On orthogonal discriminants of characters

TL;DR

This paper investigates the orthogonal discriminants of characters of finite groups, proving a conjecture for solvable groups and providing explicit formulas for p-groups, advancing understanding of quadratic forms in representation theory.

Contribution

It proves the conjecture that orthogonal discriminants are always odd square classes for finite solvable groups and derives explicit formulas for p-groups.

Findings

Proved the conjecture for finite solvable groups.

Derived explicit formulas for p-group orthogonal discriminants.

Confirmed the conjecture through experimental evidence.

Abstract

An ordinary character $\chi $ of a finite group is called orthogonally stable, if all non-degenerate invariant quadratic forms on any module affording the character $\chi $ have the same discriminant. This is the orthogonal discriminant, $\disc(\chi )$, of $\chi $, a square class of the character field. Based on experimental evidence we conjecture that the orthogonal discriminant is always an odd square class in the sense of Definition 1.4. This note proves this conjecture for finite solvable groups. For $p$-group there is an explicit formula for $\disc(\chi )$ that reads $\disc(\chi ) = (-p)^{\chi(1)/2}$ if $p\equiv 3 \pmod{4}$ and $\disc (\chi ) = (-1)^{\chi(1)/2}$ for $p=2$.