SL$_{4}(\textbf{Z})$ is not purely matricial field

Michael Magee; Mikael de la Salle

doi:10.5802/crmath.617

Article de recherche - Analyse et géométrie complexes

SL

_{4} (Z)

is not purely matricial field
[SL

_{4} (Z)

n’est pas purement MF]

Michael Magee ^1,² ; Mikael de la Salle ^3,²

¹ Department of Mathematical Sciences, Durham University, Lower Mountjoy, DH1 3LE Durham, UK
² IAS Princeton, School of Mathematics, 1 Einstein Drive, Princeton 08540, USA
³ Institut Camille Jordan, CNRS, Université Lyon 1, France

Comptes Rendus. Mathématique, Volume 362 (2024), pp. 903-910.

Abstract (VO)
Résumé

We prove that every non-zero finite dimensional unitary representation of ${SL}_{4} (Z)$ contains a non-zero ${SL}_{2} (Z)$ -invariant vector. As a consequence, there is no sequence of finite-dimensional representations of ${SL}_{4} (Z)$ that gives rise to an embedding of its reduced $C^{*}$ -algebra into an ultraproduct of matrix algebras.

Nous montrons que toute représentation unitaire de dimension finie non nulle de ${SL}_{4} (Z)$ a un vecteur ${SL}_{2} (Z)$ -invariant non nul. Il n’existe donc pas de suite de représentations de dimension finie de ${SL}_{4} (Z)$ qui permettent de réaliser sa $C^{*}$ -algèbre réduite dans un ultraproduit d’algèbres de matrices.

Reçu le : 2023-12-06
Révisé le : 2024-01-19
Accepté le : 2024-01-22
Publié le : 2024-10-07

Zbl

DOI : 10.5802/crmath.617

Classification : 20C15, 20C33, 22D25
Keywords: Special linear groups, Finite dimensionnal unitary representations, Purely MF groups, MF

C^{*}

-algebra
Mots-clés : Groupes spéciaux linéaires, représentations unitaires de dimension finie, groupes purement MF,

C^{*}

-algèbres MF

Affiliations des auteurs :

Michael Magee ^{1, 2} ; Mikael de la Salle ^{3, 2}

¹ Department of Mathematical Sciences, Durham University, Lower Mountjoy, DH1 3LE Durham, UK
² IAS Princeton, School of Mathematics, 1 Einstein Drive, Princeton 08540, USA
³ Institut Camille Jordan, CNRS, Université Lyon 1, France

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

@article{CRMATH_2024__362_G8_903_0,
     author = {Michael Magee and Mikael de la Salle},
     title = {SL$_{4}(\textbf{Z})$ is not purely matricial field},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {903--910},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {362},
     year = {2024},
     doi = {10.5802/crmath.617},
     zbl = {07929052},
     language = {en},
}

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Michael Magee; Mikael de la Salle. SL$_{4}(\textbf{Z})$ is not purely matricial field. Comptes Rendus. Mathématique, Volume 362 (2024), pp. 903-910. doi : 10.5802/crmath.617. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.617/

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