Symmetric-Group Characters Determined by Reflection Length

Definitions and mathematical statement

Let $n\geq 1$ and let $S_n$ be the symmetric group on $\{1,\ldots,n\}$. For $g\in S_n$, let $c(g)$ denote the number of cycles in its disjoint cycle decomposition, including fixed points. Define its reflection length by \[ \ell(g)=\min\{r\geq 0:g=t_1\cdots t_r,\ \text{each }t_i\text{ a transposition}\}. \] Then $\ell(g)=n-c(g)$.

Under the ordinary complex character interpretation, a character is a function $\chi_V(g)=\operatorname{tr}(\rho_V(g))$ associated with a finite-dimensional complex representation $\rho_V:S_n\to\operatorname{GL}(V)$. The requested problem is to give, for arbitrary $n$, a simple description of the collection \[ \mathcal C_n=\left\{\chi_V:\ \forall g,h\in S_n,\quad c(g)=c(h)\Longrightarrow\chi_V(g)=\chi_V(h)\right\}. \] Equivalently, these are the characters for which there exists a function $F:\{0,\ldots,n-1\}\to{\mathbb{C}}$ such that \[ \chi_V(g)=F(\ell(g))\qquad\text{for every }g\in S_n. \] Unresolved qualifications. The supplied statement says only "characters"; it does not explicitly specify the coefficient field, whether arbitrary ordinary characters or only irreducible characters are intended, or whether virtual characters are allowed. The displayed collection makes the ordinary complex character interpretation explicit, pending review of the intended scope. The word "simple" is not assigned a formal criterion in the supplied record.

Short English statement

Describe the characters of $S_n$ whose value on a permutation depends only on its total number of cycles, counting fixed points, or equivalently on its reflection length. The precise class of characters intended and the meaning of a "simple description" require source review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30003705$, title Symmetric-Group Characters Determined by Reflection Length, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The record gives proposed year 2018 and identifies no individual proposer.
• Supplied report citation: Topology of Arrangements and Representation Stability (2019), Oberwolfach Reports, pp. 43--123, DOI 10.4171/owr/2018/2. https://doi.org/10.4171/owr/2018/2. The background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. The supplied literature-source label is "Oberwolfach Report 2/2018", with the same link: https://doi.org/10.4171/owr/2018/2.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and reports that a satisfactory description of the characters depending only on cycle count remains open, with cited partial character results not resolving the classification. This is supplied dated curation derived from the OWR material, not an independently verified literature assessment. The supplied background expressly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and says that its mathematical content was preserved while notation, typography, or omitted definitions were repaired. These are supplied editorial qualifications, not independent mathematical verification of this draft.