An inequality in character algebras - Pascasio, Arlene A. (Animo Repository, 2003)
In this paper, we prove the following: Theorem. Let A=〈A0,A1,..,A(d) 〉 denote a complex character algebra with d2 which is P-polynomial with respect to the ordering A 0,A 1,..,A d of the distinguished basis. Assume that the structure constants p ijh are all nonnegative and the Krein parameters q ijh are all nonnegative. Let θ and θ′ denote eigenvalues of A 1, other than the valency k=k 1. Then the structure constants a 1=p 111 and b 1=p 121 satisfyθ+ k a1+1θ′+ k a1+1-ka1b1 (a1+1) 2.Let E and F denote the primitive idempotents of A associated with θ and θ′, respectively. Equality holds in the above inequality if and only if the Schur product E°F is a scalar multiple of a primitive idempotent of A. The above theorem extends some results of Jurišić, Koolen, Terwilliger, and the present author. These people previously showed the above theorem holds for those character algebras isomorphic to the Bose-Mesner algebra of a distance-regular graph. © 2002 Elsevier Science B.V. All rights reserved.