Fusions of tensor powers of Johnson schemes

Abstract

This paper is a follow-up to (arXiv:2203.03687), in which the first author studied primitive association schemes lying between a tensor power ${\mathcal{T}}_m^d$ of the trivial association scheme and the Hamming scheme $\mathcal{H}(m,d)$. A question which arose naturally in that study was whether all primitive fusions of ${\mathcal{T}}_m^d$ lie between ${\mathcal{T}}_{m^e}^{d/e}$ and $\mathcal{H}(m^d,d/e)$ for some $e\mid d$. This note answers this question positively provided that $m$ is large enough. We similarly classify primitive fusions of the $d$th tensor power of a Johnson scheme on $\left(\genfrac{}{}{0ex}{}{m}{k}\right)$ points provided $m$ is large enough in terms of $k$ and $d$.

5 pages. Final version incorporating referees' comments. To appear in Algebraic Combinatorics