Necessary conditions for the positivity of Littlewood-Richardson and plethystic coefficients

We give necessary conditions for the positivity of Littlewood-Richardson coefficients and SXP coefficients. We deduce necessary conditions for the positivity of the plethystic coefficients. Explicitly, our main result states that if S-lambda(V) appears as a summand in the decomposition into irreducibles of S-mu(S-nu(V)), then nu's diagram is contained in lambda's diagram.