Orthogonal Stability

TL;DR

This paper investigates the properties of orthogonally stable characters in representation theory, establishing conditions for stability, their behavior under reduction mod p, and methods for computing associated determinants.

Contribution

It introduces criteria for orthogonal stability of characters, analyzes their behavior under modular reduction, and provides computational methods and examples.

Findings

Orthogonal stability occurs when no odd-degree orthogonal constituents are present.

The determinant mod p of an orthogonally stable character's reduction matches the reduction of the original determinant.

Provides methods and examples for computing determinants of orthogonally stable characters.

Abstract

A character (ordinary or modular) is called orthogonally stable if all non-degenerate quadratic forms fixed by representations with those constituents have the same determinant mod squares. We show that this is the case provided there are no odd-degree orthogonal constituents. We further show that if the reduction mod p of an ordinary character is orthogonally stable, this determinant is the reduction mod p of the ordinary one. In particular, if the characteristic does not divide the group order, we immediately see in which orthogonal group it lies. We sketch methods for computing this determinant, and give some examples.