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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.

05E30 Association schemes, strongly regular graphs
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