Trace maps for Mackey algebras

Rognerud, Baptiste

doi:10.1016/j.jalgebra.2014.12.021

Cited by 5 publications

(3 citation statements)

References 16 publications

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“…Proof. By Corollary 4.8 of [20] then Mackey algebra µ R (D) is symmetric if and only if |D| < p 2 . Now, we just have to prove that µ 1 R (b) is a symmetric algebra if and only if µ R (D) is a symmetric algebra.…”

Section: Basic Resultsmentioning

confidence: 95%

Equivalences between blocks of p-local Mackey algebras

Rognerud

2015

Journal of Algebra

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Let G be a finite group and (K, O, k) be a p-modular system. Let R = O or k. There is a bijection between the blocks of the group algebra and the blocks of the so-called p-local Mackey algebra µ ′ are derived equivalent. Here we look at equivalences between the corresponding blocks of p-local Mackey algebras. We prove that an analogue of the Broué's conjecture is true for the p-local Mackey algebras in the following cases: for the principal blocks of p-nilpotent groups and for blocks with defect 1. We also point out the probable importance of splendid equivalences for the Mackey algebras.

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Section: Basic Resultsmentioning

confidence: 95%

Equivalences between blocks of p-local Mackey algebras

Rognerud

2015

Journal of Algebra

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“…As corollary, we have that the double Burnside algebra of a finite group G over a field k is symmetric if and only if it is a semi-simple algebra. In [22], the author studied the symmetry of the Mackey algebra. The main tool was a central linear map on the Mackey algebra which comes from the monoidal structure of the category of modules over the Mackey algebra, that is the category of Mackey functors.…”

Section: Self-injective Property Of the Double Burnside Algebramentioning

confidence: 99%

Around evaluations of biset functors

Rognerud¹

2019

Annales De l'Institut Fourier

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The evaluation functor carries important information about the category of biset functors into the category of modules over the double Burnside algebra. Our purpose here, is to study double Burnside algebras via evaluations of biset functors. In order to avoid the difficult problem of vanishing of simple functors, we look at finite groups for which there is no non-trivial vanishing and we call them non-vanishing groups. This family contains all the abelian groups, but also infinitely many others. We show that for a non-vanishing group, there is an equivalence between the category of modules over the double Burnside algebra and a specific category of biset functors. Then, we deduce results about the highest-weight structure, and the self-injective property of the double Burnside algebra. We also revisit Barker's Theorem on the semi-simplicity of the category of biset functors.

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“…The condition that Sylow p-subgroups have order 1 or p arises in characterizing other properties of Mackey functors as well: it was shown in [16] that the Mackey algebra is of finite representation type in precisely these circumstances, and also in [12,16] that this is when the Mackey algebra is symmetric. In fact, it is shown in [16] that when Sylow p-subgroups have order 1 or p (each indecomposable summand of) the Mackey algebra is a Brauer tree algebra.…”

Section: ) All Injective Cohomological Mackey Functors Have Finite Pr...mentioning

confidence: 99%

On the Projective Dimensions of Mackey Functors

Bouc

Stancu

Webb

2017

Algebr Represent Theor

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We examine the projective dimensions of Mackey functors and cohomological Mackey functors. We show over a field of characteristic p that cohomological Mackey functors are Gorenstein if and only if Sylow p-subgroups are cyclic or dihedral, and they have finite global dimension if and only if the group order is invertible or Sylow subgroups are cyclic of order 2. By contrast, we show that the only Mackey functors of finite projective dimension over a field are projective. This allows us to give a new proof of a theorem of Greenlees on the projective dimension of Mackey functors over a Dedekind domain. We conclude by completing work of Arnold on the global dimension of cohomological Mackey functors over Z.

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Trace maps for Mackey algebras

Cited by 5 publications

References 16 publications

Equivalences between blocks of p-local Mackey algebras

Equivalences between blocks of p-local Mackey algebras

Around evaluations of biset functors

On the Projective Dimensions of Mackey Functors

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