Positive Characters of Simply Connected Groups

Definitions and mathematical statement

Let $G$ be a connected simply connected compact Lie group, and let $m_G$ be normalized Haar probability. Let $R(G)$ be the integral representation ring: its elements are virtual characters \[ f=\sum_{\pi\in\operatorname{Irr}(G)}n_\pi\chi_\pi, \qquad n_\pi\in{\mathbb{Z}}, \] with finitely many nonzero coefficients. Define the character inner product by $\langle f,g\rangle=\int_G f(x)\overline{g(x)}\,{\,\mathrm{d}} m_G(x)$, and let $1$ denote the trivial character.

The question is whether, for every such $G$ and every $f\in R(G)$, the hypotheses \[ f(x)\in{\mathbb{R}}_{\geq0}\quad\text{for all }x\in G, \qquad\langle f,1\rangle=1 \] imply the existence of an irreducible finite-dimensional complex character $\chi$ with \[ f(x)=\chi(x)\overline{\chi(x)}=|\chi(x)|^2 \qquad\text{for every }x\in G. \] Pointwise nonnegativity is the hypothesis; it does not mean that the virtual coefficients $n_\pi$ are assumed nonnegative. The conclusion would identify $f$ with the character of $V\otimes V^*$ for an irreducible $V$. Such characters do satisfy the two displayed hypotheses by character orthogonality; the converse is the requested assertion.

The supplied literature triage dated 21 August 2026 reports Serre's rank-one result and says its checked primary sources located no general-rank proof or counterexample. That is a limited dated literature assessment, not an independent claim that the general statement has been resolved or that every relevant source was checked. The assumptions include connectedness, simple connectivity, compactness, and the integral virtual-character condition throughout.

Short English statement

Must every pointwise nonnegative virtual character of Haar average one be the squared absolute value of an irreducible character? The virtual coefficients are allowed to be negative; only the function values are assumed nonnegative. The supplied triage reports a rank-one result and leaves the general case open.

Sources

• Local UnsolvedMath registry, permanent ID $30000048$, exact source code catalogue entry, Oberwolfach Reports Open Problems. Statement status corrected_verified; supplied verification reports checking the official report and editing for self-contained assumptions and notation.
• "Algebraische Gruppen (2004)," Oberwolfach Reports, pp. 633--672, DOI 10.4171/owr/2004/12: https://doi.org/10.4171/owr/2004/12. Proposed year 2004.
• Supplied "Serre, compact Lie groups": https://www.college-de-france.fr/media/jean-pierre-serre/UPL8835246706135048784_On_the_values_of_the_characters_of_compact_Lie_groups.pdf.
• Supplied "OWR 12/2004": https://doi.org/10.4171/OWR/2004/12. No fuller citation or precise rank-one theorem is included in the record.
• Literature checked 21 August 2026; supplied status open, with no general-rank proof or counterexample found in the checked sources. Registry created 25 August 2026 and updated 16 September 2026. This is dated curation triage requiring independent verification before citation.
• Registry curation attribution: Ulam AI / UnsolvedMath Contributors, CC BY 4.0. Underlying report and literature retain their supplied provenance; this statement-only draft requires human review.