Strong equivariant log-concavity conjecture for Orlik–Solomon algebras

Let $M$ be a matroid of positive rank, and let $\Gamma$ be a group acting on its ground set while preserving the collection of independent sets. Let $A(M)$ be the Orlik–Solomon algebra, defined as the quotient of the exterior algebra on generators indexed by the ground set, and let $B(M)$ be the reduced Orlik–Solomon algebra, the subalgebra generated by differences of those generators. A graded representation is strongly equivariantly log concave when its graded tensor products satisfy the subrepresentation inclusions defined in the paper.

Orlik–Solomon strong log-concavity conjecture. The Orlik–Solomon algebra $A(M)$ and the reduced Orlik–Solomon algebra $B(M)$ are strongly equivariantly log concave.

When $\Gamma$ is trivial, this reduces to the known log-concavity theorem for characteristic and reduced characteristic polynomials. The equivariant statement for $A(M)$ was previously conjectured and is known for uniform matroids with the full permutation group; the general assertion, including $B(M)$, remains open.