Splitting rational incomplete Mackey functors

TL;DR

This paper extends the splitting results for rational G-Mackey functors to incomplete cases, using transfer systems to classify and analyze their structure and minimal information requirements.

Contribution

It generalizes the Greenlees--May and Thevenaz--Webb splitting results to incomplete Mackey functors, providing a new intrinsic definition of split components.

Findings

Maximal splitting of rational incomplete G-Mackey functors determined by transfer system maps

Explicit calculation of idempotents in the rational incomplete Burnside ring

Examples illustrating the complexity of split pieces with simpler transfer systems

Abstract

Inspired by equivariant homotopy theory, equivariant algebra studies generalisations of G-Mackey functors that do not have all transfer maps (also known as induction maps), for G a finite group. These incomplete Mackey functors have interesting and subtle properties that are more complicated than classical algebra. The levels of incompleteness that occur are indexed by simple combinatorial data known as transfer systems for G, which are refinements of the subgroup relation satisfying certain axioms. The aim of this paper is to generalise the Greenlees--May and Thevenaz--Webb splitting result of rational G-Mackey functors to the incomplete case. By calculating idempotents of the rational incomplete Burnside ring of G, we find the maximal splitting of the category of rational incomplete G-Mackey functors. These splittings are determined by maps of the form H to G in the transfer system. We give an intrinsic definition of the split pieces beyond the idempotent description in order to understand what is the minimal information needed to determine an arbitrary rational incomplete G-Mackey functor. We end the paper with a series of examples of possible splittings and illustrate how simpler transfer systems have fewer terms in the splitting but the split pieces are more complicated.