Symmetry of Completed Finite-Group Representation Rings

Definitions and mathematical statement

Under the positive-characteristic interpretation of the supplied modular formulation, let $p$ be a prime, let $G$ be a finite group, and let $k$ be a field of characteristic $p$. No splitting-field or algebraic-closure hypothesis, and no assumption that $p\mid |G|$, is supplied.

Let $a(G)$ denote the Green ring of finite-dimensional left $kG$-modules: its additive relations are $[M\oplus N]=[M]+[N]$, and its multiplication is $[M][N]=[M\otimes_k N]$, with diagonal $G$-action on the tensor product. Its unit is the class of the trivial module $k$. Choose representatives $(M_i)_{i\in I}$ of the isomorphism classes of nonzero indecomposable finite-dimensional $kG$-modules. By the direct-sum decomposition of finite-dimensional modules, their classes form an additive basis of $a(G)$. Set \[ a_{\mathbb C}(G)=\mathbb C\otimes_{\mathbb Z}a(G),\qquad \left\|\sum_{i\in I}a_i[M_i]\right\|=\sum_{i\in I}|a_i|\dim_k M_i, \] where only finitely many coefficients are nonzero. The indecomposable-basis interpretation is necessary to make this displayed norm unambiguous.

For a module $M$, let $M^\vee=\operatorname{Hom}_k(M,k)$, with $(g\cdot f)(m)=f(g^{-1}m)$. Interpret the involution induced by duality as the conjugate-linear map \[ \left(\sum_i a_i[M_i]\right)^*=\sum_i\overline{a_i}[M_i^\vee]. \] Let $\widehat a(G)$ be the completion in the displayed norm, with the continuous extensions of multiplication and this involution. In particular, $\widehat a(G)$ is a unital complex Banach $*$-algebra.

For a unital complex Banach algebra $A$, write \[ \sigma_A(y)=\{\lambda\in\mathbb C:y-\lambda 1_A\text{ is not invertible in }A\}. \] A Banach $*$-algebra $A$ is called symmetric if $\sigma_A(x^*x)\subseteq[0,\infty)$ for every $x\in A$.

The question is whether, for every $G$ and $k$ as above, \[ \forall x\in\widehat a(G),\qquad \sigma_{\widehat a(G)}(x^*x)\subseteq[0,\infty). \] The source writes only "characteristic $p$"; the positive-characteristic scope used here follows its modular description. Whether characteristic zero was also intended remains unresolved. This is a statement-only draft requiring human mathematical review.

Short English statement

For every finite group and every field in the supplied modular setting, form the complexified Green ring, complete it using the dimension-weighted norm on indecomposable-module coefficients, and use conjugate-linear module duality as its involution. Must every element $x^*x$ in this completion have spectrum contained in $[0,\infty)$? The supplied record labels the problem open; that status has not been independently verified.

Sources

• Local UnsolvedMath registry: supplied complete record with permanent ID $30004106$, title Symmetry of Completed Finite-Group Representation Rings, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. No individual proposer is identified in the supplied record.
• Supplied report citation: Representations of Finite Groups (2020), Oberwolfach Reports, DOI 10.4171/owr/2019/14, pp. 841--895. https://doi.org/10.4171/owr/2019/14. The record separately gives proposed year 2019 and an owr-year:2019 tag; these are retained without reconciling them with the citation's 2020 designation.
• Supplied literature source, labeled "Benson on modular representation rings": https://homepages.abdn.ac.uk/d.j.benson/pages/papers/b/benson/banach-book.pdf.
• Supplied literature source, labeled "Oberwolfach representation theory report": https://doi.org/10.4171/owr/2019/14.
• Supplied literature assessment dated exactly 2026-08-22: the question is recorded as open, with the assessment that the primary reference explicitly calls it unknown and that no later source located in that curation establishes symmetry in all cases. This is supplied dated curation and literature triage, not an independently verified assessment; time-sensitive status and cited claims require independent checking.
• The supplied statement-status label is corrected_verified; its verification note reports comparison against the official Oberwolfach report and preservation of its hypotheses, notation, and target. These are source-record claims, not verification performed for this draft. No cited source was visited in preparing this statement.