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 e d/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.