Representations of symmetric and alternating groups and their double covers that remain irreducible modulo every prime (Q6660840)

Globally irreducible representations of finite groups, introduced in [\textit{B. H. Gross}, J. Am. Math. Soc. 3, No. 4, 929--960 (1990; Zbl 0745.11035)], are representations over the field \(\mathbb{Q}\) which remain irreducible when scalars are extended to \(\mathbb{R}\), and for which certain reductions to positive characteristic \(p\) remain irreducible for every prime \(p\).\N\NIn the paper under review, the authors completely characterise irreducible representations of alternating groups and double covers of symmetric and alternating groups which reduce almost homogeneously in every characteristic. This enables them to classify irreducible representations that remain irreducible in every characteristic as well as irreducible representations of these groups that can appear as composition factors of globally irreducible representations of groups containing \(\mathfrak{A}_{n}\) or \(\widehat{\mathfrak{A}}_{n}\) as normal subgroups. In particular they show that (apart from finitely many exceptions) such representations are either \(1\)-dimensional or basic spin representations.