Orthogonal determinants of characters

TL;DR

This paper investigates the properties of orthogonal characters of even degree, establishing a unique square class in the character field that governs the determinants of invariant quadratic forms across all representations.

Contribution

It introduces the concept of a unique square class associated with irreducible orthogonal characters of even degree, linking it to invariant quadratic forms in representations.

Findings

Existence of a unique square class for even-degree orthogonal characters

Determinant of invariant quadratic forms lies in this square class

Provides a structural understanding of orthogonal characters and their quadratic forms

Abstract

For an irreducible orthogonal character $\chi $ of even degree there is a unique square class $\det({\chi })$ in the character field such that the invariant quadratic forms in any $L$-representation affording $\chi $ have determinant in $\det({\chi })(L^*)^2$.