Endotrivial modules for finite groups of Lie type A in nondefining characteristic

Carlson, Jon; Mazza, Nadia; Nakano, Daniel K.

doi:10.1007/s00209-015-1529-1

Cited by 10 publications

(9 citation statements)

References 26 publications

(46 reference statements)

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“…For finite groups of Lie type in arbitrary characteristic, the p-subgroup complex is also believed to generically be a wedge of high dimensional spheres, which would imply that generically there were no exotic Sylow-trivial modules by Theorem B. It has been verified in a number of cases [Das95,Das98,Das00], and this way, we recover very recent results of Carlson-Mazza-Nakano for the general linear group for any characteristic [CMN14,CMN16], again using Theorem B, and for symplectic groups we get the following new result.…”

Section: Introductionsupporting

confidence: 82%

Endotrivial modules for finite groups via homotopy theory

Grodal

2016

Preprint

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Classifying endotrivial kG-modules, i.e., elements of the Picard group of the stable module category for an arbitrary finite group G, has been a long-running quest, which by deep work of Dade, Alperin, Carlson, Thévenaz, and others, has been reduced to understanding the subgroup consisting of modular representations that split as the trivial module direct sum a projective module when restricted to a Sylow p-subgroup. In this paper we identify this subgroup as the first cohomology group of the orbit category on non-trivial p-subgroups with values in the units k × , 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, 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-subgroup complex and 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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Section: Introductionsupporting

confidence: 82%

Endotrivial modules for finite groups via homotopy theory

Grodal

2016

Preprint

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“…The quotient groups G/Z(G) occurring above as the classical groups PSL for simply connected simple G, i.e., the finite simple groups associated to finite groups of Lie type. Special cases of the above results can be found in [13,14,15]. Note that the rank of T F (G) depends on the characteristic of k, but not on the finer structure of k.…”

Section: Introductionmentioning

confidence: 80%

Torsion free endotrivial modules for finite groups of Lie type

et al. 2022

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“…The structure of T (kG) has been computed for a various families of groups (cf. [19,20,21,22]). When H is an arbitrary Hopf algebra, an interesting question is to determine if T (H) is finitely generated, and to work out the group structure.…”

Section: Categorical Centers Of Cohomology Rings and Fixed Point Subr...mentioning

confidence: 99%

On the spectrum and support theory of a finite tensor category

Nakano¹,

Vashaw²,

Yakimov³

2021

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Finite tensor categories (FTCs) T are important generalizations of the categories of finite dimensional modules of finite dimensional Hopf algebras, which play a key role in many areas of mathematics and mathematical physics. There are two fundamentally different support theories for them: a cohomological one and a universal one based on the noncommutative Balmer spectra of their stable (triangulated) categories T.In this paper we introduce the new notion of the categorical center C • T of the cohomology ring R • T of an FTC, T. This enables us to put forward a complete and detailed program for determining the exact relationship between the two support theories, based on C • T of the cohomology ring R • T of an FTC, T. More specifically, we construct a continuous map from the noncommutative Balmer spectrum of an FTC, T, to the Proj of the categorical center C • T , and prove that this map is surjective under a weaker finite generation assumption for T than the one conjectured by Etingof-Ostrik. Under stronger assumptions, we prove that (i) the map is homeomorphism and (ii) the two-sided thick ideals of T are classified by the specialization closed subsets of Proj C • T . We conjecture that both results hold for all FTCs. Many examples are presented that demonstrate how in important cases C • T arises as a fixed point subring of R • T and how the two-sided thick ideals of T are determined in a uniform fashion (while previous methods dealt on a case-by-case basis with case specific methods). The majority of our results are proved in the greater generality of monoidal triangulated categories.

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Endotrivial modules for finite groups of Lie type A in nondefining characteristic

Cited by 10 publications

References 26 publications

Endotrivial modules for finite groups via homotopy theory

Endotrivial modules for finite groups via homotopy theory

Torsion free endotrivial modules for finite groups of Lie type

On the spectrum and support theory of a finite tensor category

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