A variational characterisation of projective spherical designs over the quaternions
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We give an inequality on the packing of vectors/lines in quaternionic Hilbert space , which generalises those of Sidelnikov and Welch for unit vectors in and . This has a parameter t, and depends only on the vectors up to projective unitary equivalence. The sequences of vectors in 𝔽^d=ℝ^d,ℂ^d,ℍ^d that give equality, which we call spherical (t,t)-designs, are seen to satisfy a cubature rule on the unit sphere in 𝔽^d for a suitable polynomial space _(t,t). Using this, we show that the projective spherical t-designs on the Delsarte spaces P^d-1 coincide with the spherical (t,t)-designs of unit vectors in 𝔽^d. We then explore a number of examples in quaternionic space. The unitarily invariant polynomial space Hom_ℍ^d(t,t) and the inner product that we define on it so the reproducing kernel has a simple form are of independent interest.
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