Specht Modules with Abelian Vertices

Definitions and mathematical statement

Let $p$ be a prime and let $k$ be a field of characteristic $p$. In the symmetric-group setting indicated by the supplied workshop provenance, write $\mathfrak S_n$ for the symmetric group on $n$ letters. For a partition $\lambda\vdash n$, let $S_k^\lambda$ be the Specht $k\mathfrak S_n$-module: inside the permutation module on $\lambda$-tabloids, it is the submodule spanned by the polytabloids \[ e_t=\sum_{\sigma\in C_t}\operatorname{sgn}(\sigma)\,\sigma\{t\}, \] where $t$ ranges over tableaux of shape $\lambda$, $\{t\}$ is its row tabloid, and $C_t$ is its column stabilizer.

A module is indecomposable if it is nonzero and is not a direct sum of two nonzero submodules. For a finite group $G$ and an indecomposable finite-dimensional $kG$-module $M$, a vertex is a $p$-subgroup $Q\leq G$, minimal under inclusion among those for which $M$ is isomorphic to a direct summand of \[ \operatorname{Ind}_Q^G\operatorname{Res}_Q^G M. \] Vertices are conjugate in $G$, so whether a vertex is abelian is independent of its choice. The trivial subgroup is allowed as an abelian vertex.

Classification problem. For a field $k$ of prime characteristic $p$, determine all partitions $\lambda\vdash n$, across the symmetric-group degrees under consideration, such that $S_k^\lambda$ is indecomposable and a vertex $Q$ of $S_k^\lambda$ satisfies \[ [Q,Q]=\{1\}. \] The requested classification should give necessary and sufficient conditions, with any dependence on $p$ and $k$ stated explicitly.

Unresolved source qualifications. The supplied statement imposes no algebraic-closure or splitting-field hypothesis on $k$, and none is assumed here. It does not explicitly name the ambient groups; the interpretation as ordinary Specht modules for symmetric groups follows the supplied workshop context and requires source review. The degree range, including whether degree zero is included, is not specified.

Short English statement

Classify the indecomposable Specht modules in prime characteristic whose vertices are abelian. The symmetric-group interpretation follows the supplied context; the field assumptions and degree range require confirmation. The supplied dated curation labels the problem partially solved, but that assessment is not independently verified.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $30001775$, title Specht Modules with Abelian Vertices, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The record gives proposed year 2011 and names no individual proposer. Its corrected_verified label and accompanying semantic, typographic, and LaTeX review claim are supplied registry metadata, not independent verification here.
• Mini-Workshop: Modular Representations of Symmetric Groups and Related Objects (2011), Oberwolfach Reports, pp. 1177--1205, DOI 10.4171/owr/2011/21. Supplied source citation and literature-source label: original OWR report. https://doi.org/10.4171/owr/2011/21. The supplied background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature reference labeled "Lim abelian vertices": https://arxiv.org/abs/1102.2484. No fuller bibliographic details are supplied.
• Literature assessment dated 2026-08-21: the supplied curation labels the problem partially_solved and states that Lim proves necessary conditions and classifies abelian-vertex Specht modules in important cases, while the full classification requested in the report remains unresolved. This is supplied dated curation derived from the OWR record, not an independently verified assessment. Its time-sensitive status and cited claims require independent verification.