On the lifting of the Dade group

Lassueur, Caroline; Thévenaz, Jacques

doi:10.1515/jgth-2018-0145

Cited by 4 publications

(2 citation statements)

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“…Below, we translate this construction from Dade P -algebras to endo-permutation modules. In characteristic 2 the question of the existence of such a section was open for a long time and eventually proved in [58], relying on Bouc's classification of endo-permutation kPmodules.…”

Section: Lemma 73mentioning

confidence: 99%

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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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Section: Lemma 73mentioning

confidence: 99%

“…So the elements of the classes in D R (P ) are not just capped endopermutation RP -modules, but strongly capped endo-permutation RP -modules. This is one of the key arguments used in [58] which makes the construction of a section in characteristic 2 possible. To prove Assertion (a), we need to consider determinants.…”

Section: Lemma 73mentioning

confidence: 99%