Broken circuit complexes: Factorizations and generalizations

Abstract

Motivated by the question of when the characteristic polynomial of a matroid factorizes, we study join-factorizations of broken circuit complexes and rooted complexes (a more general class of complexes). Such factorizations of complexes induce factorizations not only of the characteristic polynomial but also of the Orlik-Solomon algebra of the matroid. The broken circuit complex of a matroid factors into a multiple join of zero-dimensional subcomplexes for some linear order of the ground set if and only if the matroid is supersolvable. Several other characterizations of this case are derived. It is shown that whether a matroid is supersolvable can be determined from the knowledge of its 3-element circuits and its rank alone. Also, a supersolvable matroid can be reconstructed from the incidences of its points and lines. The class of rooted complexes is introduced, and much of the basic theory for broken circuit complexes is shown to generalize. Complete factorization of rooted complexes is, however, possible also for non-supersolvable matroids, still inducing factorization of the characteristic polynomial.