Comptes Rendus
Research article - Complex analysis and geometry
SL 4 (Z) is not purely matricial field
Michael Magee1, 2; Mikael de la Salle3, 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.

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.

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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

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
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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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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