Association Schemes All of Whose Symmetric Fusions Are Integral

Let (X,S) be an association scheme where X is a finite set and S is a partition of X × X. We say that (X,S) is symmetric if σs is symmetric for each s ∈ S where σs is the adjacency matrix of s, and integral if ⋃ s∈S ev(σs) ⊆ Z where ev(σs) is the set of all eigenvalues of σs. For an association scheme (X,T ) we say that (X,T ) is a fusion of (X,S) if each element of T is a union of elements of S. In this article we aim to characterize association schemes all of whose symmetric fusions are integral, and classify those obtained from a regular action of a finite group by taking its orbitals.