Blocks of Symmetric-Group Centralizer Algebras

Definitions and mathematical statement

The supplied record leaves the hypotheses on $k$, the ranges of $n,\ell$, and the subgroup embedding unspecified. The following gives a conditional formulation using the usual symmetric-group embedding; these conventions require source confirmation.

Let $k$ be a field, let $0\leq \ell\leq n$ be integers, and embed $S_\ell$ in $S_n$ by letting it permute the first $\ell$ letters and fix the others. Identify $kS_\ell$ with its image in $kS_n$, and define \[ A=kS_n^{S_\ell}:=\{a\in kS_n: hah^{-1}=a\text{ for every }h\in S_\ell\} =\{a\in kS_n: ax=xa\text{ for every }x\in kS_\ell\}. \] For a finite-dimensional unital $k$-algebra $B$, a block idempotent is a nonzero central idempotent that cannot be written as a sum of two nonzero orthogonal central idempotents. Its associated block is the ideal $Bb$.

Question (source conjecture, conditional formulation). For the intended choices of $k,n,\ell$, does every block idempotent $b$ of $A$ satisfy \[ b=ef \] for some block idempotent $e$ of $kS_n$ and some block idempotent $f$ of $kS_\ell$? Here the product is taken in $kS_n$. This interprets the source phrase "blocks represented by products" as equality of block idempotents. The supplied record does not determine whether arbitrary fields are intended, whether positive characteristic or splitting hypotheses are required, or whether the displayed parameter boundary cases are included.

Short English statement

Under the stated conditional interpretation, the question asks whether every block idempotent of the symmetric-group centralizer algebra is a product of block idempotents from the two group algebras. The field hypotheses, parameter ranges, subgroup embedding, and precise meaning of "represented" require confirmation. The supplied open-status label is dated curation.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30001767$, source problem code catalogue entry, title Blocks of Symmetric-Group Centralizer Algebras, in the Oberwolfach Reports Open Problems collection. No individual proposer is named; the supplied proposed year is 2011.
• Supplied authoritative report citation: Mini-Workshop: Modular Representations of Symmetric Groups and Related Objects (2011), Oberwolfach Reports, pp. 1177--1205, DOI 10.4171/owr/2011/21. Source URL and supplied literature-source link: https://doi.org/10.4171/owr/2011/21. The literature-source label is Modular Representations of Symmetric Groups and Related Objects, OWR 21/2011.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open, reports that the report conjectures the assertion for all blocks, and states that no authoritative proof or counterexample for the unrestricted assertion was identified. This is supplied dated curation, not an independently verified literature assessment. Its background explicitly requests independent verification of time-sensitive status and cited claims.
• The supplied statement status is corrected_verified; its accompanying verification records a manual semantic, typographic, and LaTeX review claiming a self-contained, meaning-preserving cleanup. That supplied review does not resolve the formulation qualifications identified here. The report has not been independently consulted for this draft.