2022 ON ORTHOGONAL DISCRIMINANTS OF CHARACTERS
Gabriele Nebe
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Albanian J. Math. 16(1): 41-49 (2022). DOI: 10.51286/albjm/1658730113

Abstract

An ordinary character χ of a finite group is called orthogonally stable, if all non-degenerate invariant quadratic forms on any module affording the character χ have the same discriminant. This is the orthogonal discriminant, disc(χ), of χ, 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-groups there is an explicit formula for disc(χ) that reads disc(χ)=(p)χ(1)/2 if p3(mod 4) and disc(χ)=(1)χ(1)/2 for p=2. For rational characters χ and p1(mod 4) the discriminant is disc(χ)=pχ(1)/(p1).

Citation

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Gabriele Nebe. "ON ORTHOGONAL DISCRIMINANTS OF CHARACTERS." Albanian J. Math. 16 (1) 41 - 49, 2022. https://doi.org/10.51286/albjm/1658730113

Information

Published: 2022
First available in Project Euclid: 11 July 2023

MathSciNet: MR4466997
Digital Object Identifier: 10.51286/albjm/1658730113

Subjects:
Primary: 20C15
Secondary: 11E12

Keywords: character fields , discriminant fields , orthogonal discriminants , orthogonal representations of finite groups

Rights: Copyright © 2022 Research Institute of Science and Technology (RISAT)

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