Rational Angle Sets and Tight t-Designs

TL;DR

This paper proves that tight spherical and projective designs generally have rational angle sets, extending previous results and repairs to all cases using Jordan algebra techniques.

Contribution

It extends Lyubich's proof to all projective cases and spherical cases, confirming rationality of angle sets in tight designs with a unified algebraic approach.

Findings

Confirmed rational angle sets for all tight spherical designs.

Extended proof to all projective cases, including octonion cases.

Unified approach using Jordan algebra for all cases.

Abstract

Given a finite subset of a sphere or projective space, known as a design, we can compute the strength and angle set of that design. When the strength and angle set meet certain bounds, the design is called tight. Hoggar sought to prove that, aside from certain known cases, the angle sets of tight projective designs must be rational. Lyubich found a counter-example and provided a repair for Hoggar's proof but excluded the exceptional octonion projective cases. This note extends Lyubich's repair of Hoggar's proof to the remaining projective cases and extends the proof to all spherical cases. It does so by using Jordan algebra primitive idempotents to treat all of the cases simultaneously. We thereby confirm that tight spherical and projective designs have rational angle sets except in specific cases.