Bulletin of the Belgian Mathematical Society - Simon Stevin

Polarities of Symplectic Quadrangles

Markus Stroppel

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We give a simple proof of the known fact that the symplectic quadrangle is self-dual if and only if the ground field is perfect of characteristic~2, and that a polarity exists exactly if there is a root of the Frobenius automorphism. Moreover, we determine all polarities, characterize the conjugacy classes of polarities, and use the results to give a simple proof that the centralizer of any polarity acts two-transitively on the ovoid of absolute points. The proofs use elementary calculations in solvable subgroups of the symplectic group.

Article information

Bull. Belg. Math. Soc. Simon Stevin, Volume 10, Number 3 (2003), 437-449.

First available in Project Euclid: 12 September 2003

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 51E12: Generalized quadrangles, generalized polygons 51A10: Homomorphism, automorphism and dualities 51A50: Polar geometry, symplectic spaces, orthogonal spaces

generalized quadrangle symplectic quadrangle polarity duality ovoid elation generalized quadrangle translation generalized quadrangle symplectic group


Stroppel, Markus. Polarities of Symplectic Quadrangles. Bull. Belg. Math. Soc. Simon Stevin 10 (2003), no. 3, 437--449. https://projecteuclid.org/euclid.bbms/1063372348

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