Random generation of direct sums of finite non-degenerate subspaces

2021/11/03 by Glasby, S. P., Niemeyer, Alice C., Praeger, Cheryl E.
#05-08 #20D05 #20D06 #20F65 #68W20 #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2111.02198

Abstract

Let V be a d-dimensional vector space over a finite field \mathbbF equipped with a non-degenerate hermitian, alternating, or quadratic form. Suppose |\mathbbF|=q² if V is hermitian, and |\mathbbF|=q otherwise. Given integers e, e' such that e+e'\leqslant d, we estimate the proportion of pairs (U, U'), where U is a non-degenerate e-subspace of V and U' is a non-degenerate e'-subspace of V, such that U∩ U'=0 and U⊕ U' is non-degenerate (the sum U⊕ U' is direct and usually not perpendicular). The proportion is shown to be positive and at least 1-c/q>0 for some constant c. For example, c=7/4 suffices in both the unitary and symplectic cases. The arguments in the orthogonal case are delicate and assume that dim(U) and dim(U') are even, an assumption relevant for an algorithmic application (which we discuss) for recognising finite classical groups. We also describe how recognising a classical groups G relies on a connection between certain pairs (U,U') of non-degenerate subspaces and certain pairs (g,g')∈ G² of group elements where U=\rm im(g-1) and U'=\rm im(g'-1).

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