A note on Tutte polynomials and Orlik-Solomon algebras

Let AC = {H1, . . . , Hn} be a (central) arrangement of hyperplanes in C. Let M(AC) be the dependence matroid of the linear forms {θHi ∈ (C ) : Ker(θHi ) = Hi}. The Orlik-Solomon algebra OS(M) of a matroid M is the exterior algebra on the points modulo the ideal generated by circuit boundaries. The algebra OS(M(AC)) is isomorphic to the cohomology algebra of the manifold M = C \ ⋃ H∈AC H. The Tutte polynomial TM(x, y) is a powerful invariant of the matroid M. When M(AC) is a rank three matroid and the θHi are the complexification of real linear forms, we will prove that OS(M) determines TM(x, y). This result is sharp: we give an example of two connected rank four matroids, realizable over R, with isomorphic Orlik–Solomon algebras but different Tutte polynomials.