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Let G be a finite group and \(\rho : G \rightarrow {{\,\textrm{GL}\,}}(2n, F)\) be an absolutely irreducible orthogonal representation of even degree over a finite field F. Then \(\rho (G)\) embeds into \({{\,\textrm{GO}\,}}^+(2n, F)\) or \({{\,\textrm{GO}\,}}^-(2n, F)\) . We describe methods to decide which case holds for \(\rho \) , and use them to determine most of the orthogonal discriminants of the absolutely irreducible orthogonal representations of even degree that are listed in the ATLAS of Finite Groups [2].

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An Atlas of Orthogonal Representations

  • 章节

  • 发表日期：

    2025年

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  • Thomas Breuer,
  • Gabriele Nebe,
  • Richard Parker

摘要

Let G be a finite group and \(\rho : G \rightarrow {{\,\textrm{GL}\,}}(2n, F)\) be an absolutely irreducible orthogonal representation of even degree over a finite field F. Then \(\rho (G)\) embeds into \({{\,\textrm{GO}\,}}^+(2n, F)\) or \({{\,\textrm{GO}\,}}^-(2n, F)\) . We describe methods to decide which case holds for \(\rho \) , and use them to determine most of the orthogonal discriminants of the absolutely irreducible orthogonal representations of even degree that are listed in the ATLAS of Finite Groups [2].