Rank 5 association schemes and triality

AbstractSymmetric association schemes having a parabolic of rank 3 and corank 2 satisfying a certain uniformity condition can be interpreted on the one hand as linked strongly regular designs (analogous to linked symmetric designs) and on the other as geometries. Classical triality associated with the groups O+8(q) provides a family of examples with rank 3, corank 2 parabolics and sporadic examples are associated with the groups L3(4), U6(2) and U3(5). The triality examples and the U3(5) example are flag-transitive viewed as geometries. Results include a characterization on the level of parameters of the triality schemes leading to a characterization of the schemes using results of Cameron and Drake, and a proof (as a Cayley exercise) of simple connectivity of the example associated with U3(5)