Inequivalent representations of geometric relation algebras

Abstract

It is shown that the automorphism group of a relation algebra $\mla P$ constructed from a projective geometry $P$ is isomorphic to the collineation group of $P$. Also, the base automorphism group of a representation of $\mla P$ over an affine geometry $D$ is isomorphic to the quotient of the collineation group of $D$ by the dilatation subgroup. Consequently, the total number of inequivalent representations of $\mla P$, for finite geometries $P$, is the sum of the numbers \[ \frac{|\aut P|}{|\aut {D}|\mathbin{/}|\dil{D}|}, \] where $D$ ranges over a list of the non-isomorphic affine geometries having $P$ as their geometry at infinity. This formula is used to compute the number of inequivalent representations of relation algebras constructed over projective lines of order at most 10. For instance, the relation algebra constructed over the projective line of order 9 has 56,700 mutually inequivalent representations.