Writing projective representations over subfields

Abstract

Let G = X be an absolutely irreducible subgroup of GL(d, K), and let F be a proper subfield of the finite field K. We present a practical algorithm to decide constructively whether or not G is conjugate to a subgroup of GL(d, F ).K × , where K × denotes the centre of GL(d, K). If the derived group of G also acts absolutely irreducibly, then the algorithm is Las Vegas and costs O(|X|d 3 + d 2 log |F |) arithmetic operations in K. This work forms part of a recognition project based on Aschbacher's classification of maximal subgroups of GL(d, K).

Algorithms for matrix groups

“…The polynomial-time algorithm of [48] to decide membership in C5 requires that G ′ acts absolutely irreducibly on V . Implementations of both are available in Magma.…”

Recognition of Small Dimensional Representations of General Linear Groups

“…Let Since the exponent of ω in l i, j is a multiple of q − 1, we cannot compute ω in polynomial time. Hence we cannot compute the base change matrix B of Section 4.3, but instead use the algorithm of [8] to perform the final base change.…”

Identification of matrix generators of a Chevalley group

“…The run times of Steps 1, 2, and 3 of Algorithm 1 are bounded by O(N dim 8 V log 3 |K| log 3 |k|), O(N dim 9 L log 3 |k|), and O(dim 11 L log 3 |k| + dim 15/2 L log 4 |k|), respectively [5,8,12,18]. Hence, the run time of Algorithm 1 is bounded by a polynomial function of the length of its input data.…”

“…The main algorithm of [5] reduces the size of the field of definition of an absolutely irreducible module. It runs in Las Vegas polynomial time.…”