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Journal of the London Mathematical Society 2006 73(2):287-303; doi:10.1112/S002461070602285X
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© The London Mathematical Society

The Action of Finite Orthogonal Groups in Characteristic 2 on the Set of Anisotropic Lines

Tatsuya Fujisaki

Combinatorial and Computational Mathematics Center, Pohang University of Science and Technology San 31 Hyoja-dong, Nam-Gu, Pohang 790-784, Korea fujisaki{at}postech.ac.kr

Received 5 February 2003. Revision received 6 October 2004.

We prove that the permutation representation of the finite orthogonal group {Omega}{varepsilon}(n,q), where {varepsilon} = + or –, on the set of anisotropic lines is multiplicity-free, if q is a power of 2 and n ≥ 6 is even. This result is established by giving a description of orbitals of this action. The rank of this action is (q2 + 2q)/2 if {varepsilon} = + and n = 6, and (q2 + 2q + 2)/2 otherwise. Moreover, we compute the subdegrees of the orbitals of {Omega}{varepsilon}(n,q).


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