Abstract
The authors proved that a Weyl module for a simple algebraic group is irreducible over every field if and only if the module is isomorphic to the adjoint representation for \(E_{8}\) or its highest weight is minuscule. In this paper, we prove an analogous criteria for irreducibility of Weyl modules over the quantum group \(U_{\zeta }({\mathfrak {g}})\) where \({\mathfrak {g}}\) is a complex simple Lie algebra and \(\zeta \) ranges over roots of unity.
Dedicated to Professor David J. Benson on the occasion of his 60th birthday
The second author was partially supported by NSF grant DMS-1302886 and DMS-1600056. Research of the third author was partially supported by NSF grant DMS-1402271.
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Notes
- 1.
Note that we use the convention throughout the paper that the zero weight is minuscule.
References
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Acknowledgements
The authors thank Henning Andersen and George Lusztig for their suggestion to extend our prior work [1] to the quantum case.
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Garibaldi, S., Guralnick, R.M., Nakano, D.K. (2018). Globally Irreducible Weyl Modules for Quantum Groups. In: Carlson, J., Iyengar, S., Pevtsova, J. (eds) Geometric and Topological Aspects of the Representation Theory of Finite Groups. PSSW 2016. Springer Proceedings in Mathematics & Statistics, vol 242. Springer, Cham. https://doi.org/10.1007/978-3-319-94033-5_12
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DOI: https://doi.org/10.1007/978-3-319-94033-5_12
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