A solution to Brauer's problem 14 (Q2689193)

Let \(G\) be a finite group, an irreducible complex character \(\chi \in \mathrm{Irr}(G)\) is real (\(\chi \in \mathrm{Irr}_{\mathbb{R}}(G)\)) if it arise from a real irreducible representation, that is \(\chi\) has Frobenius-Schur indicator \(\epsilon (\chi) = 1\). Brauer's Problem 14 [\textit{R. Brauer}, in: Lect. Modern Math. 1, 133--175 (1963; Zbl 0124.26504)] asks whether one can describe the number of characters in \(\mathrm{Irr}_{\mathbb{R}}(G)\). In the paper under review, the authors provide an elegant answer to the aforementioned problem. Theorem A: Let \(G\) be a finite group with \(k_{r}(G) = | \mathrm{Irr}_{\mathbb{R}}(G)|\) real conjugacy classes. Then the multiset \(\{\chi(1) \epsilon(\chi) \mid \chi \in \mathrm{Irr}_{\mathbb{R}}(G) \}\) is determined by the sequence \[ s(n)= \big | \big \{(g_{1},\ldots, g_{n}) \in G^{n} \mid g_{1}^{2}\cdot \ldots \cdot g_{n}^{2} = 1 \big \} \big | \text{ for } n=1, \ldots, k_{r}(G)+1. \] In particular, \(k_{r}(G) = s(2) |G|^{-1}\) and the number of irreducible characters of \(G\) with Frobenius-Schur indicator 1 can be described purely in group theoretical terms. In a final remark the authors show how equality \(k_{r}(G)|G|=s(2)\) can be deduced without the use of characters.