Orthogonal representations of finite groups

• Orthogonale Darstellungen endlicher Gruppen

Braun, Oliver; Nebe, Gabriele (Thesis advisor); Hiß, Gerhard (Thesis advisor)

Aachen (2016)
Dissertation / PhD Thesis

Dissertation, RWTH Aachen University, 2016

Abstract

The subject of this thesis are orthogonal representations of finite groups. By this we mean a pair (V, q) where V is a KG-module and q is a G-invariant non-degenerate quadratic form on V . We restrict our considerations to finite groups G and totally real number fields K.The G-invariant quadratic forms on V form a K-vector space and we are mainly interested in the case where that vector space is one-dimensional. If it is, we call the KG-module V uniform.It is a well known result of ordinary representation theory that the isomorphism type of V is determined by its character χ. In other words, if V is uniform, χ determines a non-degenerate G-invariant quadratic form q on V up to scalar multiples, which begs the question: How can we determine the isometry class of q given the character χ?First results on this question were achieved by Gabriele Nebe, who devised a purely character-theoretic method to answer the above question in her habilitation thesis. However, Nebe’s method is only applicable under some favourable (and restrictive) conditions, so in general the problem is still open.This thesis contributes to this open question in several ways. First, the theoretical background and the known results are presented. After that we discuss a version of Clifford theory for orthogonal representations of normal subgroups and classify the orthogonal representations of semidirect products of the form C_p : C_(p−1). These results are then used for one of the main results of the thesis, namely the classification of the orthogonalrepresentations of the infinite series of groups SL_2(q) for prime powers q. We also discuss an independent topic, so-called Clifford orders, which are subrings of the classical Clifford algebra of a quadratic space. We define those subrings and describe some of their arithmetic properties.The thesis is concluded with a section of computational results for orthogonal representations of the finite simple groups contained in the "Atlas of finite groups".