographic shellability and orbit arrangements
cites work: Q3827224
cites work: The homology representations of the 𝑘-equal partition lattice
DOI: 10.1006/JCTA.1998.2937
Recommended article: Hyperplane arrangements with a lattice of regions
Recommended article: Q2980942
Recommended article: Lattice Theory of the Poset of Regions
Recommended article: Q4324358
Recommended article: Q5432049
Recommended article: Q4003206
Recommended article: Q3161177
Recommended article: Q3815418
Recommended article: The characteristic polynomials of subarrangements of Coxeter arrangements
Recommended article: A Note on Discriminantal Arrangements
description: scientific article; zbMATH DE number 1278832