Abstract
Let V be a d-dimensional vector space over a finite field F equipped with a non-degenerate hermitian, alternating, or quadratic form. Suppose |F|=q2 if V is hermitian, and |F|=q otherwise. Given integers e,eⲠsuch that e+eâ˛âŠ˝d, we estimate the proportion of pairs (U,Uâ˛), where U is a non-degenerate e-subspace of V and UⲠis a non-degenerate eâ˛-subspace of V, such that UâŠUâ˛=0 and UâUⲠis non-degenerate (the sum UâUⲠis direct and usually not perpendicular). The proportion is shown to be positive and at least 1âc/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 dimâĄ(U) and dimâĄ(Uâ˛) 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,Uâ˛) of non-degenerate subspaces and certain pairs (g,gâ˛)âG2 of group elements where U=im(gâ1) and Uâ˛=im(gâ˛â1).
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