Combinatorics of inductively factored arrangements

A real arrangement of hyperplanes is a finite family A of hyperplanes through the origin in a finite-dimensional real vector space V = R1. arrangement A of hyperplanes is said to be factored if there exists a partition Π = (Π1, …, Π1) of A into l disjoint subsets such that the Orlik-Solomon algebra of A factors according to this partition. A real arrangement A of hyperplanes is called inductively factored if it is factored and there exists a hyperplane H ϵ A such that the arrangement obtained by removing H from A and the arrangement on H consisting of all intersections of elements of A— H with H are both inductively factored. ber of A is a connected component of the complement of A. For a fixed base chamber, we may define a partial order on the set of chambers according to their combinatorial distances from the base chamber. Given an inductive factorization Π = (Π1,…(Π1 and a base chamber C0, we define the counting map of A with respect to C0 as a morphism from the poset of chambers to the poset 0, 1,…,|Π1| x·.x (0, 1,…,Π1). We prove that, for a suitable base chamber, the counting map is a bijection, the poset of chambers is a lattice, and its rank-generating function has a nice factorization. sider the dual decomposition of the sphere of V induced by A. We prove that, if A is inductively factored, then this cellular decomposition can be viewed as a decomposition of the boundary of the l-cube [0,|Π1|]x·.x]0, |Π1|[ by cubic cells.