Kernels, inflations, evaluations, and imprimitivity of Mackey functors


Yaraneri E.

JOURNAL OF ALGEBRA, vol.319, no.5, pp.1993-2029, 2008 (Journal Indexed in SCI) identifier identifier

  • Publication Type: Article / Article
  • Volume: 319 Issue: 5
  • Publication Date: 2008
  • Doi Number: 10.1016/j.jalgebra.2007.09.027
  • Title of Journal : JOURNAL OF ALGEBRA
  • Page Numbers: pp.1993-2029

Abstract

Let M be a Mackey functor for a finite group G. By the kernel of M we mean the largest normal subgroup N of G such that M can be inflated from a Mackey functor for G/N. We first study kernels of Mackey functors, and (relative) projectivity of inflated Mackey functors. For a normal subgroup N of G, denoting by P-H,V(G) the projective cover of a simple Mackey functor for G of the form S-H,V(G) we next try to answer the question: how are the Mackey functors P-H/N,V(G/N) and P-H,V(G) related? We then study imprimitive Mackey functors by which we mean Mackey functors for G induced from Mackey functors for proper subgroups of G. We obtain some results about imprimitive Mackey functors of the form P-H,V(G), including a Mackey functor version of Fong's theorem on induced modules of modular group algebras of p-solvable groups. Aiming to characterize subgroups H of G for which the module P-H,V(G)(H) is the projective cover of the simple K(N)over bar (G)(H)-module V where the coefficient ring K is a field, we finally study evaluations of Mackey functors. (c) 2007 Elsevier Inc. All rights reserved.