Fusions of tensor powers of Johnson schemes

TL;DR

This paper classifies primitive fusions of tensor powers of Johnson schemes and related schemes, showing that for large enough parameters, all such fusions lie within specific bounds related to tensor powers and Hamming schemes.

Contribution

It provides a positive classification of primitive fusions of tensor powers of Johnson schemes and their bounds for sufficiently large parameters.

Findings

Primitive fusions of tensor powers of Johnson schemes are classified for large parameters.

All primitive fusions lie between specific tensor power and Hamming schemes.

The results extend previous work on association schemes and their fusions.

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 $\binom{m}{k}$ points provided $m$ is large enough in terms of $k$ and $d$.