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On Donkin's tilting module conjecture II: counterexamples

Part of: Lie algebras and Lie superalgebras Linear algebraic groups and related topics

Published online by Cambridge University Press:  02 May 2024

Christopher P. Bendel ,
Daniel K. Nakano ,
Cornelius Pillen  and
Paul Sobaje
Christopher P. Bendel
Affiliation:
Department of Mathematics, Statistics and Computer Science, University of Wisconsin-Stout, Menomonie, WI 54751, USA bendelc@uwstout.edu
Daniel K. Nakano
Affiliation:
Department of Mathematics, University of Georgia, Athens, GA 30602, USA nakano@math.uga.edu
Cornelius Pillen
Affiliation:
Department of Mathematics and Statistics, University of South Alabama, Mobile, AL 36688, USA pillen@southalabama.edu
Paul Sobaje
Affiliation:
Department of Mathematical Sciences, Georgia Southern University, Statesboro, GA 30458, USA psobaje@georgiasouthern.edu
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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 ${\rm B}_{2}$.

Keywords

tilting modulesalgebraic groupsrepresentation theory

MSC classification

Primary: 20G05: Representation theory 20G10: Cohomology theory
Secondary: 17B10: Representations, algebraic theory (weights)

Information

Type
Research Article
Information
Compositio Mathematica , Volume 160 , Issue 6 , June 2024 , pp. 1167 - 1193
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Copyright
© 2024 The Author(s). The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence

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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.

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