## Bulletin of the Belgian Mathematical Society - Simon Stevin

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

### Polarities of Symplectic Quadrangles

#### Abstract

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

**Source**

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

**Dates**

First available in Project Euclid: 12 September 2003

**Permanent link to this document**

https://projecteuclid.org/euclid.bbms/1063372348

**Mathematical Reviews number (MathSciNet)**

MR2017454

**Zentralblatt MATH identifier**

1040.51005

**Subjects**

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

**Keywords**

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

#### Citation

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