## Bulletin of the Belgian Mathematical Society - Simon Stevin

### Linear representations of semipartial geometries

S. De Winter

#### Abstract

Semipartial geometries (SPG) were introduced in 1978 by Debroey and Thas. As some of the examples they provided were embedded in affine space it was a natural question to ask whether it was possible to classify all SPG embedded in affine space. In $AG(2,q)$ and $AG(3,q)$ a complete classification was obtained. Later on it was shown that if an SPG, with $\alpha>1$, is embedded in affine space it is either a linear representation or $\mathrm{TQ}(4,2^h)$. In this paper we derive general restrictions on the parameters of an SPG to have a linear representation and classify the linear representations of SPG in $AG(4,q)$, hence yielding the complete classification of SPG in $AG(4,q)$, with $\alpha>1$.

#### Article information

Source
Bull. Belg. Math. Soc. Simon Stevin, Volume 12, Number 5 (2006), 767-780.

Dates
First available in Project Euclid: 10 January 2006

https://projecteuclid.org/euclid.bbms/1136902614

Digital Object Identifier
doi:10.36045/bbms/1136902614

Mathematical Reviews number (MathSciNet)
MR2241342

Zentralblatt MATH identifier
1138.51005

#### Citation

De Winter, S. Linear representations of semipartial geometries. Bull. Belg. Math. Soc. Simon Stevin 12 (2006), no. 5, 767--780. doi:10.36045/bbms/1136902614. https://projecteuclid.org/euclid.bbms/1136902614