Splitting Subgroups of General Linear Supergroups

Definitions and mathematical statement

Write $G=\operatorname{GL}(m\mid n)$, the general linear supergroup on a super vector space of even dimension $m$ and odd dimension $n$, over a base field that the supplied record does not specify. The supplied literature assessment identifies the designated subgroup as \[ H\cong\operatorname{SL}(1\mid1)^{n}. \] Here the right-hand side denotes a direct product of $n$ copies. Its embedding into $G$ is not supplied. The term "splitting subgroup" is also not defined in the record.

Conjecture as supplied, conditional on the missing conventions. For every splitting subgroup $K\leq G$, there exists a subgroup $H'\leq K$ that is conjugate to the designated subgroup $H$ inside $G$: \[ \forall K\leq G,\qquad K\text{ is splitting} \ \Longrightarrow\ \exists H'\leq K\text{ with }H'\text{ conjugate to }H\text{ in }G. \] A fully specified statement requires the base field, the allowed values and relation of $m,n$, the embedding of $H$, the definition of splitting, the category of subgroups, and the intended meaning of conjugacy. These are unresolved in the supplied record; the displayed statement retains the source's target without choosing those conventions.

The supplied dated curation reports that $H$ is proved splitting and labels the overall problem partially solved, while reporting the universal containment conjecture as still open. These are supplied assessments, not independently verified mathematical or literature conclusions.

Short English statement

The conjecture asks whether every splitting subgroup of $\operatorname{GL}(m\mid n)$ contains a conjugate of a designated subgroup $H\cong\operatorname{SL}(1\mid1)^n$. The supplied record leaves essential definitions and ambient hypotheses unspecified, so this is a conditional statement-only draft requiring mathematical review.

Sources

• Local UnsolvedMath registry, supplied complete record: permanent ID $30004886$, title "Splitting Subgroups of General Linear Supergroups," exact source problem code catalogue entry. The proposed year is recorded as 2021; no proposer is identified.
• Supplied report citation: Enveloping Algebras and Geometric Representation Theory (2022), Oberwolfach Reports, pp. 2827--2891, DOI 10.4171/owr/2021/52. https://doi.org/10.4171/owr/2021/52. The extracted passage includes the page marker 2841.
• Supplied literature source: Serganova--Sherman. https://arxiv.org/abs/2206.07693.
• Supplied literature source: Entova-Aizenbud--Serganova--Sherman. https://doi.org/10.1016/j.jpaa.2026.108277.
• Supplied literature source: OWR report. https://doi.org/10.4171/owr/2021/52.
• Literature assessment dated exactly 2026-08-22: supplied dated curation, not independently verified. It reports that $H=\operatorname{SL}(1\mid1)^n$ is proved splitting and that the conjecture concerning every splitting subgroup remains explicitly open in the 2026 literature. The background requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports that the official report was checked directly and extraction artifacts removed. This is a supplied source-verification claim; no independent source visit or verification is asserted here.