We investigate the problem of finding a set of prime divisors of the order of a finite group, such that no two irreducible characters are in the same p-block for all primes p in the set. Our main focus is on the simple and quasi-simple groups. For results on the alternating and symmetric groups and their double covers, some combinatorial results on the cores of partitions are proved, which may be of independent interest. We also study the problem for groups of Lie type. The sporadic groups (and their relatives) are checked using GAP.

In 1998, Zalesskii proposed to classify all instances of irreducible characters of quasi- simple groups which are of prime power degree. In joint work, Malle and Zalesskii then dealt with all quasi-simple groups with the exception of the alternating groups and their double covers [5]. In an earlier article [1] we have classified all the irreducible characters of $S_n$ of prime power degree and have deduced from this also the corresponding classification for the alternating groups. In the present article we complete Zalesskii's programme by dealing with the final case left open in [5], the double covers of the alternating groups. We derive this result from a corresponding result on the double covers of the symmetric groups. If one is only interested in spin characters, one easily sees that only 2-powers can occur as prime power degrees. However from a combinatorial point of view it is natural to ask more generally: when is the number of shifted standard tableaux of a given shape a prime power? Since our method is independent of the prime, we will answer this question, showing that apart from a few accidental cases for small $n$ only the ‘obvious’ partitions satisfy the prime power condition. Thus in turn for the spin characters, apart from exceptions for small $n$ , the ‘obvious’ spin characters of 2-power degree are indeed the only ones (this confirms the conjecture stated in [5]).

In 1998, the second author of this paper raised the problem of classifying the irreducible characters of Sn of prime power degree. Zalesskii proposed the analogous problem for quasi-simple groups, and he has, in joint work with Malle, made substantial progress on this latter problem. With the exception of the alternating groups and their double covers, their work provides a complete solution. In this article we first classify all the irreducible characters of Sn of prime power degree (Theorem 2.4), and then we deduce the corresponding classification for the alternating groups (Theorem 5.1), thus providing the answer for one of the two remaining families in Zalesskii's problem. This classification has another application in group theory. With it, we are able to answer, for alternating groups, a question of Huppert: which simple groups G have the property that there is a prime p for which G has an irreducible character of p-power degree > 1 and all of the irreducible characters of G have degrees that are relatively prime to p or are powers of p?

The case of the double covers of the symmetric and alternating groups will be dealt with in a forthcoming paper; in particular, this completes the answer to Zalesskii's problem.

The paper is organized as follows. In Section 2, some results on hook lengths in partitions are proved. These results lead to an algorithm which allows us to show that every irreducible representation of Sn with prime power degree is labelled by a partition having a large hook. In Section 3, we obtain a new result concerning the prime factors of consecutive integers (Theorem 3.4). In Section 4 we prove Theorem 2.4, the main result. To do so, we combine the algorithm above with Theorem 3.4 and work of Rasala on minimal degrees. This implies Theorem 2.4 for large n. To complete the proof, we check that the algorithm terminates appropriately for small n (that is, those n [les ] 9.25 · 108) with the aid of a computer. In the last section we derive the classification of irreducible characters of An of prime power degree, and we solve Huppert's question for alternating groups.

Let F be a field, and let Σn be the symmetric group on n letters. In this paper we address the following question: given two irreducible FΣn-modules D1 and D2 of dimensions greater than 1, can it happen that D1 [otimes ] D2 is irreducible? The answer is known to be ‘no’ if char F = 0 [12] (see also [2] for some generalizations). So we assume from now on that F has positive characteristic p. The following conjecture was made in [4].

CONJECTURE. Let D1and D2be two irreducible FΣn-modules of dimensions greater than 1. Then D1 [otimes ] D2 is irreducible if and only if p = 2, n = 2 + 4l for some positive integer l; one of the modules corresponds to the partition (2l + 2, 2l) and the other corresponds to a partition of the form (n − 2j − 1, 2j + 1), 0 [les ] j < l. Moreover, in the exceptional cases, one has

formula here

The main result of this paper is the following theorem, which establishes a big part of the conjecture.

The Mullineux map is an involutory bijection on the set of p-regular partitions of any given integer n, where a partition is called p-regular if no part of it is repeated p or more times. Many combinatorial properties of the Mullineux map make it reasonable to view this map as a p-analogue of the transposition map T on the set of all partitions.

Based on the work of Kleshchev [7], it has been shown [2, 5, 14] that the Mullineux map M has the following property.

If λ is a p-regular partition of n and Dλ is the p-modular representation of the symmetric group Sn labelled by λ (see [6]) then

formula here

where sgn is the sign representation of Sn. Thus, from a representation theoretic point of view as well, the Mullineux map is a p-analogue of the transposition map T. The Mullineux map plays a vital role not only in the representation theory of symmetric groups but also in other contexts. The definition of M as given in [9] is quite complicated and to study various questions involving this map it is desirable to have other descriptions. One alternative inductive description of M using the concept of good boxes in a p-regular partition was given by Kleshchev [7] and this was used by Walker to prove a result contained in Theorem 4·5 below; this work was motivated by the investigation of Schur modules. In this paper we study a third description of M based on the operator J on the set of p-regular partitions defined in [13].

We study the characteristic p analogue of M-groups, the so-called Mp-group Generalizing this notion, we also consider the condition that the modular irreducible representations are induced from representations of dimension < p, or even weaker, of dimension not divisible by p.

In this paper periodic modules over group rings and algebras are considered. A new lower bound for the p-part of the rank of a periodic module with abeian vertex is given, and results on periodic modules with odd/even and small periods are obtained. In particular, it is shown that characters afforded by periodic lattices of odd period satisfy strong properties and that irreducible periodic lattices are always of even period.