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

Authors: Ming Fang, Kay Jin Lim and Kai Meng Tan
Journal: Represent. Theory 24 (2020), 551-579
MSC (2020): Primary 20G05, 20C30
Published electronically: October 29, 2020
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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.

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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

Kay Jin Lim
Affiliation: Division of Mathematical Sciences, Nanyang Technological University, SPMS-04-01, 21 Nanyang Link, 637371 Singapore

Kai Meng Tan
Affiliation: Department of Mathematics, National University of Singapore, Block S17, 10 Lower Kent Ridge Road, 119076 Singapore

Keywords: Jantzen filtration, Young symmetrizer, Young's seminormal basis
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.
Article copyright: © Copyright 2020 American Mathematical Society