Writing representations over minimal fields

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.

No reviews yet.

No comments yet.