  Coulomb integrals, i.e., matrix elements of bare or screened Coulomb
interaction between one-electron orbitals, are fundamental objects in many
approaches developed to tackle the challenging problem of calculating the
electronic structure of strongly correlated materials. In this paper, Coulomb
integrals are analyzed by considering both the point group symmetry of the site
occupied by the atom in the crystal or molecule and the permutation symmetries
of the orbitals in the integrals. In particular, the case where one-electron
orbitals form the basis of a general (i.e. a real, complex or pseudo-complex)
irreducible representation is considered. Explicit formulas are provided to
calculate all integrals of the interaction tensor in terms of a minimum set of
independent ones. The effect of a symmetry breaking is also investigated by
describing Coulomb integrals of a group in terms of those of one of its
subgroups. We develope the specific example of O(3) as the larger group which
can therefore be used to quantify the deviation of a specific system from the
spherical symmetry. Possible applications of the presented framework include
the calculation of solid-state and molecular spectroscopies via multiplet
techniques, dynamical mean-field theory or the GW approximation.
