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Unitary discriminants of characters

Gabriele Nebe

TL;DR

This work develops and applies a comprehensive framework for computing unitary discriminants of invariant Hermitian forms attached to ordinary and Brauer characters of finite groups. It extends the prior orthogonal-discriminant program to the unitary setting by introducing unitary stability, discriminant algebras for involutions of the second kind, and unitary condensation via suitable automorphisms. The authors present general theorems linking discriminants to local data, quasi-splits, and fixed-algebra constructions, and demonstrate the methods on ATLAS groups such as $O_{10}^+(2)$, $HN$, and $SU(3,7)$, yielding explicit discriminants and confirming global-local compatibility. The results provide a practical, scalable toolkit for determining unitary representations over number fields, with implications for database entries in OD and related character-theory computations. Overall, the paper contributes both a robust theoretical foundation and a suite of computational techniques for unitary discriminants across large and complex finite groups.

Abstract

Together with David Schlang we computed the discriminants of the invariant Hermitian forms for all indicator $o$ even degree absolutely irreducible characters of the ATLAS groups supplementing the tables of orthogonal determinants computed in collaboration with Richard Parker, Tobias Braun and Thomas Breuer. The methods that are used in the unitary case are described in this paper. A character has a well defined unitary discriminant, if and only if it is unitary stable, i.e. all irreducible unitary constituents have even degree. Computations for large degree characters are only possible because of a new method called {\em unitary condensation}. A suitable automorphism helps to single out a square class of the real subfield of the character field consisting of representatives of the discriminant of the invariant Hermitian forms. This square class can then be determined modulo enough primes.

Unitary discriminants of characters

TL;DR

This work develops and applies a comprehensive framework for computing unitary discriminants of invariant Hermitian forms attached to ordinary and Brauer characters of finite groups. It extends the prior orthogonal-discriminant program to the unitary setting by introducing unitary stability, discriminant algebras for involutions of the second kind, and unitary condensation via suitable automorphisms. The authors present general theorems linking discriminants to local data, quasi-splits, and fixed-algebra constructions, and demonstrate the methods on ATLAS groups such as , , and , yielding explicit discriminants and confirming global-local compatibility. The results provide a practical, scalable toolkit for determining unitary representations over number fields, with implications for database entries in OD and related character-theory computations. Overall, the paper contributes both a robust theoretical foundation and a suite of computational techniques for unitary discriminants across large and complex finite groups.

Abstract

Together with David Schlang we computed the discriminants of the invariant Hermitian forms for all indicator even degree absolutely irreducible characters of the ATLAS groups supplementing the tables of orthogonal determinants computed in collaboration with Richard Parker, Tobias Braun and Thomas Breuer. The methods that are used in the unitary case are described in this paper. A character has a well defined unitary discriminant, if and only if it is unitary stable, i.e. all irreducible unitary constituents have even degree. Computations for large degree characters are only possible because of a new method called {\em unitary condensation}. A suitable automorphism helps to single out a square class of the real subfield of the character field consisting of representatives of the discriminant of the invariant Hermitian forms. This square class can then be determined modulo enough primes.

Paper Structure

This paper contains 32 sections, 22 theorems, 64 equations.

Table of Contents

  1. Introduction
  2. Quadratic and Hermitian forms
  3. Hermitian forms over number fields
  4. Lattices
  5. Lattices, the local picture
  6. Algebras with involution
  7. Central simple algebras with involution
  8. Orthogonal subalgebras
  9. Stable subalgebras
  10. Discriminants of characters
  11. Characters of finite groups
  12. Schur indices
  13. The unitary discriminant of a character
  14. Unitary stability
  15. Unitary discriminants of real characters
  16. ...and 17 more sections

Key Result

Proposition 3.1

(see Scharlau) Two Hermitian spaces $(V,H)$ and $(V',H')$ are isometric if and only if the quadratic spaces $(V_K,Q_H)$ and $(V'_K,Q_{H'})$ are isometric. If $L=K[\sqrt{\delta}]$ then $\mathop{\mathrm{disc}}\nolimits(Q_H) = \delta ^n$ and the Clifford invariant of $Q_H$ is $\Delta(H)$.

Theorems & Definitions (73)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Proposition 3.1
  • Corollary 3.2
  • Lemma 3.3
  • proof
  • Remark 3.4
  • Remark 3.5
  • Proposition 3.6
  • ...and 63 more