High Performance Computations in Group Representation Theory

Abstract

The finite subgroups of GLn(Q) are classified up to dimension n = 31 by giving a system of representatives for the conjugacy classes of the maximal finite ones ([12], [9], [15],[16], [17]) cf. [11] for a survey on this and interrelations between these groups. Recently the classification has been extended to the one of absolutely irreducible maximal finite subgroups G of GLnCD),where V is a totally definite quaternion algebra and n. dimQ(V) ::; 40. As usual a subgroup G ::; GLn(V) is called absolutely irreducible, if the enveloping Q-algebra Q:i := {EgEG agg I ag E Q} ~ vnxn is the whole matrix ring vnxn (cf. [8]). The classification of these groups yields a partial classification of the rational maximal finite matrix groups in the new dimensions 32, 36, and 40 on one hand and on the other hand it gives nice Hermitian structures for interesting lattices. For example one finds eleven quaternionic structures of the Leech lattice as a Hermitian lattice of rank n > 1 over a maximal order in a definite quaternion algebra Vwith absolutely irreducible maximal finite automorphism group as displayed in Table 1. Rather than giving a survey of the classification results this note is devoted to a general structure theorem (cf. Theorem 4 below).