Unitary discriminants of characters

TL;DR

This paper details the computation of unitary discriminants for certain characters of ATLAS groups, introducing a new method called unitary condensation to handle large degrees and determine discriminants via automorphisms and prime-based methods.

Contribution

It presents the methods used to compute unitary discriminants for all indicator o even degree absolutely irreducible characters of ATLAS groups, including the novel unitary condensation technique.

Findings

Computed discriminants for all relevant characters.

Introduced the unitary condensation method.

Determined discriminants modulo primes.

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.