Fusions of tensor powers of Johnson schemes

This paper is a follow-up to [5], in which the first author studied primitive association schemes lying between a tensor power T_m^d of the trivial association scheme and the Hamming scheme H(d, m). A question which arose naturally in that study was whether all primitive fusions of T_m^d lie between T_m^d/e e and H(d/e, m^e) for some e | d. This note answers this question positively provided that m is large enough. We similarly classify primitive fusions of the dth tensor power of a Johnson scheme on (^m_k ) points when m is large enough in terms of k and d.