Characteristic and Ehrhart Polynomials

Abstract

Let A be a subspace arrangement and let χ(A,t) be the characteristic polynomial of its intersection lattice L( A). We show that if the subspaces in A are taken from \({\mathcal{L}}({\mathcal{B}}_n)\), where \({\mathcal{B}}_n\) is the type B Weyl arrangement, then χ(A,t) counts a certain set of lattice points. One can use this result to study the partial factorization of χ(A,t) over the integers and the coefficients of its expansion in various bases for the polynomial ring R[t]. Next we prove that the characteristic polynomial of any Weyl hyperplane arrangement can be expressed in terms of an Ehrhart quasi-polynomial for its affine Weyl chamber. Note that our first result deals with all subspace arrangements embedded in \({\mathcal{B}}_n\) while the second deals with all finite Weyl groups but only their hyperplane arrangements.