A note on Tutte polynomials and Orlik–Solomon algebras

Abstract

Let A C ={H 1,…,H n} be a (central) arrangement of hyperplanes in C d and M( A C ) the dependence matroid of the linear forms {θ H i ∈( C d) ∗: Ker(θ H i )=H i} . 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( A C )) is isomorphic to the cohomology algebra of the manifold M= C d⧹⋃ H∈ A C H . The Tutte polynomial T M (x,y) is a powerful invariant of the matroid M . When M( A C ) is a rank 3 matroid and the θ H i are complexifications of real linear forms, we will prove that OS( M) determines T M (x,y) . This result partially solves a conjecture of Falk.