arrangementsreflecting

Arrangements reflecting, more commonly known as reflection arrangements or Coxeter arrangements, are a class of hyperplane arrangements arising from finite reflection groups. Given a real vector space V of dimension n and a finite Coxeter group W acting on V by orthogonal transformations, the reflection arrangement A(W) is the collection of all reflecting hyperplanes corresponding to the reflections in W. Equivalently, A(W) consists of the hyperplanes fixed by reflections in W, and all hyperplanes pass through the origin, making A(W) a central arrangement.

The defining data of a reflection arrangement is the underlying root system or the Coxeter system associated with W. The group W acts on the set of hyperplanes with a high degree of symmetry, and the arrangement encodes important combinatorial and geometric information about W. The structure of A(W) is studied via its intersection lattice L(A), formed by all possible intersections of hyperplanes in A(W), partially ordered by reverse inclusion.

Key invariants include the characteristic polynomial χ_A(t) of the arrangement, which for a finite Coxeter group W factors as χ_A(t) = ∏ (t − e_i), where e_i are the exponents of W. Consequently, the number of regions of the real arrangement equals the order |W|. The complexified complement M(A) = V_C \ ∪ H_C has a cohomology ring whose Poincaré polynomial is ∏ (1 + e_i t). Reflection arrangements are central to the theory of freeness: A(W) is a free arrangement with a basis of derivations whose degrees are the exponents e_i.

Prominent examples include type A_n (the braid arrangement, hyperplanes x_i = x_j), type B_n/C_n, type D_n, and the exceptional types E6, E7, E8, F4, G2, as well as dihedral types I2(m). These arrangements connect to root systems, Lie theory, and numerous areas of combinatorics and geometry.