Induced Character Models for Hyperoctahedral Groups

Definitions and mathematical statement

Let $W_n=C_2^n\rtimes S_n$ be the hyperoctahedral group, with $S_n$ permuting the sign coordinates. A bipartition $\lambda=(\lambda^+,\lambda^-)$ of $n$ is an ordered pair of partitions with $|\lambda^+|+|\lambda^-|=n$; write $\operatorname{Bip}(n)$ for their set. Conjugacy classes are indexed by signed cycle type: the product of signs around a cycle is positive or negative and its length contributes to the corresponding partition. Let $C_\lambda$ be the class with the source's indexing convention and choose $w_\lambda\in C_\lambda$.

Let $(E_\lambda)_{\lambda\in\operatorname{Bip}(n)}$ be the stated family in the rational Mantaci--Reutenauer algebra, viewed as a subalgebra of ${\mathbb{Q}} W_n$. Its hypotheses are \[ E_\lambda^2=E_\lambda,\qquad E_\lambda E_\mu=0\ (\lambda\ne\mu),\qquad \sum_\lambda E_\lambda=1,\qquad \dim_{\mathbb{Q}}({\mathbb{Q}} W_nE_\lambda)=|C_\lambda|, \] and each $E_\lambda$ is primitive in that subalgebra. Primitivity is not asserted in the full group algebra, nor is centrality imposed. The left ideal ${\mathbb{Q}} W_nE_\lambda$ is a $W_n$-module under left multiplication.

The question is whether, for every bipartition $\lambda$, there exists a linear character \[ \zeta_\lambda:C_{W_n}(w_\lambda)\longrightarrow{\mathbb{C}}^\times \] such that ${\mathbb{Q}} W_nE_\lambda$ affords the induced character $\operatorname{Ind}_{C_{W_n}(w_\lambda)}^{W_n}\zeta_\lambda$. In the ordinary complex-character interpretation this means \[ {\mathbb{C}}\otimes_{\mathbb{Q}}({\mathbb{Q}} W_nE_\lambda) \cong\operatorname{Ind}_{C_{W_n}(w_\lambda)}^{W_n}{\mathbb{C}}_{\zeta_\lambda} \] as complex $W_n$-modules, where $C_{W_n}(w)=\{g:gw=wg\}$. It does not require the one-dimensional character itself to take values in ${\mathbb{Q}}$. The dimension condition agrees with $|C_\lambda|=[W_n:C_{W_n}(w_\lambda)]$, but equality of dimensions alone is not the asserted character identity.

The source names, but does not define, the Mantaci--Reutenauer subalgebra or construct the idempotent family. It is unclear whether the intended assertion concerns a particular family fixed in the preceding report text or every family satisfying the printed conditions; the draft keeps the stated family fixed rather than adding a new universal family quantifier. The degree range convention for $n$ and the signed-partition ordering also need confirmation.

The 2026-08-21 literature assessment describes an incomplete extraction and reports no located general resolution. The verification field instead says that the official preceding text supplies the omitted class, representative and idempotent family, now restored in the cleaned statement. This repair-versus-triage distinction is retained, and the open label is not treated as independently verified current status.

Short English statement

For each signed-cycle type, does the left ideal cut out by the stated primitive idempotent have the character induced from a one-dimensional centralizer character? All orthogonality, sum and dimension conditions are retained. The rational module's complex character is distinguished from requiring a rational-valued inducing character, and the intended idempotent-family scope remains to be checked.

Sources

• Local UnsolvedMath registry, permanent ID 30000458, source code catalogue entry, "Induced Character Models for Hyperoctahedral Groups," including the original induction question, restored family hypotheses and source-verification note.
• Representations of Finite Groups (2007), Oberwolfach Reports, pp. 895--978, DOI 10.4171/owr/2006/15: https://doi.org/10.4171/owr/2006/15. This is the sole supplied literature reference, labelled "original Oberwolfach report." The citation uses 2007, while the DOI and proposed-year metadata use 2006.
• Literature triage checked 2026-08-21; record updated 2026-09-16. The assessment describes an incomplete extraction, whereas verification reports restoration from the preceding source text. No external source has been visited or independently checked here.