On complete reducibility of tensor products of simple modules over simple algebraic groups
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- by Jonathan Gruber;
- Trans. Amer. Math. Soc. Ser. B 8 (2021), 249-276
- DOI: https://doi.org/10.1090/btran/58
- Published electronically: March 2, 2021
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Abstract:
Let $G$ be a simply connected simple algebraic group over an algebraically closed field $k$ of characteristic $p>0$. The category of rational $G$-modules is not semisimple. We consider the question of when the tensor product of two simple $G$-modules $L(\lambda )$ and $L(\mu )$ is completely reducible. Using some technical results about weakly maximal vectors (i.e. maximal vectors for the action of the Frobenius kernel $G_1$ of $G$) in tensor products, we obtain a reduction to the case where the highest weights $\lambda$ and $\mu$ are $p$-restricted. In this case, we also prove that $L(\lambda )\otimes L(\mu )$ is completely reducible as a $G$-module if and only if $L(\lambda )\otimes L(\mu )$ is completely reducible as a $G_1$-module.References
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Bibliographic Information
- Jonathan Gruber
- Affiliation: École Polytechnique Federale de Lausanne, 1015 Lausanne, Switzerland
- ORCID: 0000-0001-5975-8041
- Email: jonathan.gruber@epfl.ch
- Received by editor(s): February 12, 2020
- Received by editor(s) in revised form: July 9, 2020
- Published electronically: March 2, 2021
- Additional Notes: This work was supported by the Swiss National Science Foundation, grant number FNS 200020_175571.
- © Copyright 2021 by the author under Creative Commons Attribution-Noncommercial 3.0 License (CC BY NC 3.0)
- Journal: Trans. Amer. Math. Soc. Ser. B 8 (2021), 249-276
- MSC (2020): Primary 20G05
- DOI: https://doi.org/10.1090/btran/58
- MathSciNet review: 4223044