Abstract
By investigating the primitive idempotents of a commutative table algebra that has a table basis element with all distinct eigenvalues, we prove a necessary and sufficient condition in terms of the eigenvalues of a table basis element b ∈ B under which a real table algebra (A, B) is a P-polynomial table algebra. As direct consequences, we obtain the necessary and sufficient condition for a symmetric association scheme to be a Q-polynomial association scheme in [7] as well as a necessary and sufficient condition for a symmetric association scheme to be a P-polynomial association scheme.
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