Irreducible Specht Modules for Type-A Hecke Algebras

Definitions and mathematical statement

Use the following explicit standard conventions to interpret the supplied question. Let $n\geq 1$ be an integer, let $F$ be a field, and let $q\in F^{\times}$. Write $H_n(F,q)$ for the Iwahori--Hecke algebra of the symmetric group $\mathfrak S_n$, of Coxeter type $A_{n-1}$. It is the unital associative $F$-algebra generated by $T_1,\ldots,T_{n-1}$, with relations \[ (T_i-q)(T_i+1)=0,\qquad T_iT_{i+1}T_i=T_{i+1}T_iT_{i+1},\qquad T_iT_j=T_jT_i\quad (|i-j|>1), \] for indices for which the expressions are defined; set $H_1(F,q)=F$. A partition $\lambda\vdash n$ is a finite weakly decreasing sequence of positive integers with sum $n$. Let $S_F^{\lambda}(q)$ denote the standard Specht module of $H_n(F,q)$ indexed by $\lambda$. A module is irreducible if it is nonzero and has no nonzero proper submodule.

Classification problem. For every $n\geq 1$, classify all triples \[ (\lambda,F,q),\qquad \lambda\vdash n,\quad F\text{ a field},\quad q\in F^{\times}, \] for which $S_F^{\lambda}(q)$ is irreducible over $F$.

For the terminology in the source, put $p=\operatorname{char}F$, where $p=0$ or $p$ is prime, and define the quantum characteristic by \[ e=\min\{r\geq 2:1+q+\cdots+q^{r-1}=0\text{ in }F\}, \] with $e=\infty$ if this set is empty. The supplied record describes the classification equivalently as that of $(e,p)$-irreducible partitions. Here this phrase refers to partitions whose corresponding Specht modules are irreducible for the parameters under consideration. The claim that this property depends only on $(e,p)$, rather than on further details of $F$ and $q$, is part of the supplied equivalence; no justification or precise range of that equivalence is included in the record.

Unresolved qualifications. The presentation and the restriction $q\ne 0$ above make explicit a usual convention for Iwahori--Hecke algebras; the supplied statement does not specify a presentation, parameter normalization, or whether a zero-parameter specialization is intended. If $q=0$ is intended, its inclusion requires a separate clarification. No algebraic-closure or splitting-field hypothesis is supplied, and the stated target is irreducibility over $F$. The supplied background also refers to a general Iwahori--Hecke extension and to type $B$, whereas the extracted mathematical question explicitly concerns type $A$. It does not identify which remaining cases are intended as unresolved.

Short English statement

Classify the partitions, fields, and Hecke parameters for which a type-$A$ Specht module is irreducible. The formulation uses explicit standard nonzero-parameter conventions; the intended remaining cases and the scope of the source's equivalent $(e,p)$ terminology require review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30001452$, title Irreducible Specht Modules for Type-A Hecke Algebras, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The record gives proposed year $2010$ and supplies no individual proposer.
• Original report: Combinatorial Representation Theory (2010), Oberwolfach Reports, pp. 799--882, DOI 10.4171/owr/2010/15. Supplied source and literature link: https://doi.org/10.4171/owr/2010/15.
• Supplied literature reference: Fayers, irreducible Specht modules in type A, https://doi.org/10.1016/j.aim.2004.06.001.
• Supplied literature reference: Fayers, type B, https://arxiv.org/abs/1005.5660.
• Supplied dated literature curation, checked 2026-08-21: the record labels the problem partially solved and assesses that the symmetric-group Specht-module problem is solved, substantial type-$A$ and type-$B$ Iwahori--Hecke classifications are known, and the extracted general Iwahori--Hecke extension is not completely classified. This assessment is supplied dated curation derived from the OWR material, not independently verified here; its time-sensitive status and cited claims require independent verification.
• The registry labels the statement corrected_verified and reports that its mathematical content was checked against the official Oberwolfach report, with notation, context, and question form normalized without changing the source-level claim. This is supplied verification metadata, not an independent source check for this draft.