Non-existence of imprimitive -polynomial schemes of exceptional type with

In [H. Suzuki, Imprimitive Q -polynomial association schemes, J. Algebraic Combin. 7 (2) (1998) 165–180], it was shown that an imprimitive Q -polynomial scheme X = ( X , { R i } 0 ≤ i ≤ d ) is either dual bipartite, dual antipodal or of class 4 or 6. In this paper, it will be shown that the scheme of class 4 does not occur using the integrality conditions of the entries of the first eigenmatrix of X . These integrality conditions arise from the fact that X has exactly one Q -polynomial ordering [H. Suzuki, Association schemes with multiple Q -polynomial structures, J. Algebraic Combin. 7 (2) (1998) 181–196].