David A. Craven

The Theory of Fusion Systems: An Algebraic Approach

Preface Part I. Motivation: 1. Fusion in finite groups 2. Fusion in representation theory 3. Fusion in topology Part II. The Theory: 4. Fusion systems 5. Weakly normal subsystems, quotients, and morphisms 6. Proving saturation 7. Control in fusion systems 8. Local theory of fusion systems 9. Exotic fusion systems References Index of notation Index.

Symmetric group character degrees and hook numbers

AbstractIn this article we prove the following result: for any two natural numbers k and l, and for all sufficiently large symmetric groups S n , there are k disjoint sets of l irreducible characters of S n , such that each set consists of characters with the same degree, and distinct sets have different degrees. In particular, this resolves a conjecture most recently made by Moreto in [5]. The methods employed here are based upon the duality between irreducible characters of the symmetric groups and the partitions to which they correspond. Consequently, the paper is combinatorial in nature.

Lower bounds for representation growth

Abstract This article examines lower bounds for the representation growth of finitely generated (particularly profinite and pro-p) groups. It also considers the related question of understanding the maximal multiplicities of character degrees in finite groups, and in particular simple groups.

Normal subsystems of fusion systems

In this article, we prove that, for any saturated fusion system, the (unique) smallest weakly normal subsystem of it on a given strongly closed subgroup is actually normal. This has a variety of corollaries, such as the statement that the notion of a simple fusion system is independent of whether one uses weakly normal or normal subsystems. We also develop a theory of weakly normal maps, consider intersections and products of weakly normal subsystems, and the hypercentre of a fusion system. The theory of fusion systems is becoming an important topic in algebra, with interactions with group theory, representation theory and topology. This article is concerned with the structure of normal subsystems of fusion systems. There are two notions of a ‘normal’ subsystem in the literature, one stronger than the other. (We recall their definitions in this article.) We follow [7] and call the subsystem considered by Aschbacher in [2 ]a normal subsystem, and the subsystem considered by, among others, Linckelmann in [12] (see also [14]) a weakly normal subsystem. Our first result highlights the exact relationship between normal and weakly normal subsystems.In this article we prove that for any saturated fusion system, that the (unique) smallest weakly normal subsystem of it on a given strongly closed subgroup is actually normal. This has a variety of corollaries, such as the statement that the notion of a simple fusion system is independent of whether one uses weakly normal or normal subsystems. We also develop a theory of weakly normal maps, consider intersections and products of weakly normal subsystems, and the hypercentre of a fusion system.

Alternating subgroups of exceptional groups of Lie type

In this paper, we examine embeddings of alternating and symmetric groups into almost simple groups of exceptional type. In particular, we prove that if the alternating or symmetric group has degree equal to 5, or 8 or more, then it cannot appear as the maximal subgroup of any almost simple exceptional group of Lie type. Furthermore, in the remaining open cases of degrees 6 and 7 we give considerable information about the possible embeddings. Note that no maximal alternating or symmetric subgroups are known in the remaining cases. This is the first in a sequence of papers aiming to substantially improve the state of knowledge about the maximal subgroups of exceptional groups of Lie type.

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The Brauer trees of non-crystallographic groups of Lie type

Abstract In this article we determine the Brauer trees of the unipotent blocks with cyclic defect group in the ‘groups’ I 2 ( n , q ) , H 3 ( q ) and H 4 ( q ) . The degrees of the unipotent characters of these objects were given by Lusztig, and using the general theory of perverse equivalences we can reconstruct the Brauer trees that would be consistent with Deligne–Lusztig theory and the geometric version of Broueʼs conjecture. We construct the trees using standard arguments whenever possible, and check that the Brauer trees predicted by Broueʼs conjecture are consistent with both the mathematics and philosophy of blocks with cyclic defect groups.

Groups with a p-element acting with a single non-trivial Jordan block on a simple module in characteristic p

Abstract Let V be a vector space over a field of characteristic p. In this paper we complete the classification of all irreducible subgroups G of GL ⁢ ( V ) {\mathrm{GL}(V)} that contain a p-element whose Jordan normal form has exactly one non-trivial block, and possibly multiple trivial blocks. Broadly speaking, such a group acting primitively is a classical group acting on a symmetric power of a natural module, a 7-dimensional orthogonal group acting on the 8-dimensional spin module, a complex reflection group acting on a reflection representation, or one of a small number of other examples, predominantly with a self-centralizing cyclic Sylow p-subgroup.