Abstract
The group PGL(2,q) has an embedding into PGL(3,q) such that it acts as the group fixing a nonsingular conic in PG(2,q). This action affords a coherent configuration \({\cal R}\)(q) on the set \({\cal L}\)(q) of non-tangent lines of the conic. We show that the relations can be described by using the cross-ratio. Our results imply that the restrictions \({\cal R}\) +(q) and \({\cal R}\) −(q) of \({\cal R}\)(q) to the set \({\cal L}\) +(q) of secant (hyperbolic) lines and to the set \({\cal L}\) −(q) of exterior (elliptic) lines, respectively, are both association schemes; moreover, we show that the elliptic scheme \({\cal R}\) −(q) is pseudocyclic.
We further show that the coherent configurations \({\cal R}\)(q 2) with q even allow certain fusions. These provide a 4-class fusion of the hyperbolic scheme \({\cal R}\) +(q 2), and 3-class fusions and 2-class fusions (strongly regular graphs) of both schemes \({\cal R}\) +(q 2) and \({\cal R}\) −(q 2). The fusion results for the hyperbolic case are known, but our approach here as well as our results in the elliptic case are new.
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Hollmann, H.D.L., Xiang, Q. Association schemes from the action of PGL(2, q) fixing a nonsingular conic in PG(2, q). J Algebr Comb 24, 157–193 (2006). https://doi.org/10.1007/s10801-006-0005-8
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DOI: https://doi.org/10.1007/s10801-006-0005-8