Isotropic N-Point Basis Functions and Their Properties

Understand

Isotropic functions of positions $\hat{\bf r}_1, \hat{\bf r}_2,\ldots, \hat{\bf r}_N$, i.e.

• functions invariant under simultaneous rotations of all the coordinates, are conveniently formed using spherical harmonics and Clebsch-Gordan coefficients.
• An orthonormal basis of such functions provides a formalism suitable for analyzing isotropic distributions such as those that arise in cosmology, for instance in the clustering of galaxies as revealed by large-scale structure surveys.
• The algebraic properties of the basis functions are conveniently expressed in terms of 6-$j$ and 9-$j$ symbols.