Globally Defined Mackey Functors and Maps Between Classifying Spaces

Peter Webb http://www.math.umn.edu/∼webb

Outline of talk

1. The Deep End

2. Globally defined Mackey functors 3. Highest weight categories

4. Decomposition theory 1. The Deep End

THEOREM. Let X_{H,V} be a stable summand of (BH_{+})_{2}^{∧} corresponding to the simple 𝔽_{2} Out(H)-module V, where H is a 2-group. The matrix

rank_{ℤ_{p}} Hom(X_{H,V} , X_{K,W})

is the Cartan matrix shown, where X = all and Y = 1.

THEOREM. Over any field R, for any class of groups X closed under extensions and sections, the Cartan matrix C_{R}^{X ,X} of globally defined Mackey functors Mack_{R}^{X ,X} is symmetric and non-singular.

THEOREM. Let (K, O, k) be a p-modular system. Then the Cartan matrices of Mack_{K}^{X ,X} and Mack_{k}^{X ,X}, and decomposition matrix, satisfy C_{k} = D^{T} C_{K}D.

THEOREM. Over a field R of characteristic zero Mack_{R}^{X ,Y} is a highest weight category. Mack_{R}^{1,1} is semisimple.

2. Globally defined Mackey functors

Let X and Y be sets of finite groups, closed under taking extensions and sections.

A_{R}^{X ,Y}(G, H) = Grothendieck group over R of finite (G, H)- bisets with G-stabilizers in X and H-stablizers in Y. It is the free R-module with basis the transitive such (G, H)- bisets.

Product: _{G}Ω_{H} ∘_{H}Ψ_{K} = Ω ×_{H} Ψ = Ω × Ψ/ ∼ where (ωh, ψ) = (ω, hψ).

Special case: A_{ℤ}^{all,1}(G, G) is the double Burnside ring which appeared in the 1978 thesis of C. Witten.

The Burnside category C_{R}^{X ,Y} has objects = all finite groups in a section-closed class D, Hom(H, G) := A_{R}^{X ,Y}(G, H) with composition of morphisms given by the product. The special case C_{R}^{all,1} appeared in Adams-Gunawardena-Miller, Topology (1985).

A globally defined Mackey functor is an R-linear functor M : C_{R}^{X ,Y} → R-mod. The category of these functors is denoted Mack_{R}^{X ,Y}.

See Webb, A guide to Mackey functors, Handbook of Algebra vol 2, or Webb’s site.

Special cases: when X = all and Y = 1 these were called ‘global Mackey functors’ in tom Dieck, Transformation Groups, 1987. Webb (JPPA 1993) calls these ‘inflation functors’ and the case X = Y = 1 ‘global Mackey functors’. Bouc more recently calls these functors ‘biset functors’.

The global Mackey algebra: μ_{R}^{X ,Y} = ⨁_{G,H} A_{R}^{X ,Y}(G, H). GDMFs are the same as μ_{R}^{X ,Y}-modules.

Uses of GDMFs:

1. A method for reducing the computation of H^{∗}(G) to pgroups, by calculating composition factors of H^{∗}(G) as a GDMF (Webb 1993).

2. A proof of the theorems of Nishida, Benson-Feshbach and Martino-Priddy on multiplicities of stable summands of (BG_{+})_{p}^{∧} (Webb 1993).

3. The description of the torsion-free part of the Dade group by Bouc-Thévenaz (2000).

The simple GDMFs S_{H,V} are parametrized by pairs (H, V) where H is a group and V is a simple R Out(H)-module (Webb 1993, Bouc 1996). If R is a field or discrete valuation ring they have projective covers P_{H,V}. The representable functors Hom_{C_{R}^{X ,Y}} (H, ) are projective and

Hom_{Mack}_{R}^{X ,Y} (Hom_{C_{R}^{X ,Y}} (H, ), Hom_{C_{R}^{X ,Y}} (G, ))
≅ Hom_{C_{R}^{X ,Y}} (G, H).

Carlsson’s theorem (the Segal conjecture) implies when H is a p-group that

Hom_{C_{ℤp}^{all,1}}(G, H) = A_{ℤ_{p}}^{all,1}(H, G) ≅ Hom((BH_{+})_{p}^{∧}, (BG_{+})_{p}^{∧}).

There is an equivalence of full subcategories with the following objects: summands P_{H,V} of Hom_{C_{R}^{X ,Y}} (H, ), H a p-group summands X_{H,V} of (BH_{+})_{p}^{∧}, H a p-group.

Under this equivalence, Hom(P_{H,V} , P_{K,W}) ≅ Hom(X_{H,V} , X_{K,W}), the entries in the Cartan matrix up to dimensions of endomorphisms of simples.