Spherical 2-Designs from Finite Group Orbits

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.