Fissioned triangular schemes via the cross-ratio

A construction of association schemes is presented; these are fission schemes of the triangular schemes T (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. c © 2001 Academic Press 1. THE CONSTRUCTION This paper is a sequel to [4]. In that paper, it was observed that almost all known self-dual classical association schemes have natural fission schemes (fissioning the maximum-distance relation only); whereas in the non-self-dual case there seemed to be no analogous fission schemes. Subsequently, we found that there is at least one such non-self-dual clas-sical association scheme that admits an interesting fission scheme, namely the triangular scheme T (n) = J (n, 2) where n = q + 1 with q any prime power; this is the object of the present work. For terminology and background, we refer to Bannai and Ito [2] for asso-ciation schemes and to Hirschfeld [9] for finite geometry. Recall that the group PGL(2, q