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Orlik-Solomon algebras determine the Tutte polynomials of matroids of rank at most three
Abstract
Let ℳ be a simple matroid of rank r on the ground set [n]={1,…,n}, let Tℳ(x,y) denote its Tutte polynomial, and let OS(ℳ) denote its Orlik-Solomon ℂ-algebra. We say that OS(ℳ) determines Tℳ(x,y) if, for every simple matroid 𝒩, an isomorphism OS(ℳ)≅OS(𝒩) of ℂ-algebras, without regard to the grading, implies Tℳ(x,y)=T𝒩(x,y). Falk conjectured that the Orlik-Solomon 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.
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BibTeX
@misc{luo2026orliksolomon,
title = {{Orlik-Solomon algebras determine the Tutte polynomials of matroids of rank at most three}},
author = {Jianxuan Luo and Tingzeng Wu and Hong-Jian Lai},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.21812},
eprint = {2610.21812},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.21812},
}
RIS
TY - UNPB
TI - Orlik-Solomon algebras determine the Tutte polynomials of matroids of rank at most three
AU - Jianxuan Luo
AU - Tingzeng Wu
AU - Hong-Jian Lai
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.21812
UR - https://arxiv.org/abs/2610.21812
AB - Let ℳ be a simple matroid of rank r on the ground set [n]={1,…,n}, let T_ℳ(x,y) denote its Tutte polynomial, and let OS(ℳ) denote its Orlik-Solomon ℂ-algebra. We say that OS(ℳ) determines T_ℳ(x,y) if, for every simple matroid 𝒩, an isomorphism OS(ℳ)≅OS(𝒩) of ℂ-algebras, without regard to the grading, implies T_ℳ(x,y)=T_𝒩(x,y). Falk conjectured that the Orlik-Solomon 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.
ER -