No CrossRef data available.
Article contents
On Donkin's tilting module conjecture II: counterexamples
Published online by Cambridge University Press: 02 May 2024
Abstract
In this paper we produce infinite families of counterexamples to Jantzen's question posed in 1980 on the existence of Weyl 
$p$-filtrations for Weyl modules for an algebraic group and Donkin's tilting module conjecture formulated in 1990. New techniques to exhibit explicit examples are provided along with methods to produce counterexamples in large rank from counterexamples in small rank. Counterexamples can be produced via our methods for all groups other than when the root system is of type 
${\rm A}_{n}$ or 
${\rm B}_{2}$.
MSC classification
- Type
- Research Article
- Information
- Copyright
- © 2024 The Author(s). The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence
Footnotes
Research of the first author was supported in part by Simons Foundation Collaboration Grant 317062. Research of the second author was supported in part by NSF grants DMS-1701768 and DMS-2101941. Research of the third author was supported in part by Simons Foundation Collaboration Grant 245236.
References
Andersen, H. H., Extensions of modules for algebraic groups, Amer. J. Math. 106 (1984), 498–504.10.2307/2374311CrossRefGoogle Scholar
Andersen, H. H., 
$p$-filtrations of dual Weyl modules, Adv. Math. 352 (2019), 231–245.10.1016/j.aim.2019.05.028CrossRefGoogle Scholar
$p$-filtrations of dual Weyl modules, Adv. Math. 352 (2019), 231–245.10.1016/j.aim.2019.05.028CrossRefGoogle ScholarBendel, C. P., Nakano, D. K. and Pillen, C., Extensions for Frobenius kernels, J. Algebra 272 (2004), 476–511.10.1016/j.jalgebra.2003.04.003CrossRefGoogle Scholar
Bendel, C. P., Nakano, D. K., Pillen, C. and Sobaje, P., On tensoring with the Steinberg representation, Transform. Groups 25 (2020), 981–1008.10.1007/s00031-019-09530-xCrossRefGoogle Scholar
Bendel, C. P., Nakano, D. K., Pillen, C. and Sobaje, P., Counterexamples to the tilting and $(p,r)$-filtration conjectures, J. Reine Angew. Math. 767 (2020), 193–202.10.1515/crelle-2019-0036CrossRefGoogle Scholar
$(p,r)$-filtration conjectures, J. Reine Angew. Math. 767 (2020), 193–202.10.1515/crelle-2019-0036CrossRefGoogle ScholarBendel, C. P., Nakano, D. K., Pillen, C. and Sobaje, P., On Donkin's tilting module conjecture I: lowering the prime, Represent. Theory 26 (2022), 455–497.10.1090/ert/608CrossRefGoogle Scholar
Bendel, C. P., Nakano, D. K., Pillen, C. and Sobaje, P., On Donkin's tilting module conjecture III: new generic lower bounds, J. Algebra, doi:10.1016/j.jalgebra.2023.07.001.Google Scholar
Donkin, S., A note on decomposition numbers of reductive algebraic groups, J. Algebra 80 (1983), 226–234.10.1016/0021-8693(83)90029-7CrossRefGoogle Scholar
Donkin, S., On tilting modules for algebraic groups, Math. Z. 212 (1993), 39–60.10.1007/BF02571640CrossRefGoogle Scholar
Doty, S., GAP: Weyl Modules, Version 1.1 (2009), https://doty.math.luc.edu/weyl/version_1.1/manual.pdf.Google Scholar
Dowd, M. F. and Sin, P., On representations of algebraic groups in characteristic $2$, Comm. Algebra 28 (1996), 2597–2686.10.1080/00927879608542648CrossRefGoogle Scholar
$2$, Comm. Algebra 28 (1996), 2597–2686.10.1080/00927879608542648CrossRefGoogle ScholarThe GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.11.1 (2021), https://www.gap-system.org.Google Scholar
Humphreys, J. E., Introduction to Lie algebras and representation theory, Graduate Texts in Mathematics (Springer, New York, 1972).10.1007/978-1-4612-6398-2CrossRefGoogle Scholar
Humphreys, J. E. and Verma, D. N., Projective modules for finite Chevalley groups, Bull. Amer. Math. Soc. (N.S.) 79 (1973), 467–468.10.1090/S0002-9904-1973-13220-3CrossRefGoogle Scholar
Jantzen, J. C., Darstellungen halbeinfacher Gruppen und ihrer Frobenius-Kerne, J. Reine Angew. Math. 317 (1980), 157–199.Google Scholar
Jantzen, J. C., First cohomology groups for classical lie algebras, Progress in Mathematics, vol. 95 (Birkhäuser, Basel, 1991).Google Scholar
Jantzen, J. C., Representations of algebraic groups, second edition, Mathematical Surveys and Monographs, vol. 107 (American Mathematical Society, Providence, RI, 2003).Google Scholar
Kildetoft, T. and Nakano, D. K., On good $(p,r)$ filtrations for rational $G$-modules, J. Algebra 423 (2015), 702–725.10.1016/j.jalgebra.2014.09.030CrossRefGoogle Scholar
$(p,r)$ filtrations for rational 
$G$-modules, J. Algebra 423 (2015), 702–725.10.1016/j.jalgebra.2014.09.030CrossRefGoogle Scholarvan Leeuwen, M. A. A., Cohen, A. M. and Lisser, B., LiE, A package for Lie group computations (Computer Algebra Nederland, Amsterdam, 1992).Google Scholar
Lübeck, F., Tables of weight multiplicities, http://www.math.rwth-aachen.de/~Frank.Luebeck/chev/WMSmall/index.html.Google Scholar
Parshall, B. J. and Scott, L. L., On $p$-filtrations of Weyl modules, J. Lond. Math. Soc. (2) 91 (2015), 127–158.10.1112/jlms/jdu067CrossRefGoogle Scholar
$p$-filtrations of Weyl modules, J. Lond. Math. Soc. (2) 91 (2015), 127–158.10.1112/jlms/jdu067CrossRefGoogle ScholarPillen, C., Tensor products of modules with restricted highest weight, Comm. Algebra 21 (1993), 3647–3661.10.1080/00927879308824754CrossRefGoogle Scholar
Sin, P., Extensions of simple modules for special algebraic groups, J. Algebra 170 (1994), 1011–1034.10.1006/jabr.1994.1373CrossRefGoogle Scholar
Sobaje, P., On $(p,r)$-filtrations and tilting modules, Proc. Amer. Math. Soc. 146 (2018), 1951–1961.10.1090/proc/13926CrossRefGoogle Scholar
$(p,r)$-filtrations and tilting modules, Proc. Amer. Math. Soc. 146 (2018), 1951–1961.10.1090/proc/13926CrossRefGoogle ScholarSobaje, P., The Steinberg quotient of a tilting character, Math. Z. 297 (2021), 1733–1747.10.1007/s00209-020-02576-8CrossRefGoogle Scholar