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      Socles and radicals of Mackey functors

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      Author
      Yaraneri, E.
      Date
      2009
      Source Title
      Journal of Algebra
      Print ISSN
      218693
      Volume
      321
      Issue
      10
      Pages
      2970 - 3025
      Language
      English
      Type
      Article
      Item Usage Stats
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      Abstract
      We study the socle and the radical of a Mackey functor M for a finite group G over a field K (usually, of characteristic p > 0). For a subgroup H of G, we construct bijections between some classes of the simple subfunctors of M and some classes of the simple K over(N, -)G (H)-submodules of M (H). We relate the multiplicity of a simple Mackey functor SH, V G in the socle of M to the multiplicity of V in the socle of a certain K over(N, -)G (H)-submodule of M (H). We also obtain similar results for the maximal subfunctors of M. We then apply these general results to some special Mackey functors for G, including the functors obtained by inducing or restricting a simple Mackey functor, Mackey functors for a p-group, the fixed point functor, and the Burnside functor BK G. For instance, we find the first four top factors of the radical series of BK G for a p-group G, and assuming further that G is an abelian p-group we find the radical series of BK G. © 2009 Elsevier Inc. All rights reserved.
      Keywords
      Brauer quotient
      Burnside functor
      Loewy length
      Mackey algebra
      Mackey functor
      Maximal subfunctor
      Primordial subgroup
      Radical
      Restriction kernel
      Simple subfunctor
      Socle
      Permalink
      http://hdl.handle.net/11693/22752
      Published Version (Please cite this version)
      http://dx.doi.org/10.1016/j.jalgebra.2009.02.012
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