Imprimitive cometric association schemes: Constructions and analysis

Dualizing the “extended bipartite double” construction for distance-regular graphs, we construct a new family of cometric (or Q-polynomial) association schemes with four associate classes based on linked systems of symmetric designs. The analysis of these new schemes naturally leads to structural questions concerning imprimitive cometric association schemes, some of which we answer with others being left as open problems. In particular, we prove that any Q-antipodal association scheme is dismantlable: the configuration induced on any subset of the equivalence classes in the Q-antipodal imprimitivity system is again a cometric association scheme. Further examples are explored.