Orlik-Solomon algebras determine the Tutte polynomials of matroids of rank at most three

Abstract

Let $\mathcal{M}$ be a simple matroid of rank $r$ on the ground set $[n]=\{1,\dots ,n\}$, let $T_{\mathcal{M}}(x,y)$ denote its Tutte polynomial, and let $\mathrm{OS}(\mathcal{M})$ denote its Orlik-Solomon $\mathbb{C}$-algebra. We say that $\mathrm{OS}(\mathcal{M})$ determines $T_{\mathcal{M}}(x,y)$ if, for every simple matroid $\mathcal{N}$, an isomorphism $\mathrm{OS}(\mathcal{M})\cong \mathrm{OS}(\mathcal{N})$ of $\mathbb{C}$-algebras, without regard to the grading, implies $T_{\mathcal{M}}(x,y)=T_{\mathcal{N}}(x,y)$. Falk conjectured that the Orlik-Solomon $\mathbb{C}$-algebra of every simple matroid of rank three determines its Tutte polynomial. We prove that this holds for every simple matroid of rank at most three, thereby resolving Falk's conjecture.

arXiv comment19 pages