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The Dade group of a pgroup
 Inv. Math
"... Let p be a prime number. This paper solves the question of the structure of the group D(P) of endopermutation modules over an arbitrary finite pgroup P, that was open after Dade’s original papers in 1978 ([19], [20]), and it gives a proof of the conjectures proposed in [4] and [10]. This leads to ..."
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Cited by 29 (9 self)
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Let p be a prime number. This paper solves the question of the structure of the group D(P) of endopermutation modules over an arbitrary finite pgroup P, that was open after Dade’s original papers in 1978 ([19], [20]), and it gives a proof of the conjectures proposed in [4] and [10]. This leads to a presentation of D(P) by explicit generators and relations, generalizing the presentation obtained by Dade when P is abelian. A key result of independent interest is the explicit description of the kernel of the natural map from the Burnside group to the group of rational characters, in terms of the extraspecial group of order p 3 and exponent p if p ̸ = 2, or of all dihedral groups of order at least 8 if p = 2.
The Group of EndoPermutation Modules
, 1998
"... The group D(P ) of all endopermutation modules for a finite pgroup P is a finitely generated abelian group. We prove that its torsionfree rank is equal to the number of conjugacy classes of noncyclic subgroups of P . We also obtain partial results on its torsion subgroup. We determine next the s ..."
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Cited by 27 (11 self)
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The group D(P ) of all endopermutation modules for a finite pgroup P is a finitely generated abelian group. We prove that its torsionfree rank is equal to the number of conjugacy classes of noncyclic subgroups of P . We also obtain partial results on its torsion subgroup. We determine next the structure of Q#D() viewed as a functor, which turns out to be a simple functor SE,Q , indexed by the elementary group E of order p and the trivial Out(E)module Q . Finally we describe a rather strange exact sequence relating Q#D(P ) , Q#B(P ) , and Q#R(P ) , where B(P ) is the Burnside ring and R(P ) is the Grothendieck ring of QP modules.
Endotrivial modules for finite groups of Lie type
 J. Reine Angew. Math
"... Let G be a finite group and k be a field of characteristic p> 0. An endotrivial kGmodule is a finitely generated kGmodule M whose kendomorphism ring is isomorphic to a trivial module in the stable module category. That is, M is an endotrivial module provided Homk(M,M) ∼ = k ⊕ P where P is a pr ..."
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Cited by 17 (12 self)
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Let G be a finite group and k be a field of characteristic p> 0. An endotrivial kGmodule is a finitely generated kGmodule M whose kendomorphism ring is isomorphic to a trivial module in the stable module category. That is, M is an endotrivial module provided Homk(M,M) ∼ = k ⊕ P where P is a projective kG
Endotrivial modules for the symmetric and alternating groups
 Proc. Edinburgh Math. Soc
"... Abstract. We complete a classification of the groups of endotrivial modules for the modular group algebras of symmetric groups and alternating groups. We show that for n ≥ p2 the torsion subgroup of the group of endotrivial modules for the symmetric groups is generated by the sign representation. Th ..."
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Cited by 13 (8 self)
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Abstract. We complete a classification of the groups of endotrivial modules for the modular group algebras of symmetric groups and alternating groups. We show that for n ≥ p2 the torsion subgroup of the group of endotrivial modules for the symmetric groups is generated by the sign representation. The torsion subgroup is trivial for the alternating groups. The torsion free part of the group is free abelian of rank one if n ≥ p2 + p and has rank 2 if p2 ≤ n < p2 + p. This completes the work begun earlier by Carlson, Mazza and Nakano. 1.
GLUING IN TENSOR TRIANGULAR GEOMETRY
, 2006
"... Abstract. We discuss gluing of objects and gluing of morphisms in tensor triangulated categories. We illustrate the results by producing, among other things, a MayerVietoris exact sequence involving Picard groups. Contents ..."
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Cited by 11 (3 self)
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Abstract. We discuss gluing of objects and gluing of morphisms in tensor triangulated categories. We illustrate the results by producing, among other things, a MayerVietoris exact sequence involving Picard groups. Contents
Gluing techniques in triangular geometry
 Q. J. Math
"... Abstract. We discuss gluing of objects and gluing of morphisms in triangulated categories. We illustrate the results by producing, among other things, a MayerVietoris exact sequence involving Picard groups. Contents ..."
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Cited by 8 (3 self)
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Abstract. We discuss gluing of objects and gluing of morphisms in triangulated categories. We illustrate the results by producing, among other things, a MayerVietoris exact sequence involving Picard groups. Contents
The Dade group of (almost) extraspecial pgroups
 J. Pure and Applied Algebra
"... Abstract: In this paper, we determine a presentation by explicit generators and relations for the Dade group of all (almost) extraspecial pgroups. The proof of the main result uses the cohomological properties of the Tits building corresponding to the natural geometric structure of the lattice of ..."
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Cited by 7 (6 self)
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Abstract: In this paper, we determine a presentation by explicit generators and relations for the Dade group of all (almost) extraspecial pgroups. The proof of the main result uses the cohomological properties of the Tits building corresponding to the natural geometric structure of the lattice of subgroups of such pgroups. AMS Subject Classication: 20C20, 20D15 1.
STACKS OF GROUP REPRESENTATIONS
"... Abstract. We start with a small paradigm shift about group representations, namely the observation that restriction to a subgroup can be understood as an extensionofscalars. We deduce that, given a group G, the derived and the stable categories of representations of a subgroup H can be constructed ..."
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Cited by 7 (4 self)
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Abstract. We start with a small paradigm shift about group representations, namely the observation that restriction to a subgroup can be understood as an extensionofscalars. We deduce that, given a group G, the derived and the stable categories of representations of a subgroup H can be constructed out of the corresponding category for G by a purely triangulatedcategorical construction, analogous to étale extension in algebraic geometry. In the case of finite groups, we then use descent methods to investigate when modular representations of the subgroup H can be extended to G. We show that the presheaves of plain, derived and stable representations all form