## Bulletin of the Belgian Mathematical Society - Simon Stevin

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

### Linear representations of semipartial geometries

#### 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

**Permanent link to this document**

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

**Digital Object Identifier**

doi:10.36045/bbms/1136902614

**Mathematical Reviews number (MathSciNet)**

MR2241342

**Zentralblatt MATH identifier**

1138.51005

**Subjects**

Primary: 51Exx: Finite geometry and special incidence structures 05B25: Finite geometries [See also 51D20, 51Exx]

**Keywords**

semipartial geometry linear representation strongly regular graph

#### 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