TY - JOUR

T1 - Random generation of direct sums of finite non-degenerate subspaces

AU - Glasby, Stephen P.

AU - Niemeyer, Alice C.

AU - Praeger, Cheryl E.

N1 - Funding Information:
Acknowledgements: The authors gratefully acknowledge support from the Australian Research Council (ARC) Discovery Project DP190100450 . ACN acknowledges that this is a contribution to Project-ID 286237555 - TRR 195 - by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation). We thank the referee for suggesting shorter proofs of Lemmas 3.7 (a), 3.8 (a) and 3.11 .
Funding Information:
Acknowledgements: The authors gratefully acknowledge support from the Australian Research Council (ARC) Discovery Project DP190100450. ACN acknowledges that this is a contribution to Project-ID 286237555 - TRR 195 - by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation). We thank the referee for suggesting shorter proofs of Lemmas 3.7(a), 3.8(a) and 3.11.
Publisher Copyright:
© 2022 Elsevier Inc.

PY - 2022/9/15

Y1 - 2022/9/15

N2 - 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,e′ such that e+e′⩽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=im(g−1) and U′=im(g′−1).

AB - 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,e′ such that e+e′⩽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=im(g−1) and U′=im(g′−1).

KW - Direct sum

KW - Finite classical group

KW - Non-degenerate

KW - Proportion

UR - http://www.scopus.com/inward/record.url?scp=85131215705&partnerID=8YFLogxK

U2 - 10.1016/j.laa.2022.05.016

DO - 10.1016/j.laa.2022.05.016

M3 - Article

AN - SCOPUS:85131215705

VL - 649

SP - 408

EP - 432

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

SN - 0024-3795

ER -