TITLE: Do character tables determine association schemes up to isomorphism?
QUESTION [5 upvotes]: I am interested in commutative, but not-necessarily-symmetric association schemes. I noticed that the conjugacy class scheme of the dihedral group of order 8 has the same character table as the one for the quaternion group, and the schemes are also isomorphic.
Are there examples of non-isomorphic association schemes that have the same character table? I am not married to association schemes arising from groups.

REPLY [2 votes]: IMHO the isomorphism of association schemes is understood as isomorphism of underlying coloured digraphs. The smallest example of non-isomorphic (in this sense) association schemes with the same character table comes from a pair of nonisomorphic degree 6 strongly regular graphs on 16 vertices. One of these graphs is Shrikhande graph, the other is $4\times 4$ rook graph. 
In the example in the question one has association schemes of conjugacy classes of groups $D_8$ and $Q_8$, and they happen to be isomorphic in the sense of coloured digraph isomorphism. It is example number 9 in tables by Izumi Miyamoto and Akihide Hanaki.
In fact if isomorphism of association schemes was possible to decide from their character tables then the graph isomorphism problem would be easy to solve.