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@PHDTHESIS{Hoyer:1002384,
      author       = {Hoyer, Linda},
      othercontributors = {Nebe, Gabriele and Geck, Meinolf and Fourier, Ghislain Paul
                          Thomas},
      title        = {{O}rthogonal determinants of finite groups of {L}ie type},
      school       = {RWTH Aachen University},
      type         = {Dissertation},
      address      = {Aachen},
      publisher    = {RWTH Aachen University},
      reportid     = {RWTH-2025-00497},
      pages        = {1 Online-Ressource : Illustrationen},
      year         = {2024},
      note         = {Veröffentlicht auf dem Publikationsserver der RWTH Aachen
                      University 2025; Dissertation, RWTH Aachen University, 2024},
      abstract     = {An $orthogonal$ representation of a finite group $G$ is a
                      homomorphism $\rho:G \to \mathrm{GL}_n(K)$, for a natural
                      number $n$ and a field $K \subseteq \mathbb{R}$.
                      Analogously, we say a character $\chi$ of $G$ is orthogonal
                      if any corresponding representation is orthogonal.Nebe
                      (2022) showed that for an orthogonal character $\chi \in
                      \mathrm{Irr}(G)$ of even degree ($\chi \in
                      \mathrm{Irr}^+(G)$), there exists a unique element
                      $$\det(\chi):=d \in
                      \mathbb{Q}(\chi)^{\times}/(\mathbb{Q}(\chi)^{\times})^2,$$
                      such that for any representation $\rho:G \to
                      \mathrm{GL}_n(K)$ affording $\chi$ over an arbitrary field
                      $K/\mathbb{Q}(\chi)$ and all $\rho(G)$-invariant,
                      non-degenerate bilinear forms $\beta$, it holds that
                      $$\det(\beta)=d \cdot (K^{\times})^2.$$ We say that
                      $\det(\chi)$ is the $\textit{orthogonal determinant}$ of
                      $\chi$.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
                      $\mathrm{SL}_n(q), \mathrm{GL}_n(q)$ and $\mathrm{SU}_n(q)$
                      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 $\chi \in
                      \mathrm{Irr}^+(G)$. Given that $q$ is odd, we show that
                      there is some sort of "Jordan decomposition" of
                      $\det(\chi)=\det(\chi_U) \det(\chi_T)$, i.e., a
                      decomposition into a unipotent part $\det(\chi_U)$ and a
                      semisimple part $\det(\chi_T)$.In contrary to the relatively
                      easy determination of $\det(\chi_U)$, the calculation of
                      $\det(\chi_T)$ 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 $\mathrm{SL}_3(q)$ and
                      $G_2(q)$. In the final chapter, we handle the groups
                      $\mathrm{GL}_n(q)$, where we accomplish a complete
                      description of the orthogonal determinants.},
      cin          = {114710 / 110000},
      ddc          = {510},
      cid          = {$I:(DE-82)114710_20140620$ / $I:(DE-82)110000_20140620$},
      pnm          = {TRR 195: Symbolische Werkzeuge in der Mathematik und ihre
                      Anwendung (286237555)},
      pid          = {G:(GEPRIS)286237555},
      typ          = {PUB:(DE-HGF)11},
      doi          = {10.18154/RWTH-2025-00497},
      url          = {https://publications.rwth-aachen.de/record/1002384},
}