Green functors

Recall that the letter R denotes a commutative and associative ring with unit, and G denotes a finite group.

1. Two equivalent definitions

There are (at least) two equivalent definitions of Green functors for G over R.

1.1. Using the poset of subgroups of G. This definition of Green functors uses the poset of subgroups of G. It goes back to Green ([7])

Definition 1.1.1 : A Green functor A for G over R is a Mackey functor for G over R, together with an R-algebra structure on A(H), for each subgroup H of G. The Mackey structure and the algebras structures have to be compatible in the following sense :

• If H ⊆ K are subgroups of G, and if x ∈ G, the maps r_{H}^{K} and c_{x,H} are maps of R-algebras.

• (Frobenius relations) If H ⊆ K are subgroups of G, if a ∈ A(H) and b ∈ A(K), then

b(t_{H}^{K}a) = t_{H}^{K}((r_{H}^{K}b)a) (t_{H}^{K}a)b = t_{H}^{K}(a(r_{H}^{K}b)) .

A morphism of Green functors f : A → B is a morphism of Mackey functors such that for each subgroup H of G, the map f_{H} : A(H) → B(H) is a map of R-algebras.

Remark 1.1.2 : The R-algebras considered here are always supposed unital, and the maps of R-algebras must preserve identity elements.

1.2. Using the category of G-sets. The following definition of Green functors ([3] Section 2.2) is analogous to the Dress definition of Mackey functors.

Definition 1.2.1 : A Green functor A for G over R is a Mackey functor for G over R endowed for any G-sets X and Y with bilinear maps

A(X) × A(Y) → A(X × Y)

denoted by (a, b) ↦ a × b which are bifunctorial, associative, and unitary, in the following sense:

• (Bifunctoriality) If f : X → X′ and g : Y → Y′ are morphisms of G-sets, then the squares

A(X) × A(Y) \underrightarrow{×} A(X × Y)
A_{∗}(f) × A_{∗}(g) ↓  ↓ A_{∗}(f × g)
A(X′) × A(Y′) \overrightarrow{×} A(X′ × Y′)

A(X) × A(Y) \underrightarrow{×} A(X × Y)
A^{∗}(f) × A^{∗}(g) ↑  ↑ A^{∗}(f × g)
A(X′) × A(Y′) \overrightarrow{×} A(X′ × Y′)

are commutative.

• (Associativity) If X, Y and Z are G-sets, then the square

A(X) × A(Y) × A(Z) \underrightarrow{Id_{A(X)} × (×)} A(X) × A(Y × Z)
(×) × Id_{A(Z)} ↓  ↓ ×
A(X × Y) × A(Z)  \overrightarrow{×}  A(X × Y × Z)

is commutative, up to identifications (X × Y) × Z ≃ X × Y × Z ≃ X × (Y × Z).

• (Unitarity) If • denotes the G-set with one element, there exists an element ε_{A} ∈ A(•) such that for any G-set X and for any a ∈ A(X)

A_{∗}(p_{X})(a × ε_{A}) = a = A_{∗}(q_{X})(ε_{A} × a)

denoting by p_{X} (resp. q_{X}) the (bijective) projection from X × • (resp. from • × X) to X.

If A and B are Green functors for the group G, a morphism of Green functors f : A → B is a morphism of Mackey functors such that for any G-sets X and Y, the square

A(X) × A(Y) \underrightarrow{×} A(X × Y)
f_{X} × f_{Y} ↓  ↓ f_{X×Y}
B(X) × B(Y) \overrightarrow{×} B(X × Y)

is commutative. The composition of morphisms of Green functors is the composition of morphisms of Mackey functors. The category of Green functors for G over R is denoted by Green_{R}(G).

Remark 1.2.2 : The most concise way to express this definition (Street [9]) uses the monoidal structures on G-set and R-Mod, given respectively by direct product of G-sets and tensor product of R-modules. More precisely, let C : G-set × G-set → G-set denote the direct product functor, and T : R-Mod × R-Mod → R-Mod denote the tensor product functor. A Green functor for G over R is just a Mackey functor for G over R, viewed as a bivariant functor from G-set to RG-Mod, which is monoidal, i.e. endowed with a natural transformation T ∘ A → A ∘ C of bivariant functors, which has to be compatible with the various isomorphisms corresponding to the associativity and unit objects of the monoidal structures.

1.3. Equivalence of the definitions.

• [1 → 2] If A is a Green functor for the first definition, recall that the corresponding Mackey functor in the sense of Dress is defined for a finite G-set X by

A(X) = ( ⊕_{x∈X} A(G_{x}))^{G} .

If Y is another finite G-set, define a product map A(X)×A(Y) → A(X ×Y) in the following way : if u = (u_{x})_{x∈X} ∈ A(X) and v = (v_{y})_{y∈Y} ∈ A(Y), then the product u × v is defined by

(u × v)_{x,y} = r_{G_{x,y}}^{G_{x}}(u_{x})r_{G_{x,y}}^{G_{y}}(v_{y}) ,

for x ∈ X and y ∈ Y, where G_{x,y} = G_{x} ∩ G_{y}. The identity element ε of A is the identity element of the algebra A(G) = A(•).

• [2 → 1] If A is a Green functor in the second sense, then recall that A(H) is defined by A(H) = A(G/H) for a subgroup H of G. Then the product A(H) × A(H) → A(H) is defined by

a.b = A^{∗}(δ_{G/H})(a × b) ,

where δ_{G/H} is the diagonal inclusion G/H → (G/H) × (G/H). The identity element ε_{H} of A(H) is equal to r_{H}^{G}ε, where ε is the identity element of A.

2. Examples

2.1. Representations rings. The representation groups associated to subgroups of G have generally a natural ring structure, for which they can be viewed as Green functors :

• If k is a field, then the tensor product of RH-modules (over k) induces a ring structure on R_{k}(H), for H ⊆ G. The identity element is the image of the trivial module k. One can check that R_{k} is a Green functor for G over ℤ.

• If H is a subgroup of G, then the direct product of H-sets gives a ring structure on the Burnside group B(H). Its identity element is the image of the trivial H-set •. This endows the Burnside functor B with a Green functor structure.

Recall that if X is a finite G-set, then B(X) is the Grothendieck group of the category of finite G-sets over X. If Y is another finite G-set, then the product B(X) × B(Y) → B(X × Y) is induced by the obvious product sending a G-set U over X and a G-set V over Y to their direct product U ×V over X × Y.

More generally, the functor RB is a Green functor for G over R. It is an initial object in the category Green_{R}(G).

2.2. Cohomology rings. Let K be a subgroup of G. The cup product in cohomology defines a ring structure on

H^{⊕}(K, R) = ⊕_{l=0}^{∞} H^{l}(K, R) ,

for which H^{⊕}(−, R) becomes a Green functor. The subfunctor H^{0}(−, R) is also a Green functor, usually denoted by F P_{R}. More generally, if A is a G-algebra over R, the functor F P_{A} is a Green functor for G over R.

3. Modules over a Green functor

Green functors can be viewed as generalized R-algebras : a Green functor for the trivial group over R is nothing but an R-algebra. Similarly, there are (at least) two equivalent definitions of the notion of module over a Green functor :

3.1. Using the poset of subgroups of G.

Definition 3.1.1 : Let A be a Green functor for G over R. A module M over A (or an A-module) is a Mackey functor for G over R, together with a structure of A(H)-module on M(H), for each subgroup H of G. The Mackey structure and the module structures have to be compatible in the following sense :

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

∀a ∈ A(K), ∀m ∈ M(K), r_{H}^{K}(am) = r_{H}^{K}(a)r_{H}^{K}(m) ,

∀a ∈ A(H), ∀a ∈ M(H), c_{x,H}(am) = c_{x,H}(a)c_{x,H}(m) .

• (Frobenius relations) If H ⊆ K are subgroups of G, then

∀a_{i}nA(K), ∀m ∈ M(H), a(t_{H}^{K}m) = t_{H}^{K}((r_{H}^{K}a)m) ,
∀a ∈ A(H), ∀m ∈ M(K), (t_{H}^{K}a)m = t_{H}^{K}(a(r_{H}^{K}m)) .

A morphism of A-modules Green functors f : M → N is a morphism of Mackey functors such that for each subgroup H of G, the map f_{H} : M(H) → N(H) is a map of A(H)-modules.

3.2. Using the category of G-sets.

Definition 3.2.1 : A module M over the Green functor A for G over R is a Mackey functor for G over R, endowed for any G-sets X and Y with bilinear maps

A(X) × M(Y) → M(X × Y)

denoted by (a, m) ↦ a × m which are bifunctorial, associative, and unitary, in the following sense:

• (Bifunctoriality) If f : X → X′ and g : Y → Y′ are morphisms of G-sets, then the squares

A(X) × M(Y) \underrightarrow{×} M(X × Y)
A_{∗}(f) × M_{∗}(g) ↓  ↓ M_{∗}(f × g)
A(X′) × M(Y′) \overrightarrow{×} M(X′ × Y′)

A(X) × M(Y) \underrightarrow{×} M(X × Y)
A^{∗}(f) × M^{∗}(g) ↑  ↑ M^{∗}(f × g)
A(X′) × M(Y′) \overrightarrow{×} M(X′ × Y′)

are commutative.

• (Associativity) If X, Y and Z are G-sets, then the square

A(X) × A(Y) × M(Z) \underrightarrow{Id_{A(X)} × (×)} A(X) × M(Y × Z)
(×) × Id_{A(Z)} ↓  ↓ ×
A(X × Y) × M(Z)  \overrightarrow{×}  M(X × Y × Z)

is commutative, up to identifications (X × Y) × Z ≃ X × Y × Z ≃ X × (Y × Z).

• (Unitarity) For any G-set X and for any m ∈ M(X)

A_{∗}(q_{X})(ε_{A} × m) = m

denoting by q_{X} the (bijective) projection from • × X to X.

A morphism of A-modules f : M → N is a morphism of Mackey functors such that for any G-sets X and Y, the square

A(X) × M(Y) \underrightarrow{×} M(X × Y)
Id × f_{Y} ↓  ↓ f_{X×Y}
A(X) × N(Y) \overrightarrow{×} N(X × Y)

is commutative. The composition of morphisms of A-modules is the composition of morphisms of Mackey functors. The category of A-modules is denoted by A-Mod.

3.3. Examples.

• Let V be an RG-module. Then the functor F P_{V} is a module over the Green functor F P_{R}. More generally, if l ∈ ℕ, the cohomology functor H^{l}(−, V) and the homology functor H_{l}(−, V) are F P_{R}-modules.

Conversely, it has been observed by Puig that the F P_{R}-modules can be characterized among the Mackey functor over R as the cohomological Mackey functors (recall that it means that t_{H}^{K}r_{H}^{K} is equal to the multiplication by the index |K : H|, for any subgroups H ⊆ K of G).

• Let M and N be Mackey functors for G over R. If X is a finite G-set, define

H(M, N)(X) = Hom_{Mack_{R}(G)}(M, N_{X}) .

This definition can be extended to give a natural Mackey functor structure on H(M, N). In the case M = N, there is similarly a natural Green functor structure on H(M, M). If A is any Green functor for G over R, it is equivalent to give M a structure of A-module, or to give a morphism of Green functors A → H(M, M) ([3] 2.1.2).

Since the Green functor RB is an initial object in the category Green_{R}(G), it follows that the category of Mackey functors for G over R is equivalent to the category of RB-modules.

• The previous example shows that the category Mack_{R}(G) admits an internal Hom construction. There is also an internal tensor product in Mack_{R}(G) : if M and N are Mackey functors for G over R, this construction gives another Mackey functor M ˆ⊗N for G over R. The value of M ˆ⊗N at the finite G-set X can be defined as

(M ˆ⊗N)(X) = M̂ ⊗_{μ_{R}(G)} N̂_{X} ,

where M̂ is the μ_{R}(G)-module associated to M, viewed as a right μ_{R}(G)- module using the anti-automorphism of the Mackey algebra.

This construction is functorial in M and N, and there is an adjunction

Hom_{Mack_{R}(G)}(M ˆ⊗N, P) ≅ Hom_{Mack_{R}(G)}(N, H(M, P)) .

• If A is a Green functor for G over R, the opposite Green functor A^{op} is the Mackey functor A, equipped with the opposite product ×^{op}, defined for finite G-sets X and Y by

∀a ∈ A(X), ∀b ∈ A(Y), a ×^{op} b = A_{∗} (\begin{matrix}y,x \\ ↓ \\ x,y \end{matrix}) (b × a) ,

where (\begin{matrix}y,x \\ ↓ \\ x,y \end{matrix}) is the map from Y × X to X × Y sending (y, x) to (x, y). The identity element of A^{op} is the identity element of A.

There is an obvious notion of right module over a Green functor, and the category of right modules over A is equivalent to the category of left modules over A^{op}.

If A and B are Green functors for G over R, there is also an obvious notion of (A, B)-bimodule, or A-module-B. Moreover the tensor product A ˆ⊗B^{op} has a natural structure of Green functor, with canonical Green functor homomorphisms A → A ˆ⊗B^{op} and B^{op} → A ˆ⊗B^{op}, and the category of A-modules-B is equivalent to the category of A ˆ⊗B^{op}-modules.

• Let A be a Green functor. Then A is an A-module-A. If M is an A-module, and if X is a finite G-set, there is a natural structure of A-module on the Dress construction M_{X}. This gives an endofunctor of the category A-Mod, which is self adjoint. Moreover, if M is an A-module, then

Hom_{A-Mod}(A_{X}, M) ≅ M(X) .

In particular the modules A_{X} are projective A-modules, and the module A_{Ω} is a progenerator of the category A-Mod.

4. Associated categories and algebras

4.1. The category associated to a Green functor. Let A be a Green functor for G over R. Let C_{A} denote the following category :

• The objects of C_{A} are the finite G-sets.

• If X and Y are finite G-sets, then

Hom_{C_{A}}(X, Y) = A(Y × X) .

• If X, Y, and Z are finite G-sets, then the composition of the morphisms f ∈ A(Y ×X) and g ∈ A(Z ×Y) in C_{A} is the element g ∘f of A(Z ×X) defined by

g ∘ f = A_{∗} (\begin{matrix}z,y,x \\ ↓ \\ z,x \end{matrix}) A^{∗} (\begin{matrix}z,y,x \\ ↓ \\ z,y,y,x \end{matrix}) (g × f) ,

where (\begin{matrix}z,y,x \\ ↓ \\ z,x \end{matrix}) is the map from Z × Y × X to Z × X sending (z, y, x)

to (z, x), and (\begin{matrix}z,y,x \\ ↓ \\ z,y,y,x \end{matrix}) is the map from Z × Y × X to Z × Y × Y × X

sending (z, y, x) to (z, y, y, x).

• The identity morphism of the finite G-set X is the element

A_{∗} (\begin{matrix}x \\ ↓ \\ x,x \end{matrix}) A^{∗} (\begin{matrix}x \\ ↓ \\ • \end{matrix}) (ε)

of A(X × X), where (\begin{matrix}x \\ ↓ \\ x,x \end{matrix}) is the diagonal inclusion from X to X × X,

and (\begin{matrix}x \\ ↓ \\ • \end{matrix}) is the unique map from X to the trivial G-set •.

Proposition 4.1.1 : [[3] 3.3.5] The category of A-modules is equivalent to the category of R-linear functors from C_{A} to R-Mod.

4.2. The algebra associated to a Green functor. Since A_{Ω} is a progenerator of the category A-Mod, it follows that A-Mod is equivalent to the category of modules over the algebra

μ(A) = End_{A-Mod}(A_{Ω}) ,

which is also isomorphic to the algebra End_{C_{A}}(Ω) = A(Ω × Ω). The algebra μ(A) can also be defined by generators and relations using the following result :

Proposition 4.2.1 : Let A be a Green functor for the group G. Then μ(A) is isomorphic to the R-algebra defined by the following generators and relations :

• The generators are: – The elements t_{K}^{H} and r_{K}^{H}, for K ⊆ H ⊆ G.

– The elements c_{x,H} for x ∈ G and H ⊆ G.

– The elements λ_{K,a} for K ⊆ G and a ∈ A(K).

• The relations are : – The relations of the Mackey algebra for r_{K}^{H}, t_{K}^{H}, and c_{x,H}, i.e.

t_{K}^{H}t_{L}^{K} = t_{L}^{H}, r_{L}^{K}r_{K}^{H} = r_{L}^{H} ∀L ⊆ K ⊆ H

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

t_{H}^{H} = r_{H}^{H} = c_{h,H} ∀h ∈ H

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

∑_{H} t_{H}^{H} = ∑_{H} r_{H}^{H} = 1

r_{K}^{H}t_{L}^{H} = ∑_{x∈K\H/L} t_{K∩^{x}L}^{K}c_{x,K^{x}∩L}r_{K^{x}∩L}^{L} ∀K ⊆ H ⊇ L

the other products of r_{H}^{K}, t_{H}^{K} and c_{g,H} being zero.

– The additional following relations :

λ_{K,a}+λ_{K,a′} = λ_{K,a+a′}, λ_{K,a}λ_{K,a′} = λ_{K,aa′} ∀a, a′ ∈ A(K), ∀K ⊆ G

λ_{K,zε_{K}} = zt_{K}^{K} ∀K ⊆ G, z ∈ R

r_{K}^{H}λ_{H,a} = λ_{K,r_{K}^{H}(a)}r_{K}^{H} ∀a ∈ A(H), ∀K ⊆ H ⊆ G

λ_{H,a}t_{K}^{H} = t_{K}^{H}λ_{K,r_{K}^{H}(a)} ∀a ∈ A(H), ∀K ⊆ H ⊆ G

t_{K}^{H}λ_{K,a}r_{K}^{H} = λ_{H,t_{K}^{H}(a)} ∀a ∈ A(K), ∀K ⊆ H ⊆ G

λ_{^{x}H,c_{x,H}(a)}c_{x,H} = c_{x,H}λ_{H,a} ∀x ∈ G, ∀a ∈ A(H), ∀H ⊆ G

5. Simple modules and simple Green functors

5.1. Simple modules. The classification and description of simple Mackey functors can be generalized to an arbitrary Green functor, in the following form ([3] Chapter 11) :

Notation 5.1.1 : Let A be a Green functor for G over R. If H is a subgroup of G, denote by \overline{A}(H) the Brauer quotient of A in H, defined by

\overline{A}(H) = A(H)/ ∑_{K⊂H} t_{K}^{H}A(K) ,

and denote by a ↦ \overline{a} the projection map from A(H) to \overline{A}(H). This map induces an R-algebra structure on \overline{A}(H), together with a natural action of the group \overline{N}_{G}(H). Denote by Â(H) the semi-direct product \overline{A}(H)⊗R\overline{N}_{G}(H).

Let V be a simple Â(H)-module. If X is a finite G-set, then R(X^{H}) is a \overline{N}_{G}(H)-module. Set

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

If f : X → Y is a morphism of finite G-sets, then for α ∈ S_{H,V} (X) and y ∈ Y^{H}, set

S_{H,V} _{∗}(f)(α)(y) = ∑_{{x∈X^{H} \atop f(x)=y}} α(x)

If g : Y → X is a morphism of finite G-sets, then for α ∈ S_{H,V} (X) and y ∈ Y^{H}, set

S_{H,V}^{∗} (g)(α)(y) = αg(y)

Finally if a ∈ A(X) and f ∈ S_{H,V} (Y), then define a morphism a × f from R((X × Y)^{H}) = R(X^{H} × Y^{H}) to V by

(a × f)(x, y) = (\overline{A^{∗}(m_{x})(a)} ⊗ 1).f(y)

where m_{x} is the morphism of G-sets from G/H to X defined by m_{x}(uH) = ux.

One can show that these definitions give an A-module structure on S_{H,V}. Moreover :

Proposition 5.1.2 : Let A be a Green functor for the group G.

1. If S is a simple A-module, and H is a minimal subgroup for S, then V = S(H) is a simple Â(H)-module, and S is isomorphic to S_{H,V}.

2. Conversely, if H is a subgroup of G, and V is a simple Â(H)-module, then S_{H,V} is a simple A-module, the group H is minimal for S_{H,V}, and moreover S_{H,V} (H) ≃ V.

3. Let H and K be subgroups of G. If V is a simple Â(H)-module, and if W is a simple Â(K)-module, then the modules S_{H,V} and S_{K,W} are isomorphic if and only if the pairs (H, V) and (K, W) are conjugate under G.

5.2. Simple Green functors. A simple Green functor is a Green functor A for G over R, such that A is simple as A-module-A (or A ˆ⊗A^{op}-module). This is equivalent to requiring that A has no non-trivial functorial two-sided ideal, in the sense of Thévenaz ([10]). The simple Green functors can be described using the following result (for details, see [3] 11.5) :

Proposition 5.2.1 : [Thévenaz [10] Theorem 12.11]

1. Let A be a simple Green functor for G. Then there exists a subgroup M of G, a normal subgroup H of M, and a simple algebra S on which M/H acts projectively, such that

A ≃ Ind_{M}^{G}Inf_{M/H}^{M}F P_{S} .

The triple (M, H, S) is unique up to conjugation by G (and up to isomorphism of M/H-algebras for S).

2. Conversely, if H \underline{◁} M are subgroups of G, if S is a simple algebra on which M/H acts projectively, then Ind_{M}^{G}Inf_{M/H}^{M}F P_{S} is a simple Green functor.

6. An example of related construction

6.1. Crossed G-monoids.

Definition 6.1.1 : A G-monoid is a monoid endowed with a left G-action by monoid automorphisms. A G-monoid which is a group is called a G-group.

A morphism of G-monoids is a G-equivariant monoid homomorphism.

Example 6.1.2 : Denote by G^{c} the set G on which the group G acts by conjugation. Then the multiplication map G^{c} × G^{c} → G^{c} endows G^{c} with a structure of G-group. More generally, if N is a normal subgroup of G, then G acts on N by conjugation, and N is a G-group for this action.

Definition 6.1.3 : A crossed G-monoid is a pair (Γ, φ), where Γ is a Gmonoid, and φ : Γ → G^{c} is a morphism of G-monoids. A morphism θ : (Γ, φ) → (Γ′, φ′) of crossed G-monoids is a morphism of G-monoids θ : Γ → Γ′ such that φ′ ∘ θ = φ.

6.2. Associated Green functors.

Proposition 6.2.1 : [[4],[1]]Let (Γ, φ) be a crossed G-monoid. If A is a Green functor for G over R, let A_{Γ} denote the Mackey functor obtained by the Dress construction from the G-set Γ. If X and Y are finite G-set, define a product map ×_{Γ} : A_{Γ}(X) ⊗_{R} A_{Γ}(Y) → A_{Γ}(X × Y) by

∀a ∈ A_{Γ}(X), ∀b ∈ A_{Γ}(Y), a ⊗ b ↦ a ×_{Γ} b = A_{∗} (\begin{matrix}x,γ,y,γ′ \\ ↓ \\ x,φ(γ)y,γγ′ \end{matrix}) (a × b) ,

where (\begin{matrix}x,γ,y,γ′ \\ ↓ \\ x,φ(γ)y,γγ′ \end{matrix}) is the map from X × Γ × Y × Γ to X × Y × Γ sending

(x, γ, y, γ′) to (x, φ(γ)y, γγ′). Let moreover ε_{A_{Γ}} denote the element A_{∗} (\begin{matrix}• \\ ↓ \\ 1_{Γ} \end{matrix}) (ε_{A})

of A(Γ) ≅ A_{Γ}(•), where (\begin{matrix}• \\ ↓ \\ 1_{Γ} \end{matrix}) is the map sending the unique element of • to the identity element of Γ.

Then A_{Γ} is a Green functor for G over R, and the correspondence A ↦ A_{Γ} is an endo-functor of the category Green_{R}(G).

In particular, the evaluation A_{Γ}(•) ≅ A(Γ) of A_{Γ} at the trivial G-set is an R-algebra. One can express the product formula for this algebra in the

decomposition A(Γ) ≅ ( ⊕_{γ∈Γ} A(G_{γ}))^{G} :

Proposition 6.2.2 : Let a and b be elements of A(Γ). Then for γ ∈ Γ, the γ component of a ×_{Γ} b is given by

(a ×_{Γ} b)_{γ} = ∑_{{(α,β)∈G_{γ}\Γ×Γ \atop αβ=γ}} t_{G_{α,β}}^{G_{γ}}(r_{G_{α,β}}^{G_{α}}a_{α}.r_{G_{α,β}}^{G_{β}}b_{β}) .

6.3. Examples.

• Let B denote the Burnside Green functor, and let Γ = G^{c}. Then the ring B(Γ) = B(G^{c}) is called the crossed Burnside ring. It is the Grothendieck ring of the monoidal category of crossed G-sets, i.e. G-sets over G^{c}. This ring has been studied by Yoshida (see also [2])

• Let A denote the cohomology Green functor H^{⊕}(−, R), and let Γ = G^{c}. Then one can show that the algebra A(Γ) is isomorphic to the Hochschild cohomology algebra of the group algebra RG. In this case, the above product formula has been conjectured by Cibils ([5]) and Cibils and Solotar ([6]), and proved by Siegel and Witherspoon ([8]).

• (Cibils) Let A denote the Grothendieck ring of the category of finitely generated RG-modules, for relations given by direct sum decompositions. If Γ = G^{c}, then the ring A(Γ) is isomorphic to the Grothendieck ring of Hopf bimodules for the Hopf algebra RG.

References

[1] Serge Bouc. Hochschild constructions for Green functors. Preprint (2001), to appear in “Communications in Algebra”.

[2] Serge Bouc. The p-blocks of the Mackey algebra. Preprint (2001), to appear in “Algebras and Representation Theory”.

[3] Serge Bouc. Green-functors and G-sets, volume 1671 of Lecture Notes in Mathematics. Springer, October 1997.

[4] Serge Bouc. Green functors, crossed G-monoids, and Hochschild constructions. Algebra Montpellier Announcements, 01, 2002. http://www.emis.de/journals/AMA/AMA01-2002/index01-2002.html.

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