On orthogonal discriminants of characters

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 $p=2$.