  Combinatorial aspects of multivariate diagonal invariants of the symmetric
group are studied. As a consequence it is proved the existence of a
multivariate extension of the classical Robinson-Schensted correspondence.
Further byproduct are a pure combinatorial algorithm to describe the
irreducible decomposition of the tensor product of two irreducible
representations of the symmetric group, and new symmetry results on permutation
enumeration with respect to descent sets.
