Year: 2026
Type: preprint
Abstract: Let $\mathcal{M}$ be a simple matroid of rank $r$ on the ground set $[n]=\{1,\ldots,n\}$, 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.
Source: arXiv (Cornell University)
Authors: Luo, Jianxuan, Wu, Tingzeng, Lai, Hong-Jian
Institutions: Xiamen University, Guangdong Polytechnic Normal University
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Award: G3795835246
OpenAlex ID: W7222500028