Equivariant Log-Concavity of Orlik–Solomon Algebras

Definitions and mathematical statement

The supplied record leaves the coefficient field unspecified and does not explicitly require the matroid to have a finite ground set. The following gives the conventional finite-matroid formulation with an explicit coefficient parameter; the intended field and scope remain unresolved.

Fix a field $k$, and let $M$ be a matroid on a finite ground set $E$. Let $W$ be a finite group acting on $E$ by matroid automorphisms, meaning permutations preserving independent subsets. Form the graded exterior algebra $\Lambda_k(e_a:a\in E)$, with each generator in degree one. For distinct $a_1,\ldots,a_m$, define \[ \partial(e_{a_1}\wedge\cdots\wedge e_{a_m}) =\sum_{j=1}^{m}(-1)^{j-1}e_{a_1}\wedge\cdots\wedge\widehat{e_{a_j}}\wedge\cdots\wedge e_{a_m}, \] where the hat denotes omission and the empty product is $1$. The Orlik--Solomon algebra is \[ OS^*(M;k)=\Lambda_k(e_a:a\in E)/I_M, \] where $I_M$ is the homogeneous ideal generated by these boundaries for all nonempty dependent subsets $\{a_1,\ldots,a_m\}$ of $E$. The action $w(e_a)=e_{w(a)}$ induces a graded $k$-linear action of $W$. Write $OS^j(M;k)$ for its degree-$j$ component, and adopt the explicit boundary convention $OS^j(M;k)=0$ for $j<0$ or $j>\operatorname{rank}(M)$.

The supplied question asks whether, for every such matroid and finite symmetry group, and for every integer $i\geq 0$, there exists an injective $W$-equivariant $k$-linear map \[ OS^{i-1}(M;k)\otimes_k OS^{i+1}(M;k) \lhook\joinrel\longrightarrow OS^i(M;k)\otimes_k OS^i(M;k), \] with diagonal action $w(x\otimes y)=(wx)\otimes(wy)$ on each tensor product. This is containment as an abstract subrepresentation; no particular embedding is specified. The coefficient parameter is exposed to make the statement meaningful, and does not assert that the source intended a claim over every field. The source's phrase "for every $i$" is interpreted using the stated zero-degree conventions.

Short English statement

For each degree, does the tensor square of the Orlik--Solomon representation contain the tensor product of its two neighboring degrees as a representation of every finite matroid symmetry group? The coefficient field and the intended scope beyond finite matroids require clarification. The supplied record labels the problem open; that status is not independently verified here.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30004237$, title "Equivariant Log-Concavity of Orlik--Solomon Algebras," exact source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The record supplies proposed year $2019$ and no named proposer.
• Supplied report citation: Geometric, Algebraic, and Topological Combinatorics (2020), Oberwolfach Reports, pp. 2395--2472, DOI 10.4171/owr/2019/39. https://doi.org/10.4171/owr/2019/39. The supplied background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature reference attributed to Matherne, Miyata, Proudfoot, and Ramos: https://doi.org/10.1093/imrn/rnab316. No article title or precise supporting passage is supplied.
• Supplied literature reference labeled "Oberwolfach report": https://doi.org/10.4171/owr/2019/39.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the general equivariant Orlik--Solomon log-concavity inclusion a conjecture and states that published work gives only low-degree computer-assisted cases for the braid arrangement. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment. The supplied background expressly calls for independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and explains that mathematical content was preserved while notation, typography, and sentence structure were repaired for publication. This supplied editorial label does not establish independent mathematical or source verification of this draft.