Simple components of \(\mathbb{Q}[\text{Sp}_4(F_q)]\). (Q2565485)

The author continues his investigations, begun in 1977, of the Schur indices of complex irreducible characters of finite groups of Lie type in the subject matter of this review. The specific group under consideration here is the symplectic group \(G=\text{Sp}_4(q)\), where \(q\) is a power of an odd prime. The reviewer has already shown that the rational Schur index of any irreducible character of a symplectic group is at most 2 and that when \(q\equiv 1\bmod 4\), all faithful irreducible characters have Schur index 2 over the real numbers and hence over the rational numbers. These rather general observations do not provide the detail required to investigate the Schur index of any specific family of characters and the author's results indicate that no simple answers can be expected to the question of, say, computing the local Schur indices of characters. It should be noted that \textit{A. Przygocki} [Commun. Algebra 10, 279-310 (1982; Zbl 0486.20026)] determined the rational Schur indices of the irreducible characters of \(G\) and showed in particular that there exist real-valued irreducible characters of Schur index 1 over the real field which nonetheless have Schur index 2 over the rationals. The author of the present paper points out some errors in Przygocki's reasoning and in his findings. He presents his information on the Schur indices of the irreducible characters of \(G\) in the form of calculations of the Hasse invariants of the corresponding division algebra components of the rational group algebra of \(G\) (rather than using the language of the local Schur index). It cannot be said that the paper makes easy reading, as the results are often presented in the form of detailed lists of technical information, often occupying a page of text. The impression created is that attempts to bring this level of specific calculation to bear on larger groups may be infeasible and that only general principles, say reflecting properties of unipotent or cuspidal characters, may be forthcoming.