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

Álvaro Gutiérrez; Mercedes H. Rosas

doi:10.5802/crmath.468

Combinatoire, Théorie des représentations

Necessary conditions for the positivity of Littlewood–Richardson and plethystic coefficients
[Conditions nécessaires de positivité pour les coefficients de Littlewood–Richardson et du pléthysme]

Álvaro Gutiérrez ¹ ; Mercedes H. Rosas ¹

¹ Universidad de Sevilla, Spain

Comptes Rendus. Mathématique, Volume 361 (2023), pp. 1163-1173.

Résumé
Abstract

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 $S^{λ} (V)$ apparaît dans la décomposition en irréductibles de $S^{μ} (S^{ν} (V))$ , alors le diagramme de $ν$ est contenu dans celui de $λ$ .

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 $S^{λ} (V)$ appears as a summand in the decomposition into irreducibles of $S^{μ} (S^{ν} (V))$ , then $ν$ ’s diagram is contained in $λ$ ’s diagram.

Reçu le : 2021-09-28
Révisé le : 2022-12-30
Accepté le : 2023-01-26
Publié le : 2023-10-24

DOI : 10.5802/crmath.468

Classification : 05E05, 05E18, 05A17

Affiliations des auteurs :

Álvaro Gutiérrez ¹ ; Mercedes H. Rosas ¹

¹ Universidad de Sevilla, Spain

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {\'Alvaro Guti\'errez and Mercedes H. Rosas},
     title = {Necessary conditions for the positivity of {Littlewood{\textendash}Richardson} and plethystic coefficients},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1163--1173},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {361},
     year = {2023},
     doi = {10.5802/crmath.468},
     language = {en},
}

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Álvaro Gutiérrez; Mercedes H. Rosas. Necessary conditions for the positivity of Littlewood–Richardson and plethystic coefficients. Comptes Rendus. Mathématique, Volume 361 (2023), pp. 1163-1173. doi : 10.5802/crmath.468. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.468/

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