## Jordan blocks of unipotent elements in some irreducible representations of classical groups in good characteristic

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## Abstract:

Let $G$ be a classical group with natural module $V$ over an algebraically closed field of good characteristic. For every unipotent element $u$ of $G$, we describe the Jordan block sizes of $u$ on the irreducible $G$-modules which occur as composition factors of $V \otimes V^*$, $\wedge ^2(V)$, and $S^2(V)$. Our description is given in terms of the Jordan block sizes of the tensor square, exterior square, and the symmetric square of $u$, for which recursive formulae are known.## References

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## Additional Information

**Mikko Korhonen**- Affiliation: School of Mathematics, The University of Manchester, Manchester M13 9PL, United Kingdom
- MR Author ID: 1211748
- Email: korhonen_mikko@hotmail.com
- Received by editor(s): August 27, 2018
- Received by editor(s) in revised form: January 19, 2019
- Published electronically: June 10, 2019
- Additional Notes: Some of the results in this paper were obtained during the author’s doctoral studies at École Polytechnique Fédérale de Lausanne, supported by a grant from the Swiss National Science Foundation (grant number $200021 \_ 146223$).
- Communicated by: Pham Huu Tiep
- © Copyright 2019 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**147**(2019), 4205-4219 - MSC (2010): Primary 20G05
- DOI: https://doi.org/10.1090/proc/14570
- MathSciNet review: 4002536