Tower-Equivalence for Irreducible Spetsial Groups

Definitions and mathematical statement

A class function on a group $W$, with values in a field $k$, is a function $\varphi\colon W\to k$ satisfying $\varphi(xwx^{-1})=\varphi(w)$ for all $x,w\in W$. The supplied excerpt does not specify the intended field $k$, the precise vector space of class functions, the operator $f$, or the definition of tower equivalence.

To display the conjecture without supplying these missing definitions, let $V_W$ denote the intended class-function space, let $f_W$ denote the source's operator on that space, and write $\sim_{\mathrm{tower}}$ for the source's tower-equivalence relation. These are placeholders for source-defined objects, not independently specified constructions. Set \[ K_{\mathrm{tower}}(W):=\{\varphi\in V_W: \varphi\sim_{\mathrm{tower}}0\}. \] The excerpt calls this set the kernel of tower equivalence and describes it as a vector space.

The proposed statement, identified in the excerpt as Conjecture 1, is: for every irreducible spetsial group $W$ other than $G_{32}$, \[ \operatorname{im}\left(\operatorname{Id}_{V_W}-f_W\right) = K_{\mathrm{tower}}(W). \] Equivalently, every $\varphi\in V_W$ tower-equivalent to zero should admit $\psi\in V_W$ such that $\varphi=\psi-f_W(\psi)$, together with the inclusion in the other direction reported by the source. The excerpt attributes that inclusion to an earlier, unsupplied first item and reports computational support for the equality. It does not assert the equality for $G_{32}$, nor state that $G_{32}$ is a counterexample. The definitions of an irreducible spetsial group and the identification of $G_{32}$ are also not provided. Thus this is a conditional transcription of the intended conjecture; the missing source definitions must be recovered before it becomes a fully specified mathematical statement.

Short English statement

For each irreducible spetsial group other than $G_{32}$, the conjecture says that the class functions tower-equivalent to zero are exactly those of the form $\psi-f(\psi)$. The supplied excerpt leaves the operator, tower equivalence, and ambient class-function space undefined. Its dated curation labels the conjecture open and reports one inclusion as proved; these assessments have not been independently verified.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30005516$, title "Tower-Equivalence for Irreducible Spetsial Groups," collection "Oberwolfach Reports Open Problems," exact source problem code catalogue entry. The record gives the proposed year as 2023 and supplies no proposer attribution.
• Representations of Finite Groups (2023), Oberwolfach Reports, pp. 1031--1084, DOI 10.4171/owr/2023/19: https://doi.org/10.4171/owr/2023/19. The supplied passage identifies the assertion as Conjecture 1. Oberwolfach Reports document mathematical workshops and their open-problem discussions; the record provides no narrower page reference for this passage.
• Supplied literature reference labelled "Michel primary paper": https://doi.org/10.1090/bproc/167. The record supplies no fuller bibliographic information or precise supporting location.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open, reports that the image of $\operatorname{Id}-f$ is contained in the kernel of tower equivalence, and describes equality for irreducible spetsial groups other than $G_{32}$ as an explicit open question. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment; its status and cited claims require independent verification.
• The registry assigns the statement status corrected_verified and says it was reconstructed and checked against the official Oberwolfach report. This is supplied provenance, not verification performed for this draft.