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Hard Lefschetz for the Gorenstein ring A(M)

Determine whether, for every matroid M of rank d, the Gorenstein ring A(M) associated to the basis generating polynomial of M satisfies the Hard Lefschetz property in all degrees k ≤ d/2 for some choice of a ∈ ℝ^E_{>0}.

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Background

The thesis studies the Gorenstein ring A(M) attached to the basis generating polynomial of a matroid M and its Hodge-theoretic properties. Prior work by Murai–Nagaoka–Yazawa establishes Hard Lefschetz and Hodge–Riemann relations in degree 1. The cited conjecture asks for Hard Lefschetz in all degrees up to half the rank, which would significantly strengthen current results and align with broader Kähler package phenomena for matroidal cohomology rings.

References

Conjecture~\ref{conj:mny} Let $M = (E, \mcI)$ be a matroid of rank $d$. The ring $\A(M)$ satisfies $\HL_k$ for some $a \in \RRE_{> 0}$ for all $k \leq \frac{d}{2}$.

— Log-concavity in Combinatorics (2404.10284 - Yan, 16 Apr 2024) in Section 6 (The Gorenstein Ring associated to the Basis Generating Polynomial of a Matroid)