Spherical 2-Designs from Finite Group Orbits

TL;DR

This paper classifies all spherical 2-designs formed by finite group orbits in real inner product spaces, linking geometric configurations with representation theory to provide a complete characterization.

Contribution

It offers a complete classification of spherical 2-designs from finite group orbits using isotypic decomposition and moment conditions, unifying geometric and algebraic perspectives.

Findings

Identifies conditions for orbits to form spherical 2-designs

Provides explicit classification via isotypic decomposition

Connects geometric configurations with representation theory

Abstract

We classify all spherical 2-designs that arise as orbits of finite group actions on real inner product spaces. Although it is well known that such designs can occur in representations without trivial components, we give a complete characterization of the orbits that satisfy the second-moment condition. In particular, we show that these orbits correspond to projections of compact group orbits within the regular representation, and we provide an explicit classification via isotypic decomposition and moment conditions. This approach unifies geometric and representation-theoretic viewpoints on highly symmetric point configurations.