In a recent paper, Külshammer, Olsson, and Robinson proved a deep generalization of the Nakayama conjecture for symmetric groups. We provide a similar but a shorter and relatively elementary proof of their result. Our method enables us to obtain a more general $H$-analogue of the Nakayama conjecture where $H$ is a set of positive integers.

Let $F$ be an algebraically closed field of characteristic $p$, and let $H$ be a finite group. A natural and important problem in representation theory is to classify the pairs $(G, D)$, where $G$ is a subgroup of $H$, and $D$ is an irreducible $FH$-module that remains irreducible when restricted to $G$. For example, if $H$ is an almost simple group or a central extension of an almost simple group, then one encounters this problem in the attempt to determine the maximal subgroups of the finite classical groups. We solve the above problem for Schur's double covers $H = {\widehat S}_n$ and ${\widehat A}_n$ of the symmetric and alternating groups, and $G$ a Young-type subgroup. The answer is given in terms of the combinatorics of restricted, $p$-strict partitions; such partitions are used to parameterize the irreducible spin representations of ${\widehat S}_n$ and ${\widehat A}_n$. We exploit the connections, recently developed by Ariki, Grojnowski, Brundan, Kleshchev, and others, between modular branching rules, crystal graphs of affine Kac–Moody algebras, and affine Hecke algebras and related objects.