Probability of Vanishing Character Values

Definitions and mathematical statement

Let $G$ be a finite group. The supplied question asks for the probability that a randomly chosen character value $\chi(g)$ is zero. It does not specify the characters being considered or the probability distribution used to choose the value, so it does not determine a unique probability.

For a conditional formulation using ordinary complex characters, a character is a function $\chi\colon G\to{\mathbb{C}}$ of the form $\chi(g)=\operatorname{tr}(\rho(g))$, where $\rho\colon G\to\operatorname{GL}(V)$ is a representation on a finite-dimensional complex vector space. Let $\mathcal X$ be a specified nonempty finite collection of such characters, and let $\mu\colon\mathcal X\times G\to[0,1]$ be a specified probability mass function, so that \[ \sum_{\chi\in\mathcal X}\sum_{g\in G}\mu(\chi,g)=1. \] Define \[ Z(G,\mathcal X)=\{(\chi,g)\in\mathcal X\times G:\chi(g)=0\}. \] The corresponding precise version of the question is: determine or describe, for finite groups $G$ and the intended choices of $\mathcal X$ and $\mu$, the probability \[ p_0(G;\mathcal X,\mu) =\mathbb P_{\mu}\left(\chi(g)=0\right) =\sum_{(\chi,g)\in Z(G,\mathcal X)}\mu(\chi,g). \] This expression defines the quantity under a specified sampling rule; it does not answer the source's question.

For example, if a character $\chi$ is fixed and $g$ is uniform on $G$, the quantity is \[ \frac{|\{g\in G:\chi(g)=0\}|}{|G|}. \] If instead $\chi$ is uniform among the ordinary irreducible complex characters $\operatorname{Irr}(G)$ and $g$ is independently uniform on $G$, the quantity is \[ \frac{|\{(\chi,g)\in\operatorname{Irr}(G)\times G:\chi(g)=0\}|} {|\operatorname{Irr}(G)|\,|G|}. \] These are conditional interpretations, not choices established by the supplied passage. Uniform sampling of conjugacy classes, rather than elements, gives another sampling rule. The source does not resolve these alternatives, specify a family of groups or an asymptotic regime, or state whether an exact formula, bounds, or a distributional description is sought.

Short English statement

For a finite group, ask how often a character value is zero. A precise probability requires specifying which characters are sampled and how characters and group elements are chosen. The supplied record leaves these choices unresolved; the displayed formulations make their dependence explicit.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30005222$, title Probability of Vanishing Character Values, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection.
• Supplied report citation: Character Theory and Categorification (2023), Oberwolfach Reports, pp. 2267--2333, DOI 10.4171/owr/2022/39. https://doi.org/10.4171/owr/2022/39. The record gives proposed year 2022 and identifies the report as 39/2022; the relationship to the citation's 2023 date requires verification. No individual proposer is supplied.
• Supplied literature source: Oberwolfach Report 39/2022. https://ems.press/content/serial-article-files/46974.
• Supplied literature source: Miller, Dense proportions of zeros. https://doi.org/10.1016/j.crma.2019.10.006. No additional bibliographic details are supplied.
• The supplied original passage contains the reference label [3], whose bibliographic entry and relationship to the listed references are not supplied.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open and reports that known results describe substantial zero sets for character values, but that no general probability law answering the question was found. This is supplied dated curation, not an independently verified literature assessment. The record itself requests independent verification of time-sensitive status and cited claims.