Writing representations over minimal fields

TL;DR

This paper presents a probabilistic algorithm to find minimal field representations of finite group representations over finite fields, optimizing the field size for computational efficiency.

Contribution

It introduces a new probabilistic method based on Hilbert's Theorem 90 to produce minimal field representations of finite group representations over finite fields.

Findings

Algorithm efficiently finds minimal field representations.

Method applies to finite soluble groups and their irreducible representations.

Expected runtime is proportional to the field extension degree and cube of the representation dimension.

Abstract

The chief aim of this paper is to describe a procedure which, given a $d$-dimensional absolutely irreducible matrix representation of a finite group over a finite field $\mathbb{E}$, produces an equivalent representation such that all matrix entries lie in a subfield $\mathbb{F}$ of $\mathbb{E}$ which is as small as possible. The algorithm relies on a matrix version of Hilbert's Theorem 90, and is probabilistic with expected running time ${\rm O}(|\mathbb{E}:\mathbb{F}|d^3)$ when $|\mathbb{F}|$ is bounded. Using similar methods we then describe an algorithm which takes as input a prime number and a power-conjugate presentation for a finite soluble group, and as output produces a full set of absolutely irreducible representations of the group over fields whose characteristic is the specified prime, each representation being written over its minimal field.