Representation Growth of Finite Quasisimple Groups

In this thesis, we establish concrete numerical upper bounds for the representation growth of various families of finite quasisimple groups. Let G be a finite quasisimple group and let rn(G) denote the number of inequivalent irreducible n-dimensional linear representations of G. We describe certain infinite collections C of finite quasisimple groups and derive upper bounds to the growth of rn(G) as a function of n; the bounds hold for any G in C. We also bound the total number sn(C) of inequivalent faithful irreducible n-dimensional representations of groups in C. Three cases are examined: the complex representation growth of alternating groups and their Schur covers, the complex representation growth of groups of Lie type, and the cross-characteristic modular representation growth of groups of Lie type. In all the cases, it is necessary to find lower bounds for the minimal dimensions of irreducible representations, and also to classify the representations of some of the smallest possible dimensions. The main results are in all cases upper bounds to the growth of rn(G) or sn(C) for a given collection C. All bounds have the form cns, where c and s are some constants that depend on the collection under study, with s being always at most 1. The results are applied to a known problem concerning the number of conjugacy classes of maximal subgroups in classical groups. By Aschbacher’s Theorem, the maximal subgroups of finite classical groups can be classified into so-called geometrical types, but there are some additional almost simple subgroups that do not fit into this classification. However, these almost simple subgroups are obtained from representations of quasisimple groups, and the number of conjugacy classes of such subgroups can be estimated by counting the number of irreducible representations