Schur indices of perfect groups (Q2758971)

It has been conjectured that the Schur index over the rational numbers of every irreducible character of every covering group of every finite simple group \(G\) is either 1 or 2. The conjecture is true when \(G\) is a sporadic simple group by work of \textit{W. Feit} [Isr. J. Math. 46, 274-300 (1983; Zbl 0528.20009), ibid. 93, 229-251 (1996; Zbl 0845.20009)]. The conjecture is also true when \(G\) is the alternating group or when \(G\) is a Chevalley group of type \(A\) or \(C\), except possibly when \(G\) has an exceptional Schur multiplier. It might seem plausible to conjecture that the Schur index of a finite perfect group is at most 2, but the author shows in the paper under review that no such conjecture is true. He shows that for any integer \(n\), there is a finite perfect group \(G\) that possesses an irreducible character of Schur index \(n\) over the rational numbers.NEWLINENEWLINENEWLINEThe author's construction of an appropriate group \(G\) is surprisingly elementary. \(G\) is the split extension of an elementary Abelian \(q\)-group \(N\) of order \(q^p\) by a subgroup \(S\) isomorphic to the simple group \(\text{PSL}(2,p)\). Here, \(q\) and \(p\) are different primes chosen to satisfy certain congruence conditions modulo \(n^2\) (Dirichlet's theorem guarantees that there are infinitely many choices for \(p\) and \(q\)). As a module for \(S\) over the field of order \(q\), \(N\) is isomorphic to the Steinberg module of dimension \(p\). Using a result of \textit{R. Brauer} [Math. Z. 31, 733-747 (1930; JFM 56.0865.04)], the author shows that \(G\) has an irreducible character of degree \(np(p-1)\) that has Schur index \(n\).