A variational characterisation of projective spherical designs over the quaternions

Shayne Waldron


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

Keywords: Sidelnikov inequality, Welch bound/inequality, Delsarte space, projective spherical $t$-designs, spherical $(t,t)$-designs, harmonic polynomials, reproducing kernels, apolar inner product, Bombieri inner product, finite tight frames, quaternionic equiangular lines, projective unitary equivalence over the quaternions.

Math Review Classification: Primary 05B25, 05B30, 15B33, Secondary 42C15, 51M20, 65D30.

Length: 28 pages

Last Updated: 17 November 2020