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

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.