Nonexistence of exceptional imprimitive QQ-polynomial association schemes with six classes

Suzuki (1998) [9] showed that an imprimitive QQ-polynomial association scheme with first multiplicity at least 3 is QQ-bipartite, or is QQ-antipodal, or has four or six classes. The exceptional case with four classes has recently been ruled out by Cerzo and Suzuki (2009) [5]. In this paper, we show the nonexistence of the last case with six classes. Hence Suzuki’s theorem now exactly mirrors its well-known counterpart for imprimitive distance-regular graphs.