Fissioned Triangular Schemes via the Cross-ratio

A construction of association schemes is presented; these are fission schemes of the triangular schemesT (n) where n = q + 1 with q any prime power. The key observation is quite elementary, being that the natural action of PGL(2, q) on the 2-element subsets of the projective line PG(1, q) is generously transitive. In addition, some observations on the intersection parameters and fusion schemes of these association schemes are made.