Association schemes from the action of PGL(2, q) fixing a nonsingular conic in PG(2, q)

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 R(q) on the set 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 R+(q) and R-(q) of R(q) to the set L+(q) of secant (hyperbolic) lines and to the set L-(q) of exterior (elliptic) lines, respectively, are both association schemes; moreover, we show that the elliptic scheme R-(q) is pseudocyclic. We further show that the coherent configurations R(q(2)) with q even allow certain fusions. These provide a 4-class fusion of the hyperbolic scheme R+(q(2)), and 3-class fusions and 2-class fusions (strongly regular graphs) of both schemes R+(q(2)) and 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.