Trivial-source endotrivial modules for sporadic groups

Craven, David

doi:10.1007/s13366-020-00521-8

Cited by 3 publications

(1 citation statement)

References 21 publications

(49 reference statements)

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“…Lassueur-Malle in [52], Lassueur-Mazza in [55] and Craven in [32]. • Schur covering groups of the symmetric and alternating groups: Lassueur-Mazza in [54].…”

Section: Endo-trivial Modules Over Arbitrary Finite Groupsmentioning

confidence: 99%

A Tour of $p$-Permutation Modules and Related Classes of Modules

Lassueur¹

2023

Jahresber. Dtsch. Math. Ver.

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This survey provides an overview of numerous results on $p$ p -permutation modules and the closely related classes of endo-trivial, endo-permutation and endo-$p$ p -permutation modules. These classes of modules play an important role in the representation theory of finite groups. For example, they are important building blocks used to understand and parametrise several kinds of categorical equivalences between blocks of finite group algebras. For this reason, there has been, since the late 1990’s, much interest in classifying such modules. The aim of this manuscript is to review classical results as well as all the major recent advances in the area. The first part of this survey serves as an introduction to the topic for non-experts in modular representation theory of finite groups, outlining proof ideas of the most important results at the foundations of the theory. Simultaneously, the connections between the aforementioned classes of modules are emphasised. In this respect, results, which are dispersed in the literature, are brought together, and emphasis is put on common properties and the role played by the $p$ p -permutation modules throughout the theory. Finally, in the last part of the manuscript, lifting results from positive characteristic to characteristic zero are collected and their proofs sketched.

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“…Lassueur-Malle in [52], Lassueur-Mazza in [55] and Craven in [32]. • Schur covering groups of the symmetric and alternating groups: Lassueur-Mazza in [54].…”

Section: Endo-trivial Modules Over Arbitrary Finite Groupsmentioning

confidence: 99%

A Tour of $p$-Permutation Modules and Related Classes of Modules

Lassueur¹

2023

Jahresber. Dtsch. Math. Ver.

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Trivial source endo-trivial modules for finite groups with semi-dihedral Sylow 2-subgroups

Koshitani

Lassueur

2021

Beitr Algebra Geom

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Endotrivial modules for finite groups via homotopy theory

Grodal

2022

J. Amer. Math. Soc.

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Classifying endotrivial k G kG -modules, i.e., elements of the Picard group of the stable module category for an arbitrary finite group G G , has been a long-running quest. By deep work of Dade, Alperin, Carlson, Thévenaz, and others, it has been reduced to understanding the subgroup consisting of modular representations that split as the trivial module k k direct sum a projective module when restricted to a Sylow p p -subgroup. In this paper we identify this subgroup as the first cohomology group of the orbit category on non-trivial p p -subgroups with values in the units k × k^\times , viewed as a constant coefficient system. We then use homotopical techniques to give a number of formulas for this group in terms of the abelianization of normalizers and centralizers in G G , in particular verifying the Carlson–Thévenaz conjecture—this reduces the calculation of this group to algorithmic calculations in local group theory rather than representation theory. We also provide strong restrictions on when such representations of dimension greater than one can occur, in terms of the p p -subgroup complex and p p -fusion systems. We immediately recover and extend a large number of computational results in the literature, and further illustrate the computational potential by calculating the group in other sample new cases, e.g., for the Monster at all primes.

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Trivial-source endotrivial modules for sporadic groups

Cited by 3 publications

References 21 publications

A Tour of $p$-Permutation Modules and Related Classes of Modules

A Tour of $p$-Permutation Modules and Related Classes of Modules

Trivial source endo-trivial modules for finite groups with semi-dihedral Sylow 2-subgroups

Endotrivial modules for finite groups via homotopy theory

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