## Jantzen filtration of Weyl modules, product of Young symmetrizers and denominator of Young’s seminormal basis

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- by Ming Fang, Kay Jin Lim and Kai Meng Tan PDF
- Represent. Theory
**24**(2020), 551-579 Request permission

## Abstract:

Let $G$ be a connected reductive algebraic group over an algebraically closed field of characteristic $p>0$, $\Delta (\lambda )$ denote the Weyl module of $G$ of highest weight $\lambda$ and $\iota _{\lambda ,\mu }:\Delta (\lambda +\mu )\to \Delta (\lambda )\otimes \Delta (\mu )$ be the canonical $G$-morphism. We study the split condition for $\iota _{\lambda ,\mu }$ over $\mathbb {Z}_{(p)}$, and apply this as an approach to compare the Jantzen filtrations of the Weyl modules $\Delta (\lambda )$ and $\Delta (\lambda +\mu )$. In the case when $G$ is of type $A$, we show that the split condition is closely related to the product of certain Young symmetrizers and, under some mild conditions, is further characterized by the denominator of a certain Young’s seminormal basis vector. We obtain explicit formulas for the split condition in some cases.## References

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## Additional Information

**Ming Fang**- Affiliation: HLM, HCMS, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, People’s Republic of China; and School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, 100049, People’s Republic of China
- MR Author ID: 715486
- Email: fming@amss.ac.cn
**Kay Jin Lim**- Affiliation: Division of Mathematical Sciences, Nanyang Technological University, SPMS-04-01, 21 Nanyang Link, 637371 Singapore
- MR Author ID: 865544
- ORCID: 0000-0002-2605-0482
- Email: limkj@ntu.edu.sg
**Kai Meng Tan**- Affiliation: Department of Mathematics, National University of Singapore, Block S17, 10 Lower Kent Ridge Road, 119076 Singapore
- MR Author ID: 656415
- Email: tankm@nus.edu.sg
- Received by editor(s): December 12, 2019
- Received by editor(s) in revised form: July 20, 2020, and August 24, 2020
- Published electronically: October 29, 2020
- Additional Notes: The first author was supported by NSFC (No. 11688101, 11471315 and 11321101), while the second and third authors were supported by Singapore MOE Tier 2 AcRF MOE2015-T2-2-003.
- © Copyright 2020 American Mathematical Society
- Journal: Represent. Theory
**24**(2020), 551-579 - MSC (2020): Primary 20G05, 20C30
- DOI: https://doi.org/10.1090/ert/553
- MathSciNet review: 4168181