The torsion group of endotrivial modules

Carlson, Jon; Thévenaz, Jacques

doi:10.2140/ant.2015.9.749

Cited by 9 publications

(20 citation statements)

References 19 publications

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“…The statement proved in Theorem 3.1 below is sufficient for this paper. A somewhat different version of the method is contained in the paper [16].…”

Section: The Main Methodsmentioning

confidence: 99%

“…In all but a few examples of small Lie rank and small characteristic we show that the torsion part of T (G) equals the isomorphism classes of one-dimensional modules. There are a couple of instances in this paper when it is necessary to call upon a somewhat more sophisticated variation of the method developed in [16].…”

Section: Introductionmentioning

confidence: 99%

“…In Sections 9 and 10 we compute of the groups of endotrivial modules when p = 2 and n = 2 or 3, excluded from Theorem 1.1 by condition (d). These sections use results of [16], that show the existence of trivial source endotrivial modules of dimension greater than one. Table 1 summarizes the cases when such modules occur.…”

Section: Introductionmentioning

confidence: 99%

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

2015

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Let G be a finite group such that SL(n, q) ⊆ G ⊆ GL(n, q) and Z be a central subgroup of G. In this paper we determine the group T (G/Z) consisting of the equivalence classes of endotrivial k(G/Z)-modules where k is an algebraically closed field of characteristic p such that p does not divide q. The results in this paper complete the classification of endotrivial modules for all finite groups of (untwisted) Lie Type A, initiated earlier by the authors.

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“…The statement proved in Theorem 3.1 below is sufficient for this paper. A somewhat different version of the method is contained in the paper [16].…”

Section: The Main Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

2015

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“…This result has already found interesting applications, for instance the computation of T k (G, P ) for new classes of groups by Carlson-Mazza-Nakano [CMN14] and Carlson-Thévenaz [CT15]. Here, we will use the complex version A C (G, P ) to build a homomorphism L : A C (G, P ) → Pic G (S p (G)) which will yield the isomorphism of Theorem 1.1 when suitably restricted to torsion.…”

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confidence: 90%

Endotrivial representations of finite groups and equivariant line bundles on the Brown complex

Balmer¹

2018

Geom. Topol.

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We relate endotrivial representations of a finite group in characteristic p to equivariant line bundles on the simplicial complex of non-trivial p-subgroups, by means of weak homomorphisms.Dedicated to Serge Bouc on the occasion of his 60 th birthday IntroductionLet G be a finite group, p a prime dividing the order of G and k a field of characteristic p. For the whole paper, we fix a Sylow p-subgroup P of G.Consider the endotrivial kG-modules M , i.e. those finite dimensional k-linear representations M of G which are ⊗-invertible in the stable category kG -stab = kG -mod / kG -proj; this means that the kG-module End k (M ) is isomorphic to the trivial module k plus projective summands. The stable isomorphism classes of these endotrivial modules form an abelian group, T k (G), under tensor product. This important invariant has been fully described for p-groups in celebrated work of Carlson and Thévenaz [CT04, CT05]. Therefore, for general finite groups G, the focus has moved towards studying the relative version: T k (G, P ) := Ker T k (G) → T k (P ) .

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“…Moreover many contributions towards a general classification of endotrivial modules have been obtained over the past ten years for several families of finite groups (see e.g. [7,29,8,9,6,32,10,22,13,27] and the references therein). However, the problem of describing the structure of T (G) and its elements for an arbitrary finite group G remains open in general.…”

Section: Introductionmentioning

confidence: 99%

Endo-trivial modules for finite groups with dihedral Sylow 2-subgroup

Koshitani

Lassueur

2016

Journal of Group Theory

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Abstract. Let k be an algebraically closed field of characteristic p > 0 and G a finite group. We provide a description of the torsion subgroup T T (G) of the finitely generated abelian group T (G) of endo-trivial kG-modules when p = 2 and G has a dihedral Sylow 2-subgroup P . We prove that, in the case |P | ≥ 8, T T (G) ∼ = X(G) the group of one-dimensional kG-modules, except possibly when G/O 2 ′ (G) ∼ = A 6 , the alternating group of degree 6; in which case G may have 9-dimensional simple torsion endo-trivial modules. We also prove a similar result in the case |P | = 4, although the situation is more involved. Our results complement the tame-representation type investigation of endotrivial modules started by Carlson-Mazza-Thévenaz in the cases of semi-dihedral and generalized quaternion Sylow 2-subgroups. Furthermore we provide a general reduction result, valid at any prime p, to recover the structure of T T (G) from the structure of T T

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The torsion group of endotrivial modules

Cited by 9 publications

References 19 publications

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

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

Endotrivial representations of finite groups and equivariant line bundles on the Brown complex

Endo-trivial modules for finite groups with dihedral Sylow 2-subgroup

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