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PreprintarXiv licencearXiv

On Donkin's Tilting Module Conjecture IV: New Frontiers

  • Christopher P. Bendel
  • Daniel K. Nakano
  • Cornelius Pillen
  • Paul Sobaje
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Abstract

Let G be a simple, simply connected algebraic group scheme defined over 𝔽p, and let Gr be the rth Frobenius kernel. Donkin's famous Tilting Module Conjecture purports that a given indecomposable injective Gr-module can be realized as the restriction of a specific tilting module. The conjecture was first stated in 1990 and withstood proof for nearly 30 years. The authors discovered a counterexample in 2019. This paper provides some indication about the rich mathematics surrounding the Tilting Module Conjecture that also discusses an older conjecture by Humphreys and Verma. New versions of the Tilting Module Conjectures are formulated given multiple families of new counterexamples. Finally, through a new calculation, the authors present further evidence that the Tilting Module Conjecture should hold for reductive algebraic groups with an underlying root system of Type An for all fields of characteristic p > 0.

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Cite

BibTeXRIS
BibTeX
@misc{bendel2026donkins,
  title = {{On Donkin's Tilting Module Conjecture IV: New Frontiers}},
  author = {Christopher P. Bendel and Daniel K. Nakano and Cornelius Pillen and Paul Sobaje},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.13483},
  eprint = {2609.13483},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.13483},
}
RIS
TY  - UNPB
TI  - On Donkin's Tilting Module Conjecture IV: New Frontiers
AU  - Christopher P. Bendel
AU  - Daniel K. Nakano
AU  - Cornelius Pillen
AU  - Paul Sobaje
PY  - 2026
DA  - 2026/09/11
PB  - arXiv
DO  - 10.48550/arxiv.2609.13483
UR  - https://arxiv.org/abs/2609.13483
AB  - Let G be a simple, simply connected algebraic group scheme defined over 𝔽_p, and let G_r be the rth Frobenius kernel. Donkin's famous Tilting Module Conjecture purports that a given indecomposable injective G_r-module can be realized as the restriction of a specific tilting module. The conjecture was first stated in 1990 and withstood proof for nearly 30 years. The authors discovered a counterexample in 2019. This paper provides some indication about the rich mathematics surrounding the Tilting Module Conjecture that also discusses an older conjecture by Humphreys and Verma. New versions of the Tilting Module Conjectures are formulated given multiple families of new counterexamples. Finally, through a new calculation, the authors present further evidence that the Tilting Module Conjecture should hold for reductive algebraic groups with an underlying root system of Type A_n for all fields of characteristic p > 0.
ER  -