Abstract
Given positive integers \(e_1,e_2\), let \(X_i\) denote the set of \(e_i\)-dimensional subspaces of a fixed finite vector space\(V=({\mathbb F}_q)^{e_1+e_2}\). Let \(Y_i\) be a non-empty subset of \(X_i\) and let \(\alpha _i = |Y_i|/|X_i|\). We give a positive lower bound, depending only on \(\alpha _1,\alpha _2,e_1,e_2,q\), for the proportion of pairs \((S_1,S_2)\in Y_1\times Y_2\) which intersect trivially. As an application, we bound the proportion of pairs of non-degenerate subspaces of complementary dimensions in a finite classical space that intersect trivially. This problem is motivated by an algorithm for recognizing classical groups. By using techniques from algebraic graph theory, we are able to handle orthogonal groups over the field of order 2, a case which had eluded Niemeyer, Praeger, and the first author.
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Acknowledgements
We thank the referee for their very helpful comments. The first author is supported by the Australian Research Council Discovery Grant DP190100450. The second author is supported by a postdoctoral fellowship of the Research Foundation—Flanders (FWO).
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Communicated by K.-U. Schmidt.
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Glasby, S.P., Ihringer, F. & Mattheus, S. The proportion of non-degenerate complementary subspaces in classical spaces. Des. Codes Cryptogr. 91, 2879–2891 (2023). https://doi.org/10.1007/s10623-023-01235-3
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DOI: https://doi.org/10.1007/s10623-023-01235-3