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On Perfect P-polynomial Table Algebras

The Bose–Mesner algebra of the association scheme of the ordinary n-gon has the following remarkable properties: (i) It has a P-polynomial structure with respect to every faithful basis element; and (ii) Any closed subset generated by a basis element has a P-polynomial structure with respect to this basis element. C-algebras or table algebras that have these two properties are called perfect P-polynomial C-algebras or table algebras. By applying and extending some of the techniques developed in Xu (2006), we will give a classification of perfect P-polynomial table algebras in terms of intersection matrices. As a direct consequence of this classification, we will prove that a standard real integral table algebra (A, B) with |B| ≥ 6 is a perfect P-polynomial table algebra if and only if it is exactly isomorphic to the Bose–Mesner algebra of the association scheme of the ordinary (2|B|−2)-gon or (2|B|−1)-gon. This result generalizes part of the main theorem in Xu (2006). We will present examples revealing that this result is not true if |B| ≤ 5.