Nice Graphical Arrangements Are Chordal

Abstract: Supersolvable lattices, introduced by Stanley in 1972, are geometric lattices that contain a maximal modular chain. The characteristic polynomials of these lattices can be factored into a product of linear terms. Moreover, he showed that a graphical arrangement is supersolvable if and only if the graph is chordal. In 1992, Terao defined a class of hyperplane arrangements with a ``nice partition''. The Orlik-Solomon algebra of these arrangements admits a tensor product of desirable submodules of rank one, which leads to a factorization of its Poincar\'e polynomial into linear terms. In this paper, we prove that a graphical arrangement has a nice partition if and only if the graph is chordal. Moreover, for any chordal graph, every nice partition of its graphical arrangement can be induced by a maximal modular chain of the corresponding supersolvable lattice.