Bulletin of the Belgian Mathematical Society - Simon Stevin

Linear representations of semipartial geometries

S. De Winter

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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

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

First available in Project Euclid: 10 January 2006

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

semipartial geometry linear representation strongly regular graph


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

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