h1

%0 Thesis
%A Hoyer, Linda
%T Orthogonal determinants of finite groups of Lie type
%I RWTH Aachen University
%V Dissertation
%C Aachen
%M RWTH-2025-00497
%P 1 Online-Ressource : Illustrationen
%D 2024
%Z Veröffentlicht auf dem Publikationsserver der RWTH Aachen University 2025
%Z Dissertation, RWTH Aachen University, 2024
%X An orthogonal representation of a finite group G is a homomorphism ρ:G → GL<sub>n</sub>(K), for a natural number n and a field K  ⊆ \mathbbR. Analogously, we say a character χ of G is orthogonal if any corresponding representation is orthogonal.Nebe (2022) showed that for an orthogonal character χ ∈ Irr(G) of even degree (χ ∈ Irr<sup>+</sup>(G)), there exists a unique element 
%X <br clear="all" /><table border="0" width="100%"><tr><td>
%X <table align="center" cellspacing="0"  cellpadding="2"><tr><td nowrap="nowrap" align="center">
%X </td><td nowrap="nowrap" align="center">
%X det<br />
%X </td><td nowrap="nowrap">(χ):=d  ∈ \mathbbQ(χ)<sup>×</sup>/(\mathbbQ(χ)<sup>×</sup>)<sup>2</sup>,</td></tr></table>
%X </td></tr></table>
%X 
%X  such that for any representation ρ:G → GL<sub>n</sub>(K) affording χ over an arbitrary field K/\mathbbQ(χ) and all ρ(G)-invariant, non-degenerate bilinear forms β, it holds that 
%X <br clear="all" /><table border="0" width="100%"><tr><td>
%X <table align="center" cellspacing="0"  cellpadding="2"><tr><td nowrap="nowrap" align="center">
%X </td><td nowrap="nowrap" align="center">
%X det<br />
%X </td><td nowrap="nowrap">(β)=d ·(K<sup>×</sup>)<sup>2</sup>.</td></tr></table>
%X </td></tr></table>
%X 
%X  We say that det(χ) is the <i>orthogonal determinant</i> of χ.As part of the classification of finite simple groups, the groups of Lie type form the largest class among them. Examples of finite groups of Lie type include SL<sub>n</sub>(q), GL<sub>n</sub>(q) and SU<sub>n</sub>(q) for q a prime power. The goal of this thesis is to present methods for the calculation of the orthogonal determinants of the finite groups of Lie type. Let G:=G(q) be a finite group of Lie type with parameter q and let χ ∈ Irr<sup>+</sup>(G). Given that q is odd, we show that there is some sort of "Jordan decomposition" of det(χ)=det(χ<sub>U</sub>) det(χ<sub>T</sub>), i.e., a decomposition into a unipotent part  det(χ<sub>U</sub>) and a semisimple part det(χ<sub>T</sub>).In contrary to the relatively easy determination of det(χ<sub>U</sub>), the calculation of det(χ<sub>T</sub>) proves to be a challenge. For that, we apply the theory of Iwahori-Hecke algebras, which are deformations of Coxeter groups, and extensions thereof.The thesis consists of 6 chapters. After the introduction, the following two chapters establish the theory of orthogonal determinants and finite groups of Lie type. Afterwards we will consider Coxeter groups, where the orthogonal determinants of all Coxeter groups, as well as the alternating groups and some Iwahori-Hecke algebras, are covered. In the fifth chapter, we will describe orthogonal determinants of finite groups of Lie type, where we will also consider some examples like SL<sub>3</sub>(q) and G<sub>2</sub>(q). In the final chapter, we handle the groups GL<sub>n</sub>(q), where we accomplish a complete description of the orthogonal determinants.
%F PUB:(DE-HGF)11
%9 Dissertation / PhD Thesis
%R 10.18154/RWTH-2025-00497
%U https://publications.rwth-aachen.de/record/1002384