A class of hypergraph arrangements with shellable intersection lattice
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.
scientific article; zbMATH DE number 1278832
scientific article