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Classify when Orlik–Solomon algebras are quadratic

Determine all geometric lattices L for which the Orlik–Solomon algebra OS(L) is quadratic, equivalently when the defining ideal of relations of OS(L) is generated in degree 2.

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Background

The morphism I_L: MD(L) → OS(L) constructed in Proposition \ref{propmor} factors through the quadratic envelope of OS(L). Consequently, if OS(L) is not quadratic, I_L cannot be a quasi-isomorphism. This highlights the importance of identifying exactly which Orlik–Solomon algebras are quadratic.

The authors explicitly note that this classification is an open question in the literature, with references to Falk (2001) and Yuzvinsky (2001).

References

Determining which Orlik--Solomon algebras are quadratic is an interesting open question (see Section 4 Section 6) which we will avoid in this article.

— Matroid complexes and Orlik-Solomon algebras (2506.15048 - Coron, 18 Jun 2025) in Section 4 (The combinatorics), after Proposition \ref{propmor}