The irreducible representations of the alternating group which remain irreducible in characteristic \(p\). (Q2790716)

In this article, the author proves his conjecture [in Represent. Theory 14, 601-626 (2010; Zbl 1236.20009)], classifying the ordinary irreducible modules for the alternating group, \(\mathfrak A_n\), which remain irreducible in characteristic \(p\).NEWLINENEWLINE In [loc. cit.], the author solved the problem when \(p=2\), and for \(p\geq 3\) proved that these modules are precisely the ones which are either indexed by a \textit{JM-partition} or by a self-conjugate partition \(\lambda\) such that the Specht module \(S^\lambda\) for the symmetric group \(\mathfrak S_n\) has exactly two composition factors. The author gave a conjectural description of all such partitions (when \(p\geq 5\), they are the partitions of \(p\)-weight 1 and so-called \(R\)-partitions; the \(p=3\) case is slightly more complicated), and proved one half of the conjecture; namely that all the \(R\)-partitions do indeed give Specht modules with exactly two composition factors. The main result of the paper under review is that there are no others. That is, given that all irreducble Specht modules have been classified by the author and others, and are indexed by \textit{JM-partitions} the author proves that any Specht module labelled by a self-conjugate partition which is neither a JM-partition nor an \(R\)-partition has at least three composition factors.NEWLINENEWLINE The author's approach to proving this statement makes use of a wide range of machinery. He spends roughly half of the paper giving an excellent account of the necessary background material, including the \(i\)-restriction and \(i\)-induction functors, James's Regularisation Theorem, abacus combinatorics, and semistandard homomorphisms. This background material is peppered with many new additions to the established theory, which is all put into the proof of the main result in the second half of the paper.NEWLINENEWLINE The main proof basically proceeds by an induction argument, with some case analysis and the use of the tools developed in the first half of the paper.