The proportion of non-degenerate complementary subspaces in classical spaces

S. P. Glasby; Ferdinand Ihringer; Sam Mattheus

doi:10.1007/s10623-023-01235-3

The proportion of non-degenerate complementary subspaces in classical spaces

S. P. Glasby, Ferdinand Ihringer, Sam Mattheus

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Given positive integers e₁, e₂ , let X_i denote the set of e_i -dimensional subspaces of a fixed finite vector space V=(Fq)e1+e2 . Let Y_i be a non-empty subset of X_i and let α_i= | Y_i| / | X_i| . We give a positive lower bound, depending only on α₁, α₂, e₁, e₂, q , for the proportion of pairs (S₁, S₂) ∈ Y₁× Y₂ 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.

Originele taal-2	English
Pagina's (van-tot)	2879-2891
Aantal pagina's	13
Tijdschrift	Designs, Codes, and Cryptography
Volume	91
Nummer van het tijdschrift	9
DOI's	https://doi.org/10.1007/s10623-023-01235-3
Status	Published - 23 mei 2023

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Funding Information:
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).

Publisher Copyright:
© 2023, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.

Copyright:
Copyright 2023 Elsevier B.V., All rights reserved.

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abstract = "Given positive integers e1, e2 , let Xi denote the set of ei -dimensional subspaces of a fixed finite vector space V=(Fq)e1+e2 . Let Yi be a non-empty subset of Xi and let αi= | Yi| / | Xi| . We give a positive lower bound, depending only on α1, α2, e1, e2, q , for the proportion of pairs (S1, S2) ∈ Y1× Y2 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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AU - Ihringer, Ferdinand

AU - Mattheus, Sam

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PY - 2023/5/23

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N2 - Given positive integers e1, e2 , let Xi denote the set of ei -dimensional subspaces of a fixed finite vector space V=(Fq)e1+e2 . Let Yi be a non-empty subset of Xi and let αi= | Yi| / | Xi| . We give a positive lower bound, depending only on α1, α2, e1, e2, q , for the proportion of pairs (S1, S2) ∈ Y1× Y2 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.

AB - Given positive integers e1, e2 , let Xi denote the set of ei -dimensional subspaces of a fixed finite vector space V=(Fq)e1+e2 . Let Yi be a non-empty subset of Xi and let αi= | Yi| / | Xi| . We give a positive lower bound, depending only on α1, α2, e1, e2, q , for the proportion of pairs (S1, S2) ∈ Y1× Y2 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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The proportion of non-degenerate complementary subspaces in classical spaces

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