  Kronecker coefficients encode the tensor products of complex irreducible
representations of symmetric groups. Their stability properties have been
considered recently by several authors (Vallejo, Pak and Panova, Stembridge).
We describe a geometric method, based on Schur-Weyl duality, that allows to
produce huge series of instances of this phenomenon. Moreover the method gives
access to lots of extra information. Most notably, we can often compute the
stable Kronecker coefficients, sometimes as number of points in very explicit
polytopes. We can also describe explicitely the moment polytope in the
neighbourhood of our stable triples. Finally, we explain an observation of
Stembridge on the behaviour of certain rectangular Kronecker coefficients, by
relating it to the affine Dynkin diagram of type $E\_6$.
