Writing representations over minimal fields

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 E, produces an equivalent representation such that all matrix entries lie in a subfield F of 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 O(|E : F|d 3 ) when |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.

Constructive recognition of 𝑃𝑆𝐿(2,𝑞)

“…The complexity of writing an irreducible representation in smaller dimension over a larger field is O(d 3 ) field operations in the smaller field [15]. The complexity of writing an absolutely irreducible representation in the same dimension over a smaller field is O(d 3 ) field operations in the larger field [13]; we observe that this also assumes the existence of a discrete logarithm oracle for the smaller field. Here the smaller field is GF(q), and we already use a discrete logarithm oracle for this field for the explicit recognition of PSL(2, q) in the natural representation.…”

“…If G, as an absolutely irreducible subgroup of GL(d, p a ), is conjugate to a subgroup of GL(d, p b ) for some proper divisor b of a, then we find such a conjugating matrix using the algorithm of [13], and thus write G over GF(p b ). We may now apply the following theorem of Brauer & Nesbitt [7, §30].…”

Writing projective representations over subfields

“…These are used to contrast the performance of our new algorithm for this task with that of Glasby and Howlett [6]. In the column entitled "Time," we list the CPU time in seconds needed to construct the conjugation matrix using the algorithm of Section 3; in the column labelled "G & H" we record the CPU time taken by our implementation of the Glasby and Howlett algorithm to construct this matrix.…”

“…Glasby and Howlett [6] present an algorithm to answer this special case, which has similar complexity, given an oracle to construct discrete logarithms in F . For a description of discrete logarithm algorithms, see [15,Chapter 4].…”

Constructing class invariants

“…(3.2) holds? We will use a version of the Glasby-Howlett probabilistic algorithm [8]. We consider the sum…”