Splitting rational incomplete Mackey functors
David Barnes, Michael A. Hill, Magdalena Kedziorek
TL;DR
The paper studies rational incomplete Mackey functors for a finite group $G$ parameterized by a transfer system $\\mathcal{O}$, aiming to extend the Greenlees–May type splitting to the incomplete setting. Central to the approach is the rational $\\mathcal{O}$-Burnside ring $\\underline{A}^{\\mathcal{O}}(G/G)\\otimes\\mathbb{Q}$, whose orthogonal idempotents $e_{[H]^{\\mathcal{O}}}$ induce a maximal decomposition of the category of $\\mathcal{O}$-Mackey functors into a product over inseparability classes $[H]^{\\mathcal{O}}$. The authors develop intrinsic characterizations of the split pieces via subcategories above inseparability classes, left and right adjoints (Ind and CoInd) relative to $[H]^{\\mathcal{O}}$-Mackey functors, and Frobenius reciprocity in the rational setting. A key simplification occurs for disk-like transfer systems, where the splitting reduces to a product of categories of modules over Weyl groups, recovering Greenlees–May in the complete case. The paper provides explicit formulas for induction, restriction, and transfer maps within the split components and illustrates the theory with concrete examples for cyclic groups, highlighting how transfer structure controls the complexity of the split pieces and the overall decomposition.
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.