Simple Specht Modules for Hecke Algebras

Definitions and mathematical statement

Let $k$ be a field of characteristic $p$, where $p$ is a prime. A nonzero module $M$ over a $k$-algebra $H$ is called simple if its only $H$-submodules are $0$ and $M$.

The supplied source asks for a classification of the simple Specht modules for "the corresponding Hecke algebra" over $k$. The antecedent specifying that algebra is absent from the supplied record. Its type, rank, defining parameters, and the intended construction and indexing of its Specht modules are therefore unresolved. No assumption that $k$ is algebraically closed is supplied.

Statement schema, pending these specifications. For each intended Hecke algebra $H$ over such a field $k$, write $(S^\lambda)_{\lambda\in\Lambda}$ for its specified family of Specht modules. Determine exactly the set \[ \left\{\lambda\in\Lambda : S^\lambda\ne 0\ \text{and every }H\text{-submodule }N\subseteq S^\lambda\text{ satisfies }N=0\text{ or }N=S^\lambda\right\}. \] Here $\Lambda$ and $S^\lambda$ are placeholders for source data that have not been supplied. This schema expresses the classification target but does not specify a unique mathematical problem. In particular, neither a restriction to type $A$ nor a quantification over every family of Hecke algebras is justified by the supplied wording.

Short English statement

Determine precisely which Specht modules are nonzero and simple over fields of prime characteristic. The source does not identify the intended Hecke algebra, its parameters, or its Specht-module conventions, so a definitive classification problem requires further source clarification. The supplied status is partially solved; this is a curation label, not an independently verified conclusion.

Sources

• Supplied local registry record: permanent UnsolvedMath ID $30001765$, title Simple Specht Modules for Hecke Algebras, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The record assigns the year $2011$ and supplies no individual proposer.
• Supplied report citation: Mini-Workshop: Modular Representations of Symmetric Groups and Related Objects (2011), Oberwolfach Reports, pp. 1177--1205, DOI 10.4171/owr/2011/21. Literature-source label: Modular Representations of Symmetric Groups and Related Objects, OWR 21/2011. https://doi.org/10.4171/owr/2011/21. The supplied background describes these reports as documenting workshops and their open-problem discussions; no specific page locating this problem is given.
• Supplied literature source: Fayers, irreducible Specht modules at $q=-1$. https://arxiv.org/abs/0911.0342.
• Supplied literature source: Ariki, classification of simple modules for cyclotomic Hecke algebras. https://arxiv.org/abs/math/9908004.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem partially solved and attributes major type-$A$ cases for simple Specht modules to classification results of Fayers--Lyle and Ariki. It also states that the broad formulation extracted from the 2011 remark is insufficiently specified to classify safely as completely closed. This is supplied dated curation derived from the OWR material, 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 manual semantic, typographic, and LaTeX review of a meaning-preserving cleanup, which it describes as self-contained. These are supplied verification claims; they do not resolve the missing algebra specifications identified in this draft.