Arrangement of Hyperplanes - Real Arrangements

Real Arrangements

In real affine space, the complement is disconnected: it is made up of separate pieces called cells or regions or chambers, each of which is either a bounded region that is a convex polytope, or an unbounded region that is a convex polyhedral region which goes off to infinity. Each flat of A is also divided into pieces by the hyperplanes that do not contain the flat; these pieces are called the faces of A. The regions are faces because the whole space is a flat. The faces of codimension 1 may be called the facets of A. The face semilattice of an arrangement is the set of all faces, ordered by inclusion. Adding an extra top element to the face semilattice gives the face lattice.

In two dimensions (i.e., in the real affine plane) each region is a convex polygon (if it is bounded) or a convex polygonal region which goes off to infinity.

• As an example, if the arrangement consists of three parallel lines, the intersection semilattice consists of the plane and the three lines, but not the empty set. There are four regions, none of them bounded.
• If we add a line crossing the three parallels, then the intersection semilattice consists of the plane, the four lines, and the three points of intersection. There are eight regions, still none of them bounded.
• If we add one more line, parallel to the last, then there are 12 regions, of which two are bounded parallelograms.

A typical problem about an arrangement in n-dimensional real space is to say how many regions there are, or how many faces of dimension 4, or how many bounded regions. These questions can be answered just from the intersection semilattice. For instance, two basic theorems are that the number of regions of an affine arrangement equals (−1)np_A(−1) and the number of bounded regions equals (−1)np_A(1). Similarly, the number of k-dimensional faces or bounded faces can be read off as the coefficient of xn−k in (−1)n w_A (−x, −1) or (−1)nw_A(−x, 1).

Meiser (1993) designed a fast algorithm to determine the face of an arrangement of hyperplanes containing an input point.

Another question about an arrangement in real space is to decide how many regions are simplices (the n-dimensional generalization of triangles and tetrahedra). This cannot be answered based solely on the intersection semilattice.

A real linear arrangement has, besides its face semilattice, a poset of regions, a different one for each region. This poset is formed by choosing an arbitrary base region, B₀, and associating with each region R the set S(R) consisting of the hyperplanes that separate R from B. The regions are partially ordered so that R₁ ≥ R₂ if S(R₁, R) contains S(R₂, R). In the special case when the hyperplanes arise from a root system, the resulting poset is the corresponding Weyl group with the weak Bruhat order. In general, the poset of regions is ranked by the number of separating hyperplanes and its Möbius function has been computed (Edelman 1984).