Modular elements in the lattice L(A) when A is a real reflection arrangement Original Research Article

Let W be a real reflection group, and let LW denote the lattice consisting of all possible intersections of reflecting hyperplanes of reflections in W. Let pW(t) be the characteristic polynomial of LW. To every element X of LW there corresponds a parabolic subgroup of W denoted by Gal(X). If W is irreducible, we show that an element X of LW is modular if and only if pGal(X)(t) divides pW(t). This characterization is not true if W is not irreducible. Also, we show that if W is neither An nor Bn, then the only modular elements are 0, 1 and the atoms of LW.