Combinatorics of inductively factored arrangements

Abstract

A real arrangement of hyperplanes is a finite family $\text{A}$ of hyperplanes through the origin in a finite-dimensional real vector space V = R¹.

A real arrangement $\text{A}$ of hyperplanes is said to be factored if there exists a partition Π = (Π₁, …, Π₁) of $\text{A}$ into l disjoint subsets such that the Orlik-Solomon algebra of $\text{A}$ factors according to this partition. A real arrangement $\text{A}$ of hyperplanes is called inductively factored if it is factored and there exists a hyperplane H ϵ $\text{A}$ such that the arrangement obtained by removing H from $\text{A}$ and the arrangement on H consisting of all intersections of elements of $\text{A}$— H with H are both inductively factored.

A chamber of $\text{A}$ is a connected component of the complement of $\text{A}$. 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 $\text{A}$ 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.

We consider the dual decomposition of the sphere of V induced by $\text{A}$. We prove that, if $\text{A}$ 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.