A note on Tutte polynomials and Orlik–Solomon algebras

Abstract

Let $\text{A}{}_{\mathrm{<mtext>C</mtext>}}\text{={H}{}_1\text{,…,H}{}_{\mathrm{n}}\text{}}$ be a (central) arrangement of hyperplanes in $\text{C}{}^{\mathrm{d}}$ and $\text{M}\text{(}\text{A}{}_{\mathrm{<mtext>C</mtext>}}\text{)}$ the dependence matroid of the linear forms . The Orlik–Solomon algebra $\text{OS}\text{(}\text{M}\text{)}$ of a matroid $\text{M}$ is the exterior algebra on the points modulo the ideal generated by circuit boundaries. The graded algebra $\text{OS}\text{(}\text{M}\text{(}\text{A}{}_{\mathrm{<mtext>C</mtext>}}\text{))}$ is isomorphic to the cohomology algebra of the manifold . The Tutte polynomial $\text{T}{}_{\mathrm{<mtext>M</mtext>}}\text{(x,y)}$ is a powerful invariant of the matroid $\text{M}$. When $\text{M}\text{(}\text{A}{}_{\mathrm{<mtext>C</mtext>}}\text{)}$ is a rank 3 matroid and the θ_Hi are complexifications of real linear forms, we will prove that $\text{OS}\text{(}\text{M}\text{)}$ determines $\text{T}{}_{\mathrm{<mtext>M</mtext>}}\text{(x,y)}$. This result partially solves a conjecture of Falk.