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.