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We characterize that the image of the embedding of the $Q$-polynomial association scheme into eigenspace by primitive idempotent $E_1$ is a spherical $t$-design in terms of the Krein numbers.
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E. Bannai, E. Bannai, On antipodal spherical t t -designs of degree s s with t ≤ 2 s − 3 t\leq 2s-3 , arXiv:math/0802.2905v1[math.CO]
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S. Suda, Coherent configurations and triply regular association schemes obtained from spherical designs, arXiv:math/0903.5169v1[math.CO]
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M. Aigner, A Course in Enumeration, Graduate Texts in Mathematics, 238. Springer, Berlin, 2007
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