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An inequality in character algebras. (English) Zbl 1014.05076
Summary: We prove the following theorem: Let $${\mathcal A}= \langle A_0,A_1,\dots, A_d\rangle$$ denote a complex character algebra with $$d\geq 2$$ which is $$P$$-polynomial with respect to the ordering $$A_0,A_1,\dots, A_d$$ of the distinguished basis. Assume that the structure constants $$p^h_{ij}$$ are all nonnegative and the Krein parameters $$q^k_{ij}$$ are all nonnegative. Let $$\theta$$ and $$\theta'$$ denote eigenvalues of $$A_1$$, other than the valency $$k= k_1$$. Then the structure constants $$a_1= p^1_{11}$$ and $$b_1= p^1_{12}$$ satisfy $\Biggl(\theta+{k\over a_1+ 1}\Biggr) \Biggl(\theta'+ {k\over a_1+ 1}\Biggr)\geq -{ka_1 b_1\over (a_1+ 1)^2}.$ Let $$E$$ and $$F$$ denote the primitive idempotents of $${\mathcal A}$$ associated with $$\theta$$ and $$\theta'$$, respectively. Equality holds in the above inequality if and only if the Schur product $$E\circ F$$ is a scalar multiple of a primitive idempotent of $${\mathcal A}$$.
The above theorem extends some results of A. Jurišić, J. Koolen and P. Terwilliger [J. Algebr. Comb. 12, 163-197 (2000; Zbl 0959.05121)], and the present author [J. Algebr. Comb. 10, 47-59 (1999; Zbl 0927.05085)]. These people previously showed the above theorem holds for those character algebras isomorphic to the Bose-Mesner algebra of a distance-regular graph.

##### MSC:
 5e+30 Association schemes, strongly regular graphs
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