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

TL;DR

This paper establishes necessary conditions for the positivity of Littlewood-Richardson and plethystic coefficients, linking the shapes of partitions involved in their representation-theoretic decompositions.

Contribution

It provides new necessary conditions for positivity, especially relating the diagrams of partitions in plethystic coefficients, advancing understanding of their combinatorial structure.

Findings

Necessary conditions for positivity of Littlewood-Richardson coefficients

Necessary conditions for positivity of plethystic coefficients

Diagram containment condition for irreducible summands

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.