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Unitary Discriminants of SL3(q) and SU3(q)

Linda Hoyer, Gabriele Nebe

TL;DR

This work delivers a complete enumeration of the unitary discriminants for the even-degree indicator '$o$' ordinary irreducible characters of the Lie-type groups $SL_3(q)$ and $SU_3(q)$. It combines a robust methodological framework—leveraging Hermitian/quadratic forms, unitary stability, Schur indices, and Brauer theory—with explicit Harish-Chandra induction from parabolic Levi factors (notably $GL_2(q)$) and detailed analysis of Borel and metabelian subgroups. For $SL_3(q)$, unitary discriminants are determined via Harish-Chandra induction from the parabolic, with explicit results depending on whether $q$ is even or odd. For $SU_3(q)$, the paper treats even and odd $q$ separately, computing unitary discriminants through a mixture of restriction to the Borel, analysis of $A_0$ and $B$, and a comprehensive case-by-case table for $q$ odd, including the dependence on parameters such as $u,v,w$ and the function $f(u)$. These results extend the program of understanding unitary discriminants in finite groups of Lie type and provide concrete data for SL$_3(q)$ and SU$_3(q)$ across all prime powers $q$.

Abstract

We give a full list of the unitary discriminants of the even degree indicator 'o' ordinary irreducible characters of SL3(q) and SU3(q).

Unitary Discriminants of SL3(q) and SU3(q)

TL;DR

This work delivers a complete enumeration of the unitary discriminants for the even-degree indicator '' ordinary irreducible characters of the Lie-type groups and . It combines a robust methodological framework—leveraging Hermitian/quadratic forms, unitary stability, Schur indices, and Brauer theory—with explicit Harish-Chandra induction from parabolic Levi factors (notably ) and detailed analysis of Borel and metabelian subgroups. For , unitary discriminants are determined via Harish-Chandra induction from the parabolic, with explicit results depending on whether is even or odd. For , the paper treats even and odd separately, computing unitary discriminants through a mixture of restriction to the Borel, analysis of and , and a comprehensive case-by-case table for odd, including the dependence on parameters such as and the function . These results extend the program of understanding unitary discriminants in finite groups of Lie type and provide concrete data for SL and SU across all prime powers .

Abstract

We give a full list of the unitary discriminants of the even degree indicator 'o' ordinary irreducible characters of SL3(q) and SU3(q).

Paper Structure

This paper contains 22 sections, 24 theorems, 71 equations.

Table of Contents

  1. Introduction
  2. Methods
  3. Quadratic and Hermitian forms
  4. Unitary stable characters
  5. Induced representations
  6. Schur indices
  7. Unitary discriminants of real characters
  8. Orthogonal subalgebras
  9. A generalisation of the $Q_8$-trick
  10. The special linear group
  11. The characters of $\mathop{\mathrm{GL}}\nolimits_2(q)$
  12. The characters of $P$
  13. The characters of $\mathop{\mathrm{SL}}\nolimits_3(q)$
  14. The special unitary group, general results
  15. The special unitary group
  16. ...and 7 more sections

Key Result

Lemma 2.6

(see BOI) Let $F$ be an odd degree extension of $K$. Put $E:=FL$ and extend $\sigma$ to a field automorphism $\sigma$ of $E$ with fixed field $F$. Let $(V,H)$ and $(W,H')$ be two $L/K$ Hermitian spaces. If the $E/F$ Hermitian spaces $(V\otimes_L E,H_E)$ and $(W\otimes _L E, H'_E)$ are isometric, the

Theorems & Definitions (57)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Lemma 2.6
  • Definition 2.7
  • Definition 2.8
  • Definition 2.9
  • Remark 2.10
  • ...and 47 more