Mackey functors

Throughout this presentation of Mackey functors, the letter R denotes a commutative and associative ring with unit, and G denotes a finite group.

1. Three equivalent definitions

There are (at least) three equivalent definitions of Mackey functors for G over R. Each of them uses a different object associated to G.

1.1. Using the poset of subgroups of G. This definition of Mackey functors uses the poset of subgroups of G. It goes back to Green ([2])

Definition 1.1.1 : A Mackey functor M for G over R consists of the following data :

• An R-module M(H), for any subgroup H of G.

• Maps of R-modules

t_{H}^{K} : M(H) → M(K) r_{H}^{K} : M(K) → M(H) c_{x,H} : M(H) → M(^{x}H) ,

whenever H ⊆ K are subgroups of G, and x ∈ G.

These maps are subject to the following conditions :

• (transitivity) If H ⊆ K ⊆ L are subgroups of G, then

t_{K}^{L}t_{H}^{K} = t_{H}^{L} r_{H}^{K}r_{K}^{L} = r_{H}^{L} .

If x and y are elements of G, and if H is a subgroup of G, then

c_{y,^{x}H}c_{x,H} = c_{yx,H} .

• (compatibility) If H ⊆ K are subgroups of G, and if x ∈ G, then

c_{x,K}t_{H}^{K} = t_{^{x}H}^{^{x}K}c_{x,H} c_{x,H}r_{H}^{K} = r_{^{x}H}^{^{x}K}c_{x,K} .

• (triviality) If H is a subgroup of G and x is an element of H, then

t_{H}^{H} = r_{H}^{H} = c_{x,H} = Id_{M(H)} .

• (Mackey axiom) If H ⊆ K ⊇ L are subgroups of G, then

r_{H}^{K}t_{L}^{K} = ∑_{x∈[H\K/L]} t_{H∩^{x}L}^{H}c_{x,H^{x}∩L}r_{H^{x}∩L}^{L} ,

where [H\K/L] is a set of representatives of the double cosets modulo H and L in K.

If M and M′ are Mackey functors for G over R, a morphism of Mackey functors f : M → M′ is a collection of morphisms f_{H} : M(H) → M′(H), where H is a subgroup of G, such that for any subgroups H ⊆ K of G and any x ∈ G

f_{K}t_{H}^{K} = t_{H}^{K}f_{H}  f_{H}r_{H}^{K} = r_{H}^{K}f_{K}  f_{^{x}H}c_{x,H} = c_{x,H}f_{H} .

If f : M → M′ and f′ : M′ → M” are morphisms of Mackey functors, the composition f′f is the morphism of Mackey functors defined by (f′f)_{H} = f′_{H}f_{H}, for H ⊆ G. The Mackey functors for G over R, with these morphisms and this composition of morphism, form a category, denoted by Mack_{R}(G).

It follows in particular from these definitions that if H is a subgroup of G and M is a Mackey functor for G over R, then M(H) is an R\overline{N}_{G}(H)-module, where \overline{N}_{G}(H) denotes the group N_{G}(H)/H.

1.2. Using the category of G-sets. The second definition of Mackey functors for G over R is due to Dress ([1]). It uses the category G-set of finite G-sets : its objects are the finite sets with a left G-action, its morphisms are the G-equivariant maps, and composition of morphisms is composition of maps.

Definition 1.2.1 : A Mackey functor M̃ for G over R is a bivariant functor from G-set to the category RG-Mod of left RG-modules, i.e. a pair of functors (M̃_{∗}, M̃^{∗}) from G-set to RG-Mod, with M̃_{∗} covariant, and M̃^{∗} contravariant, which coincide on objects (i.e. M̃_{∗}(X) = M̃^{∗}(X) for any finite G-set X, and this common value is the denoted by M̃(X)). This bivariant functor is subject to the two following additional conditions :

• (additivity) If X and Y are finite G-sets, and if i_{X} and i_{Y} are the respective inclusion maps from X and Y to their disjoint union X ⊔ Y, then the maps

M̃(X) ⊕ M̃(Y) \underrightarrow{(M̃_{∗}(i_{X}), M̃_{∗}(i_{Y}))} M̃(X ⊔ Y) \underrightarrow{\binom{M̃^{∗}(i_{X})}{M̃^{∗}(i_{Y})}} M̃(X) ⊕ M̃(Y)

are mutual inverse isomorphisms.

• (cartesian squares) If

X \underrightarrow{a} Y
b ↓  ↓ c
Z \overrightarrow{d} T

is a cartesian (i.e. pull-back) square of finite G-sets, then

M̃_{∗}(b)M̃^{∗}(a) = M̃^{∗}(d)M̃_{∗}(c) .

A morphism of Mackey functors is a natural transformation of bivariant functors, and composition of morphisms is composition of natural transformations. The category of Mackey functors for this definition will be denoted

by { \atop Mack}_{R}(G).

1.3. Using the Mackey algebra. The following definition is due to Thévenaz and Webb ([4]) :

Definition 1.3.1 : The Mackey algebra μ_{R}(G) of G over R is the associative R-algebra defined by generators t̂_{H}^{K}, r̂_{H}^{K}, and ĉ_{x,H}, for subgroups H ⊆ K of G and x ∈ G, subject to the following relations :

• (transitivity) If H ⊆ K ⊆ L are subgroups of G, then

t̂_{K}^{L} t̂_{H}^{K} = t̂_{H}^{L} r̂_{H}^{K}r̂_{K}^{L} = r̂_{H}^{L} .

If x and y are elements of G, and if H is a subgroup of G, then

ĉ_{y,^{x}H} ĉ_{x,H} = ĉ_{yx,H} .

• (compatibility) If H ⊆ K are subgroups of G, and if x ∈ G, then

ĉ_{x,K} t̂_{H}^{K} = t̂_{^{x}H}^{^{x}K} ĉ_{x,H}  ĉ_{x,H}r̂_{H}^{K} = r̂_{^{x}H}^{^{x}K} ĉ_{x,K} .

• (triviality) If H is a subgroup of G and x is an element of H, then

t̂_{H}^{H} = r̂_{H}^{H} = ĉ_{x,H} .

• (Mackey axiom) If H ⊆ K ⊇ L are subgroups of G, then

r̂_{H}^{K} t̂_{L}^{K} = ∑_{x∈[H\K/L]} t̂_{H∩^{x}L}^{H}ĉ_{x,H^{x}∩L}r̂_{H^{x}∩L}^{L} ,

where [H\K/L] is a set of representatives of the double cosets modulo H and L in K.

• (vanishing) All products of generators, different from those appearing in the previous four relations, are zero.

• (identity) The sum of the elements t̂_{H}^{H}, over all subgroups H of G, is equal to the identity element of μ_{R}(G).

Definition 1.3.2 : A Mackey functors M̂ for G over R is a μ_{R}(G)-module, and a morphism of Mackey functors is a morphism of μ_{R}(G)-modules.

Remark 1.3.3 : The Mackey algebra μ_{R}(G) has a natural anti-automor-phism σ_{G}, defined by R-linearity from

t_{H}^{K} ↦ r_{H}^{K}  r_{H}^{K} ↦ t_{H}^{K}  c_{x,H} ↦ c_{x^{−1},H} ,

for H ⊆ K ⊆ G and x ∈ G. In particular, any left μ_{R}(G)-module can be viewed as a right μ_{R}(G)-module via this anti-automorphism.

1.4. Equivalence of the definitions.

• [1 → 2] If M is a Mackey functor in the first sense, and X is a finite G-set, define

M̃(X) = ( ⊕_{x∈X} M(G_{x}))^{G} ,

where the group G permutes the components of the direct sum via its action on X, and via the maps c_{g,G_{x}} : M(G_{x}) → M(G_{gx}), for g ∈ G and x ∈ X.

If f : X → Y is a map of G-sets, and if u = (u_{x})_{x∈X} is an element of M̃(X), define the element M_{∗}(f)(u) of M̃(Y) by

∀y ∈ Y, M_{∗}(f)(u)_{y} = ∑_{x∈G_{y}\f^{−1}(y)} t_{G_{x}}^{G_{y}}u_{x} .

Conversely, if v = (v_{y})_{y∈Y} is an element of M̃(Y), define the element M^{∗}(f)(v) of M̃(X) by

∀x ∈ X, (M^{∗}(f)(v)_{x} = r_{G_{x}}^{G_{f(x)}}v_{f(x)} .

Then M̃ is a Mackey functor in the second sense, and the correspondence M ↦ M̃ is an equivalence of category from Mack_{R}(G) to { \atop Mack}_{R}(G).

• [2 → 1] If M̃ is a Mackey functor in the second sense, and if H is a subgroup of G, set

M(H) = M̃(G/H) .

If H ⊆ K are subgroups of G, let p_{H}^{K} : G/H → G/K denote the natural projection map. Define then

t_{H}^{K} = M̃_{∗}(p_{H}^{K}) r_{H}^{K} = M̃^{∗}(p_{H}^{K}) .

If x ∈ G, and if H is a subgroup of G, let γ_{x,H} denote the map of G-sets from G/H to G/^{x}H defined by γ_{x,H}(gH) = gx^{−1}^{x}H. Then set

c_{x,H} = M̃_{∗}(γ_{x,H}) .

Then M is a Mackey functor in the first sense, and the correspondence M̃ ↦ M extends to an equivalence of categories from { \atop Mack}_{R}(G) to Mack_{R}(G), which is a quasi-inverse to the equivalence 1 → 2.

• [1 → 3] If M is a Mackey functor in the first sense, set

M̂ = ⊕_{H⊆G} M(H) .

Let the generator t̂_{H}^{K} of μ_{R}(G) act on M̂ by sending the component M(L), for L ≠ H, to 0, and the component M(H) to M(K) via the map t_{H}^{K}. Similarly, let r̂_{H}^{K} send M(L) to 0 if L ≠ K, and to M(H) via the map r_{H}^{K} otherwise. Finally, if x ∈ G, let ĉ_{x,H} send the component M(L) to 0 if L ≠ H, and to M(^{x}H) via the map c_{x,H} otherwise.

Then M̂ is a μ_{R}(G)-module, and the correspondence M ↦ M̂ extends to an equivalence of categories from Mack_{R}(G) to μ_{R}(G)-Mod.

• [3 → 1] Let M̂ be a μ_{R}(G)-module. If H is a subgroup of G, define

M(H) = t̂_{H}^{H} M̂ .

The relations on the generators of μ_{R}(G) imply that if H ⊆ K and x ∈ G, then

t̂_{H}^{K}M(H) ⊆ M(K)  r̂_{H}^{K}M(K) ⊆ M(H)  ĉ_{x,H}M(H) ⊆ M(^{x}H) ,

and this defines respectively a map t_{H}^{K} : M(H) → M(K), a map r_{H}^{K} : M(K) → M(H), and a map c_{x,H} : M(H) → M(^{x}H).

Now M is a Mackey functor in the first sense, and the correspondence M̂ ↦ M extends to an equivalence of categories from μ_{R}(G)-Mod to Mack_{R}(G), which is a quasi-inverse to the equivalence 1 → 3.

Notation 1.4.1 : In view of these equivalence, the three categories Mack_{R}(G), { \atop Mack}_{R}(G) and μ_{R}(G)-Mod will sometimes be identified, and a Mackey functor will be considered as an object of any of them. This will lead in particular to equalities such as M(H) = M(G/H) for a Mackey functor M and a subgroup H of G.

2. Examples

2.1. Representation groups. The various representation groups attached to subgroups of G can generally be viewed as Mackey functors :

• Let k be a field. If H is a subgroup of G, denote by R_{k}(G) the Grothendieck group of the category of finitely generated kG-modules, for relations given by short exact sequences.

If H ⊆ K are subgroups of G, the restriction of modules induces a map r_{H}^{K} = Res_{H}^{K} : R_{k}(K) → R_{k}(H), whereas induction of modules induces a map t_{H}^{K} = Ind_{H}^{K} : R_{k}(H) → R_{k}(K). If x ∈ G, there is an obvious conjugation map c_{x,H} : R_{k}(H) → R_{k}(^{x}H). With this notation R_{k} becomes a Mackey functor for G over ℤ.

One can check easily that if X is a finite G-set, then R_{k}(X) is the Grothendieck group of G-equivariant k-vector bundles over X.

Similarly, one can consider the Grothendieck group P_{k}(G) of the category of finitely generated projective kG-modules, for relations given by direct sum decomposition. It is also a Mackey functor for the previous operations of restriction, induction, and conjugation.

• If H is a subgroup of G, denote by B(H) the Burnside group of H, i.e. the Grothendieck group of the category of finite H-sets, for relations given by disjoint unions decomposition. Then if H ⊆ K are subgroups of G, the restriction of K-sets gives a map r_{H}^{K} = Res_{H}^{K} : B(K) → B(H). Similarly, induction of H-sets (defined by Ind_{H}^{K}X = K ×_{H} X) induces a map B(H) → B(K). If x ∈ G, there is again an obvious conjugation map c_{x,H} : B(H) → B(^{x}H), and B is a Mackey functor for G over ℤ.

One can show that for any finite G-set X, the group B(X) is the Grothendieck group of the category of finite G-sets over X, for relations given by disjoint union decomposition.

2.2. (Co)homology functors. Let M be an RG-module, and l be a nonnegative integer. If K is a subgroup of G, define M(K) as the cohomology group H^{l}(K, Res_{K}^{G}M) (or H_{l}(K, Res_{K}^{G}M)). Then the operations of transfer, restriction, and conjugation, define a structure of Mackey functor on M. If X is a finite G-set, then M̃(X) ≅ H^{l}(G, RX ⊗_{R} M) (or H_{l}(G, RX ⊗_{R} M)). Similarly, the Tate (co)homology groups have a natural structure of Mackey functor.

These Mackey functors have the following additional property : the composition t_{H}^{K}r_{H}^{K} is equal to the multiplication by |K : H|, for any subgroups H ⊆ K of G. Mackey functors with this property are called cohomological.

3. Fixed points, fixed quotients, and simple Mackey functors

Since the category of Mackey functors for G over R is equivalent to the category of modules over an R-algebra, it is an abelian category. One can try to classify and describe its simple and projective objects.

3.1. Fixed points and fixed quotients. Let V be an RG-module. The zero-th cohomology functor H^{0}(−, V) is denoted by F P_{V}, and it is called the fixed points functor associated to V. Similarly, the zero-th homology functor H_{0}(−, V) is denoted by F Q_{V}, and it is called the fixed quotient functor associated to V. If H is a subgroup of G

F P_{V} (H) = V^{H} F Q_{V} (H) = V_{H} .

If H ⊆ K are subgroups of G, then the maps t_{H}^{K} and r_{H}^{K} for F P_{V} are the relative trace maps and restriction maps, i.e.

∀v ∈ V^{H}, t_{H}^{K}(v) = ∑_{x∈K/H} xv ∀w ∈ V^{K}, r_{H}^{K}(v) = v .

If g ∈ G, then the conjugation map c_{g,H} is given by

∀v ∈ V^{H}, c_{g,H}(v) = gv .

Similarly, the maps t_{H}^{K} and r_{H}^{K} for F Q_{V} are given by

∀v ∈ V_{H}, t_{H}^{K}(v) = v ∀w ∈ V_{K}, r_{H}^{K}(w) = ∑_{x∈H\K} xw ,

and the map c_{g,H} is given by c_{g,H}(v) = gv, for v ∈ V_{H}.

If X is a finite G-set, one can show that

F P_{V} (X) = Hom_{RG}(RX, V) F Q_{V} (X) = RX ⊗_{RG} V ,

where RX is the permutation module associated to X, i.e. the free R-module with basis X.

If f : X → Y is a map of finite G-sets, let Rf : RX → RY and ^{t}Rf : RY → RX be the R-linear maps defined by

∀x ∈ X, (Rf)(x) = f(x) ∀y ∈ Y, (^{t}Rf)(y) = ∑_{x∈f^{−1}(y)} x .

Then the map (F P_{V})_{∗}(f) is given by composition with ^{t}Rf, and the map (F P_{V})^{∗}(f) is given by composition with Rf. Similarly, the map (F Q_{V})_{∗}(f) is the map Rf ⊗_{RG} Id_{V}, and (F Q_{V})^{∗}(f) = ^{t}Rf ⊗_{RG} Id_{V}.

The main property of these constructions is the following :

Proposition 3.1.1 : The correspondence V ↦ F P_{V} (resp. V ↦ F Q_{V}) is a functor from RG-Mod to Mack_{R}(G), which is right adjoint (resp. left adjoint) to the evaluation functor M ↦ M(1).

Notation 3.1.2 : Let H be a subgroup of G, and let V be an R\overline{N}_{G}(H)- module. The composite bivariant functor

X ↦ X^{H} ↦ F P_{V} (X^{H}) = Hom_{R\overline{N}_{G}(H)}(R(X^{H}), V )

from G-set to R-Mod is denoted by F P_{H,V}. If X is a finite G-set, set

S_{H,V} (X) = T r_{1}^{\overline{N}_{G}(H)}(Hom_{R}(R(X^{H}), V )) ⊂ F P_{H,V} (X) ,

where T r_{1}^{\overline{N}_{G}(H)} is the relative trace map.

One can can show that F P_{H,V} is a Mackey functor, and that S_{H,V} is a sub-Mackey functor of F P_{H,V}.

Definition 3.1.3 : Let M be a Mackey functor for G over R. A minimal subgroup for M is a subgroup H of G, such that M(H) ≠ 0, but M(K) = 0 for every proper subgroup K of H.

Proposition 3.1.4 : (Thévenaz-Webb [3])

1. Let H be a subgroup of G, and V be a simple R\overline{N}_{G}(H)-module. Then S_{H,V} is a simple Mackey functor for G over R. Moreover H is a minimal subgroup for S_{H,V}, and S_{H,V} (H) ≅ V.

2. Let S be a simple Mackey functor for G over R. Let H be a minimal subgroup for S, and set V = S(H). Then V is a simple R\overline{N}_{G}(H)- module, and S is isomorphic to S_{H,V}.

3. If K is a subgroup of G and W is a simple R\overline{N}_{G}(K)-module, the simple functors S_{H,V} and S_{K,W} are isomorphic if and only if the pairs (H, V) and (K, W) are conjugate under G.

4. The Dress construction and projective Mackey functors

4.1. The Dress construction. Let M be a Mackey functor for G over R, and let X be a fixed finite G-set. The functor π_{X} : Y ↦ Y × X is an endofunctor of the category G-set, which commutes to disjoint unions, and preserves cartesian squares. If M is a Mackey functor for G over R, it follows that the functor M ∘ π_{X} is also a Mackey functor for G over R, which is denoted by M_{X}, and called the Dress construction associated to M and X.

The construction M ↦ M_{X} is an endofunctor of the category Mack_{R}(G). One can show that this functor is self-adjoint.

4.2. The Burnside functor. Let RB be the Burnside functor over R, defined by composing the Burnside functor B from G-set to ℤ-Mod with the functor A ↦ R ⊗_{ℤ} A from ℤ-Mod to R-Mod. The main properties of RB are summarized in the following proposition :

Proposition 4.2.1 : 1. Let M be a Mackey functor for G over R. Then

Hom_{Mack_{R}(G)}(RB, M) ≅ M(G) .

2. More generally, if X is a finite G-set, then

Hom_{Mack_{R}(G)}(RB_{X}, M) ≅ M(X) .

3. The functor B_{X} is a projective Mackey functor, and any Mackey functor is isomorphic to a quotient of a direct sum of such Mackey functors.

4. Let Ω = ⊔_{H⊆G} G/H. Then RB_{Ω} is a progenerator in Mack_{R}(G), and

there is an isomorphism of R-algebras

End_{Mack_{R}(G)}(RB_{Ω}) ≅ μ_{R}(G) .

5. There is an isomorphism End_{Mack_{R}(G)}(RB_{Ω}) ≅ RB(Ω × Ω), and the corresponding algebra structure on RB(Ω × Ω) is obtained by linearity from the pull-back product (X, Y) ↦ X ×_{Ω} Y in the following diagram

\xymatrix{ & & X ×_{Ω} Y \ar[dl] \ar[dr] \\ & X \ar[dl] \ar[dr] & & Y \ar[dl] \ar[dr] \\ Ω & & Ω & & Ω} .

With this identification μ_{R}(G) ≅ RB(Ω × Ω), the anti-automorphism σ_{G} of μ_{R}(G) sends the G-set (X, (f, g)) over Ω × Ω to (X, (g, f)).

References

[1] A.W.M Dress. Contributions to the theory of induced representations, volume 342 of Lecture Notes in Mathematics, pages 183–240. Springer-Verlag, 1973.

[2] J.A. Green. Axiomatic representation theory for finite groups. J. Pure Appl. Algebra, 1:41–77, 1971.

[3] Jacques Thévenaz and Peter Webb. Simple Mackey functors. In Proceedings of the 2nd International group theory conference Bressanone 1989, volume 23 of Rend. Circ. Mat. Palermo, pages 299–319, 1990. Serie II.

[4] Jacques Thévenaz and Peter Webb. The structure of Mackey functors. Trans. Amer. Math. Soc., 347(6):1865–1961, June 1995.