A variational characterisation of projective spherical designs over the quaternions

Abstract: 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 ${\mathbb{F}}^d={\mathbb{R}}^d,{\mathbb{C}}^d,{\mathbb{H}}^d$ that give equality, which we call spherical $(t,t)$-designs, are seen to satisfy a cubature rule on the unit sphere in ${\mathbb{F}}^d$ 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 ${\mathbb{F}}^d$. We then explore a number of examples in quaternionic space. The unitarily invariant polynomial space ${\mathop{\rm Hom}\nolimits}_{{\mathbb{H}}^d}(t,t)$ and the inner product that we define on it so the reproducing kernel has a simple form are of independent interest.