SL$_{4}(\textbf{Z})$ is not purely matricial field
We prove that every non-zero finite dimensional unitary representation of $\mathrm{SL}_{4}(\mathbf{Z})$ contains a non-zero $\mathrm{SL}_{2}(\mathbf{Z})$-invariant vector. As a consequence, there is no sequence of finite-dimensional representations of $\mathrm{SL}_{4}(\mathbf{Z})$ that gives rise to...
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| Language: | English |
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Académie des sciences
2024-10-01
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| Series: | Comptes Rendus. Mathématique |
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| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.617/ |
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| author | Magee, Michael de la Salle, Mikael |
| author_facet | Magee, Michael de la Salle, Mikael |
| author_sort | Magee, Michael |
| collection | DOAJ |
| description | We prove that every non-zero finite dimensional unitary representation of $\mathrm{SL}_{4}(\mathbf{Z})$ contains a non-zero $\mathrm{SL}_{2}(\mathbf{Z})$-invariant vector. As a consequence, there is no sequence of finite-dimensional representations of $\mathrm{SL}_{4}(\mathbf{Z})$ that gives rise to an embedding of its reduced $C^*$-algebra into an ultraproduct of matrix algebras. |
| format | Article |
| id | doaj-art-3bc6a356ac584cc3a9a3e5f1e1eb26b3 |
| institution | Kabale University |
| issn | 1778-3569 |
| language | English |
| publishDate | 2024-10-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj-art-3bc6a356ac584cc3a9a3e5f1e1eb26b32025-02-07T11:22:49ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692024-10-01362G890391010.5802/crmath.61710.5802/crmath.617SL$_{4}(\textbf{Z})$ is not purely matricial fieldMagee, Michael0de la Salle, Mikael1Department of Mathematical Sciences, Durham University, Lower Mountjoy, DH1 3LE Durham, UK; IAS Princeton, School of Mathematics, 1 Einstein Drive, Princeton 08540, USAIAS Princeton, School of Mathematics, 1 Einstein Drive, Princeton 08540, USA; Institut Camille Jordan, CNRS, Université Lyon 1, FranceWe prove that every non-zero finite dimensional unitary representation of $\mathrm{SL}_{4}(\mathbf{Z})$ contains a non-zero $\mathrm{SL}_{2}(\mathbf{Z})$-invariant vector. As a consequence, there is no sequence of finite-dimensional representations of $\mathrm{SL}_{4}(\mathbf{Z})$ that gives rise to an embedding of its reduced $C^*$-algebra into an ultraproduct of matrix algebras.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.617/Special linear groupsFinite dimensionnal unitary representationsPurely MF groupsMF $C^*$-algebra |
| spellingShingle | Magee, Michael de la Salle, Mikael SL$_{4}(\textbf{Z})$ is not purely matricial field Comptes Rendus. Mathématique Special linear groups Finite dimensionnal unitary representations Purely MF groups MF $C^*$-algebra |
| title | SL$_{4}(\textbf{Z})$ is not purely matricial field |
| title_full | SL$_{4}(\textbf{Z})$ is not purely matricial field |
| title_fullStr | SL$_{4}(\textbf{Z})$ is not purely matricial field |
| title_full_unstemmed | SL$_{4}(\textbf{Z})$ is not purely matricial field |
| title_short | SL$_{4}(\textbf{Z})$ is not purely matricial field |
| title_sort | sl 4 textbf z is not purely matricial field |
| topic | Special linear groups Finite dimensionnal unitary representations Purely MF groups MF $C^*$-algebra |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.617/ |
| work_keys_str_mv | AT mageemichael sl4textbfzisnotpurelymatricialfield AT delasallemikael sl4textbfzisnotpurelymatricialfield |