Splitting fields of real irreducible representations of finite groups

TL;DR

The paper proves that real irreducible representations of finite groups can be realized over a specific real cyclotomic field and provides an algorithm to achieve this transformation.

Contribution

It establishes the minimal field of realization for real irreducible representations of finite groups and offers an explicit algorithm for the field reduction process.

Findings

Irreducible real representations are realizable over the intersection of cyclotomic and real fields.

An explicit algorithm transforms realizations over cyclotomic fields to the minimal real subfield.

The minimal field of realization is the intersection of the cyclotomic field with the real numbers.

Abstract

We show that any irreducible representation $\rho$ of a finite group $G$ of exponent $n$, realisable over $\mathbb{R}$, is realisable over the field $E:=\mathbb{Q}(\zeta_n)\cap\mathbb{R}$ of real cyclotomic numbers of order $n$, and describe an algorithmic procedure transforming a realisation of $\rho$ over $\mathbb{Q}(\zeta_n)$ to one over $E$.