# Source for: a permutation group is multiplicity-free if and only if its 2-orbits define an association scheme

I have recently proven the following (at least, so I believe):

Theorem. Given a permutation group $$\Sigma\subseteq\mathrm{Sym}(\Omega)$$ on the set $$\Omega:=\{1,...,n\}$$, the following are equivalent:

1. the permutation character of $$\Sigma$$ is multiplicity-free, that is, it decomposes into distinct irreducible characters.
2. the (self-paired) 2-orbits of $$\Sigma$$ (that is, its orbits on $$\Omega\times \Omega$$) define an association scheme.

I believe that this is known (if true). Has this result a name? Can someone point me to the literature proving/discussing/using this result? I am also grateful for a vague direction.

Update

As pointed out by Tom De Medts, an alternative formulation is the following:

Theorem. Given a permutation group $$\Sigma\subseteq\mathrm{Sym}(\Omega)$$, the following are equivalent:

1. the permutation character of $$\Sigma$$ is multiplicity-free.
2. the association scheme formed by the 2-orbits of $$\Sigma$$ is commutative.

This claim can also be found in the beginning of Section 3 of Commutative Association Schemes by William J. Martin & Hajime Tanaka.

• That is strange. On the one page available here, Example 2.3 states that the 2-orbits form an association scheme if $\Sigma$ acts transitively, no matter whether it is multiplicity-free. Isn't this in contradiction to what I claim in the question? Feb 24, 2021 at 15:01