## 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
- Represent. Theory
**24**(2020), 551-579 - DOI: https://doi.org/10.1090/ert/553
- 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.## References

- Kaan Akin, David A. Buchsbaum, and Jerzy Weyman,
*Schur functors and Schur complexes*, Adv. in Math.**44**(1982), no. 3, 207–278. MR**658729**, DOI 10.1016/0001-8708(82)90039-1 - Henning Haahr Andersen,
*Filtrations of cohomology modules for Chevalley groups*, Ann. Sci. École Norm. Sup. (4)**16**(1983), no. 4, 495–528 (1984). MR**740588**, DOI 10.24033/asens.1458 - Henning Haahr Andersen,
*Jantzen’s filtrations of Weyl modules*, Math. Z.**194**(1987), no. 1, 127–142. MR**871225**, DOI 10.1007/BF01168012 - Jonathan Brundan and Alexander Kleshchev,
*On translation functors for general linear and symmetric groups*, Proc. London Math. Soc. (3)**80**(2000), no. 1, 75–106. MR**1719176**, DOI 10.1112/S0024611500012132 - S. Donkin,
*The $q$-Schur algebra*, London Mathematical Society Lecture Note Series, vol. 253, Cambridge University Press, Cambridge, 1998. MR**1707336**, DOI 10.1017/CBO9780511600708 - M. Fang, K. J. Lim, and K. M. Tan,
*Young’s seminormal basis vectors and their denominators*, arXiv:2007.11188v1 (2020). - William Fulton,
*Young tableaux*, London Mathematical Society Student Texts, vol. 35, Cambridge University Press, Cambridge, 1997. With applications to representation theory and geometry. MR**1464693** - Eugenio Giannelli, Kay Jin Lim, William O’Donovan, and Mark Wildon,
*On signed Young permutation modules and signed $p$-Kostka numbers*, J. Group Theory**20**(2017), no. 4, 637–679. MR**3667114**, DOI 10.1515/jgth-2017-0007 - James A. Green,
*Polynomial representations of $\textrm {GL}_{n}$*, Lecture Notes in Mathematics, vol. 830, Springer-Verlag, Berlin-New York, 1980. MR**606556**, DOI 10.1007/BFb0092296 - David J. Hemmer and Daniel K. Nakano,
*Specht filtrations for Hecke algebras of type A*, J. London Math. Soc. (2)**69**(2004), no. 3, 623–638. MR**2050037**, DOI 10.1112/S0024610704005186 - G. D. James,
*The representation theory of the symmetric groups*, Lecture Notes in Mathematics, vol. 682, Springer, Berlin, 1978. MR**513828**, DOI 10.1007/BFb0067708 - Jens C. Jantzen,
*Darstellungen halbeinfacher Gruppen und kontravariante Formen*, J. Reine Angew. Math.**290**(1977), 117–141. MR**432775**, DOI 10.1515/crll.1977.290.117 - Jens Carsten Jantzen,
*Representations of algebraic groups*, 2nd ed., Mathematical Surveys and Monographs, vol. 107, American Mathematical Society, Providence, RI, 2003. MR**2015057** - E. E. Kummer,
*Über die Ergänzungssätze zu den allgemeinen Reciprocitätsgesetzen*, J. Reine Angew. Math.**44**(1852), 93–146 (German). MR**1578793**, DOI 10.1515/crll.1852.44.93 - Sinéad Lyle and Andrew Mathas,
*Carter-Payne homomorphisms and Jantzen filtrations*, J. Algebraic Combin.**32**(2010), no. 3, 417–457. MR**2721060**, DOI 10.1007/s10801-010-0222-z - Andrew Mathas,
*Iwahori-Hecke algebras and Schur algebras of the symmetric group*, University Lecture Series, vol. 15, American Mathematical Society, Providence, RI, 1999. MR**1711316**, DOI 10.1090/ulect/015 - G. E. Murphy,
*On the representation theory of the symmetric groups and associated Hecke algebras*, J. Algebra**152**(1992), no. 2, 492–513. MR**1194316**, DOI 10.1016/0021-8693(92)90045-N - Claudiu Raicu,
*Products of Young symmetrizers and ideals in the generic tensor algebra*, J. Algebraic Combin.**39**(2014), no. 2, 247–270. MR**3159252**, DOI 10.1007/s10801-013-0447-8 - S. Ryom-Hansen,
*On the denominators of Young’s seminormal basis*, arXiv:0904.4243v3. - Steen Ryom-Hansen,
*Young’s seminormal form and simple modules for $S_n$ in characteristic $p$*, Algebr. Represent. Theory**16**(2013), no. 6, 1587–1609. MR**3127349**, DOI 10.1007/s10468-012-9372-0 - Peng Shan,
*Graded decomposition matrices of $v$-Schur algebras via Jantzen filtration*, Represent. Theory**16**(2012), 212–269. MR**2915315**, DOI 10.1090/S1088-4165-2012-00416-2 - Burt Totaro,
*Projective resolutions of representations of $\textrm {GL}(n)$*, J. Reine Angew. Math.**482**(1997), 1–13. MR**1427655**, DOI 10.1515/crll.1997.482.1 - Nanhua Xi,
*Irreducible modules of quantized enveloping algebras at roots of $1$*, Publ. Res. Inst. Math. Sci.**32**(1996), no. 2, 235–276. MR**1382803**, DOI 10.2977/prims/1195162964

## Bibliographic 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