Representations of SO(3) and angular polyspectra

We characterize the angular polyspectra, of arbitrary order, associated with isotropic fields defined on the sphere S 2 = { ( x , y , z ) : x 2 + y 2 + z 2 = 1 } . Our techniques rely heavily on group representation theory, and specifically on the properties of Wigner matrices and Clebsch–Gordan coefficients. The findings of the present paper constitute a basis upon which one can build formal procedures for the statistical analysis and the probabilistic modelization of the Cosmic Microwave Background radiation, which is currently a crucial topic of investigation in cosmology. We also outline an application to random data compression and “simulation” of Clebsch–Gordan coefficients.