Murphy-Element Description of Centralizer-Algebra Centers

Definitions and mathematical statement

The supplied question concerns the center of $kS_n^{S_\ell}$. Its coefficient domain, parameter range, and polynomial symmetry action are not specified sufficiently to give a uniquely determined conjecture.

To explain the notation conditionally, suppose that $k$ is a field, $n\geq 1$, and $0\leq\ell\leq n$, and embed $S_\ell$ in $S_n$ by permuting the first $\ell$ letters. These are explanatory conventions, not hypotheses explicitly supplied by the record. Writing $kS_n$ for the group algebra, the usual centralizer interpretation is \[ A_{n,\ell}=kS_n^{S_\ell} =\{a\in kS_n:ha=ah\text{ for every }h\in S_\ell\}. \] Its center is \[ Z(A_{n,\ell})=\{z\in A_{n,\ell}:za=az\text{ for every }a\in A_{n,\ell}\}. \] Under the standard additive Jucys--Murphy convention, the Murphy elements are the commuting elements \[ J_1=0,\qquad J_i=\sum_{j=1}^{i-1}(j\ i)\in kS_n\quad(2\leq i\leq n). \] The record does not explicitly specify this normalization or the indexing of the Murphy elements.

The source asks whether $Z(A_{n,\ell})$ is the algebra of polynomials in the Murphy elements that are symmetric under $S_\ell\times S_{n-1}$. More precisely, if an intended action $\rho$ of $H=S_\ell\times S_{n-1}$ on $k[x_1,\ldots,x_n]$ were supplied, the corresponding candidate equality would be \[ Z(A_{n,\ell})\stackrel{?}{=} \{f(J_1,\ldots,J_n):f\in k[x_1,\ldots,x_n]^{H,\rho}\}, \] where the superscript denotes polynomials fixed by every element of $H$ under $\rho$. The action $\rho$ is missing: the displayed product does not specify how its two factors permute the variables or act on polynomial expressions. In particular, no replacement of the supplied $S_{n-1}$ by another symmetric group is justified by this record. Thus this display describes the unresolved target conditionally; it is not a fully specified, universally quantified conjecture. The intended coefficient assumptions, parameter range, Murphy convention, and symmetry action require clarification.

Short English statement

The supplied problem asks whether the center of a symmetric-group centralizer algebra consists exactly of suitably symmetric polynomial expressions in Murphy elements. The expression $S_\ell\times S_{n-1}$ is retained exactly as supplied, but its action and the coefficient hypotheses are unspecified. The record labels the problem open; that label is supplied curation, not an independently verified mathematical conclusion.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30001771$, source problem code catalogue entry, title: Murphy-Element Description of Centralizer-Algebra Centers. The record belongs to the Oberwolfach Reports Open Problems collection and records the proposed year as 2011; no individual proposer is supplied.
• Supplied source 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: https://doi.org/10.4171/owr/2011/21. The background describes these reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature source, labeled "original OWR report": https://doi.org/10.4171/owr/2011/21.
• Literature assessment checked at 2026-08-21: the supplied curation labels the problem open and states that the report poses the center and Jucys--Murphy description as an open question, with later work handling special cases but not the stated general problem. This is supplied dated curation, not independently verified. The supplied background expressly requests independent verification of time-sensitive status and cited claims.