A note on Tutte polynomials and Orlik–Solomon algebras

Let AC={H1,…,Hn} be a (central) arrangement of hyperplanes in Cd and M(AC) the dependence matroid of the linear forms {θHi∈(Cd)∗: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 graded algebra OS(M(AC)) is isomorphic to the cohomology algebra of the manifold M=Cd⧹⋃H∈ACH. The Tutte polynomial TM(x,y) is a powerful invariant of the matroid M. When M(AC) is a rank 3 matroid and the θHi are complexifications of real linear forms, we will prove that OS(M) determines TM(x,y). This result partially solves a conjecture of Falk.