# A variational characterisation of projective spherical
designs over the quaternions

## Shayne Waldron

## 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 $\Fd=\Rd,\Cd,\Hd$ 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

## Availability: