Abstract
Motivated by a recent result of Robinson showing that simple endotrivial modules essentially come from quasi-simple groups we classify such modules for finite special linear and unitary groups as well as for exceptional groups of Lie type. Our main tool is a lifting result for endotrivial modules obtained in a previous paper which allows us to apply character theoretic methods. As one application we prove that the \(\ell \)-rank of quasi-simple groups possessing a faithful simple endotrivial module is at most 2. As a second application we complete the proof that principal blocks of finite simple groups cannot have Loewy length 4, thus answering a question of Koshitani, Külshammer and Sambale. Our results also imply a vanishing result for irreducible characters of special linear and unitary groups.
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Acknowledgments
We thank Frank Lübeck for information on the Brauer trees of \(6.{}^2\!E_6(2)\) for the prime 13, and Shigeo Koshitani for drawing our attention to [12].
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The authors gratefully acknowledge financial support by ERC Advanced Grant 291512. The first author also acknowledges financial support by SNF Fellowship for Prospective Researchers PBELP2_143516.
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Lassueur, C., Malle, G. Simple endotrivial modules for linear, unitary and exceptional groups. Math. Z. 280, 1047–1074 (2015). https://doi.org/10.1007/s00209-015-1465-0
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DOI: https://doi.org/10.1007/s00209-015-1465-0
Keywords
- Simple endotrivial modules
- Quasi-simple groups
- Special linear and unitary groups
- Loewy length
- Zeroes of characters