M ar 2 00 5 Association schemes from the action of PGL ( 2 , q ) fixing a nonsingular conic in PG ( 2 , q )
نویسنده
چکیده
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 configurationsR(q2) 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.
منابع مشابه
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)...
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