Mackey functors, induction from restriction functors and coinduction from transfer functors

Abstract

Boltje's plus constructions extend two well-known constructions on Mackey functors, the fixed-point functor and the fixed-quotient functor. In this paper, we show that the plus constructions are induction and coinduction functors of general module theory. As an application, we construct simple Mackey functors from simple restriction functors and simple transfer functors. We also give new proofs for the classification theorem for simple Mackey functors and semisimplicity theorem of Mackey functors. © 2007 Elsevier Inc. All rights reserved

Alcahestic subalgebras of the alchemic algebra and a correspondence of simple modules

“…Our first main result in this direction concerns the construction of simple biset functors using the techniques in [12]. In fact our result extends to classification and description of simple modules of certain unital subalgebras.…”

“…We shall not introduce this triangle-structure in this paper. But all of the results in [12,Section 3] hold in this case with some modifications. Furthermore it is easy to describe the coordinate modules of the functors induced (or coinduced) from these subalgebras.…”

“…Further, each of these subalgebras has two special types of alcahestic subalgebras. These special subalgebras allow us to introduce a triangle having similar properties as the Mackey triangle introduced in [12]. We shall not introduce this triangle-structure in this paper.…”

“…Boltje [2] generalized this concept to a morphism connecting his plus constructions. It is shown in [12] that the plus constructions are usual induction and coinduction functors and the mark morphism can be constructed by applying a series of adjunctions. In Section 6, we further generalize the mark morphism to biset functors in a similar way.…”

Projective resolutions of globally defined Mackey functors in characteristic zero

“…[4, Theorem 4.1] There are bijective correspondences between 1. the isomorphism classes of simple X , Y -modules, 2. the isomorphism classes of simple 0, 0 -modules,3. the isomorphism classes of simple pairs (H, V ) as H runs over all subquotients of G.We denote by SX ,Y H,Vthe simple X , Y -module corresponding to the simple pair (H, V ).…”