Nonnegativity of Fusion-System Multiplicities

Definitions and mathematical statement

Let $p$ be a prime, let $S$ be a finite $p$-group, and let $\mathcal F$ be a saturated fusion system on $S$, using the standard terminology of fusion systems: its objects are the subgroups of $S$, its morphisms are injective group homomorphisms, it contains the homomorphisms induced by conjugation in $S$, its morphisms factor as isomorphisms followed by inclusions, and it satisfies the Sylow and extension saturation axioms. Let $d\in\mathbb Z_{\geq 0}$.

The supplied conjecture is the universal nonnegativity assertion \[ m(\mathcal F,d)\geq 0 \qquad\text{for every saturated fusion system }\mathcal F \text{ and every }d\in\mathbb Z_{\geq 0}. \] The prime and underlying finite group are implicit in the source terminology; no restriction on the prime is stated.

Unresolved definition. The supplied record does not define $m(\mathcal F,d)$, specify its codomain, or give a formula identifying this invariant. The displayed inequality therefore preserves the source assertion but cannot yet be a self-contained mathematical conjecture. Its precise interpretation requires the intended definition of $m$ and any hypotheses or auxiliary data belonging to that definition. In particular, the title alone does not justify interpreting $m$ as the cardinality of a set or as an actual multiplicity in a representation.

Short English statement

The conjecture asserts that the source's quantity $m(\mathcal F,d)$ is nonnegative for every saturated fusion system and every nonnegative integer $d$, including $d=0$. The supplied record omits the definition of $m$, so the mathematical target remains incomplete and requires human review.

Sources

• Local UnsolvedMath registry, supplied record with permanent ID $30005514$, title Nonnegativity of Fusion-System Multiplicities, source problem code catalogue entry; collection: Oberwolfach Reports Open Problems. The record gives the proposed year as 2023 and names no proposer.
• Supplied primary-source citation: Representations of Finite Groups (2023), Oberwolfach Reports, DOI 10.4171/owr/2023/19, pp. 1031--1084. https://doi.org/10.4171/owr/2023/19. The supplied original excerpt explicitly conjectures the displayed inequality, but does not define $m$.
• Kessar--Malle--Semeraro, weight conjectures for ell-compact groups and spetses, supplied literature reference: https://arxiv.org/abs/2003.07213.
• Semeraro, A 2-compact group as a spets, supplied literature reference: https://doi.org/10.1080/10586458.2021.1926004.
• Supplied dated literature curation, checked 2026-08-22: status labelled open; the assessment reports computational support and proofs in selected spets and fusion-system families, while reporting no established proof for all saturated fusion systems and nonnegative integers $d$. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment; its time-sensitive status and cited claims require independent verification.
• The registry labels the statement corrected_verified and reports that it was reconstructed and checked against the official Oberwolfach report. These are supplied provenance claims, not verification performed for this draft.