On P-Polynomial Table Algebras and Applications to Association Schemes

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.

Key Words:

• Eigenvalues
• P-polynomial association schemes
• P-polynomial table algebras
• Primitive idempotents
• Q-polynomial association schemes

2000 Mathematics Subject Classification:

• 20C99
• 13A99
• 05E30

ACKNOWLEDGMENTS

The author is sincerely grateful to the referee for correcting an error and simplifying the proof of Lemma 2.1. The referee's valuable suggestions have improved the quality of the article.

Notes

Communicated by A. Turull.