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

Abstract

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.