Hyperplane arrangements with a lattice of regions

Abstract

A hyperplane arrangement is a finite set of hyperplanes through the origin in a finite-dimensional real vector space. Such an arrangement divides the vector space into a finite set of regions. Every such region determines a partial order on the set of all regions in which these are ordered according to their combinatorial distance from the fixed base region. We show that the base region is simplicial whenever the poset of regions is a lattice and that conversely this condition is sufficient for the lattice property for three-dimensional arrangements, but not in higher dimensions. For simplicial arrangements, the poser of regions is always a lattice. In the case of supersolvable arrangements (arrangements for which the lattice of intersections of hyperplanes is supersolvable), the poset of regions is a lattice if the base region is suitably chosen. We describe the geometric structure of such arrangements and derive an expression for the rank-generating function similar to a known one for Coxeter arrangements. For arrangements with a lattice of regions we give a geometric interpretation of the lattice property in terms of a closure operator defined on the set of hyperplanes. The results generalize to oriented matroids. We show that the adjacency graph (and poset of regions) of an arrangement determines the associated oriented matroid and hence in particular the lattice of intersections.

Lattice and order properties of the poset of regions in a hyperplane arrangement

“…The doublings in Theorem 1 correspond to adjoining hyperplanes one by one to form the arrangement. Björner, Edelman and Ziegler [2] showed that the poset of regions (with respect to a canonical base region) of a supersolvable hyperplane arrangement is a lattice. The proof given here of Theorem 1 provides a different proof of the lattice property, but uses several key observations made in [2].…”

“…Björner, Edelman and Ziegler [2] showed that the poset of regions (with respect to a canonical base region) of a supersolvable hyperplane arrangement is a lattice. The proof given here of Theorem 1 provides a different proof of the lattice property, but uses several key observations made in [2]. that one can always take P to be Irr(Con(L)), the poset of join-irreducibles of the congruence lattice of L. The usefulness of Theorem 4 is that it makes possible a non-inductive proof of the congruence normality of a class of posets.…”

“…The supersolvable arrangements include Coxeter arrangements of types A and B, as well as graphic arrangements associated to chordal graphs. For more information on hyperplane arrangements, see [2,9,19].…”

“…Examples of posets of regions are given later, in the figures. For more details on this poset, see [2,9]. For now, note only the following: in P(A, B), the region B is the unique minimal element, and complementation of separating sets is an antiautomorphism which is denoted by R → −R.…”

“…For an arbitrary hyperplane arrangement A and base region B there is the following "change of base" isomorphism. Another useful result from [2] characterizes the join operation when P(A, B) is a lattice. There is a closure operator defined on subsets of A in Section 5 of [2].…”