  A geometric theory of the irreducible tensor operators of quantum spin
systems. It is based upon the Maxwell-Sylvester geometric representation of the
multipolar electrostatic potential. In the latter, an order-$\ell$ multipolar
potential is represented by a collection of $\ell$ equal length vectors, i.e.
by $\ell$ points on a sphere, instead of by its components on some fixed (but
arbitrary) basis. The geometric representation offers a much more appropriate
tool for getting physical insight on specific characteristics of a multipole,
such as its symmetries, or its departure from ideal symmetry. We derive
explicit expressions enabling to perform any calculations we may need to
perform on multipoles. All relevant quantities are eventually expressed in
terms of scalar products of pairs of vectors (i.e., in terms of geometric
quantities such as lengths and angles). The whole formalism is entirely
independent of any particular choice of coordinate, and needs no use of the
somehow abstract formalism traditionally used when dealing with angular
momenta.
  The formalism is then applied to treat the problem of the irreducible tensor
operators of quantum spin systems. It enables to completely dispense with the
calculation and use of the Stevens operators, which can be quite complicated
even for moderate values of $\ell$. Explicit expressions for the calculation of
expectations values of physical observables are derived. They essentially
consist in combinations of scalar products of vector pairs. Together with the
coherent state representation of the quantum states of spin systems, this
provides a complete geometric, coordinate-free, description of the states,
dynamics and physical properties of these systems.
