An inequality in character algebras Original Research Article

In this paper, we prove the following: Theorem. Let A=〈A0,A1,…,Ad〉 denote a complex character algebra with d⩾2 which is P-polynomial with respect to the ordering A0,A1,…,Ad of the distinguished basis. Assume that the structure constants pijh are all nonnegative and the Krein parameters qijh are all nonnegative. Let θ and θ′ denote eigenvalues of A1, other than the valency k=k1. Then the structure constants a1=p111 and b1=p121 satisfyθ+ka1+1θ′+ka1+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.