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

Gutiérrez, Álvaro; Rosas, Mercedes H.

Comptes Rendus. Mathématique, Tome 361 (2023) no. G7, pp. 1163-1173

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 $S^{\lambda }\left(V\right)$ appears as a summand in the decomposition into irreducibles of $S^{\mu }\left(S^{\nu }\left(V\right)\right)$, then $\nu$’s diagram is contained in $\lambda$’s diagram.

Nous donnons des conditions nécessaires de positivité pour les coefficients de Littlewood–Richardson et pour les coefficients SXP. Nous en déduisons la condition nécessaire de positivité suivante pour les coefficients du pléthysme : si $S^{\lambda }\left(V\right)$ apparaît dans la décomposition en irréductibles de $S^{\mu }\left(S^{\nu }\left(V\right)\right)$, alors le diagramme de $\nu$ est contenu dans celui de $\lambda$.

Classification : 05E05, 05E18, 05A17

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