Nonexistence of exceptional imprimitive -polynomial association schemes with six classes

Suzuki (1998) [9] showed that an imprimitive Q -polynomial association scheme with first multiplicity at least 3 is Q -bipartite, or is Q -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.