Bisets and associated functors

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

1. Functors between categories of G-sets

In view of Dress’s definition of Mackey functors, it is natural to consider the following problem : if G and H are finite groups, find all functors from H-set to G-set which preserve disjoint unions and cartesian squares. If F is such a functor and M is a Mackey functor for G over R, then the composition M ∘ F will be a Mackey functor for H over R.

1.1. Bisets and related products.

Definition 1.1.1 : If G and H are groups, a (G, H)-biset, or G-set-H, is a set U with a left G action and a right H action which commute, i.e. are such that

∀g ∈ G, ∀u ∈ U, ∀h ∈ H, (gu)h = g(uh) .

If G, H, and K are groups, if U is a G-set-H, and V is an H-set-K, the product U ×_{H} V is the quotient of the direct product U ×V by the right action of H given by

∀(u, v) ∈ U × V, ∀h ∈ H, (u, v)h = (uh, h^{−1}v) .

It is a G-set-K for the action induced by

∀g ∈ G, ∀k ∈ K, ∀(u, v) ∈ U × V, g(u, v)k = (gu, vk) .

The product U ∘V is the set of pairs (u, v) in ×V such that whenever uh = u for some h ∈ H, there exists k ∈ K with hv = vk. The set U ∘V is invariant under the right action of H on U × V, and the set of H-orbits on U ∘ V is denoted by U ∘_{H} V. It is a sub-G-set-K of U ×_{H} V.

Proposition 1.1.2 : Let G, H, K and L be groups. Let U be a (G, H)-biset, let V be an (H, K)-biset, and let W be a (K, L)-biset. Then the canonical bijection

(U × V) × W → U × (V × W)

induces isomorphisms of (G, L)-bisets

(U ×_{H} V) ×_{K} W ≅ U ×_{H} (V ×_{K} W)
(U ∘_{H} V) ∘_{K} W ≅ U ∘_{H} (V ∘_{K} W) .

Remark 1.1.3 : Let G and H be finite groups, and let U be a finite G-set-H. If X is a finite H-set, consider X as an H-set-{1}. Then U ∘_{H} X is a G-set-{1}, i.e. a G-set. Note that in this case

U ∘_{H} X = {(u, x) ∈ U × X | ∀h ∈ H, uh = u ⇒ hx = x}/H ⊆ U ×_{H} X ,

If f : X → Y is a morphism of G-sets, denote by U ∘_{H} f the map from U ∘_{H} X to U ∘_{H} Y defined by (U ∘_{H} f)(u, x) = (u, f(x)). This shows that any finite (G, H)-biset U gives rise to a functor U ∘_{H} − from H-set to G-set.

1.2. Classification of functors.

Proposition 1.2.1 : [[1] Théorème 1] Let G and H be finite groups.

1. If F : H-set → G-set is a functor which commutes with disjoint unions, and preserves cartesian squares, then there exists a finite (G, H)- biset U such that F is isomorphic to the functor U ∘_{H} −. Such a biset U is unique up to isomorphism.

2. Conversely, if U is a finite (G, H)-biset, then the functor U ∘_{H} − commutes with disjoint unions and preserves cartesian squares.

3. If K is another finite group, and if V is a finite (H, K)-biset, then the composition of the functors V ∘_{K} − and U ∘_{H} − is isomorphic to the functor (U ∘_{H} V) ∘_{K} −.

1.3. Examples.

• Let H be a subgroup of G, and let U be the set G, viewed as an (H, G)- biset by left and right multiplication. Then the functor U ∘_{G} − is isomorphic to the restriction functor Res_{H}^{G} : G-set → H-set.

• Let H be a subgroup of G, and let U be the set G, viewed as a (G, H)-biset by left and right multiplication. Then the functor U ∘_{H} − is isomorphic to the induction functor Ind_{H}^{G} : H-set → G-set.

• Let N be a normal subgroup of G, and set H = G/N. Let U be the set H, viewed as a (G, H)-biset, by right multiplication, and by projection G → H followed by left multiplication. Then the functor U ∘_{H} − is isomorphic to the inflation functor Inf_{H}^{G} : H-set → G-set.

• Let N be a normal subgroup of G, and set H = G/N. Let U be the set H, viewed as a (H, G)-biset, by left multiplication, and by projection G → H followed by right multiplication. Then the functor U ∘_{G} − is isomorphic to the fixed points by N functor Fix(N) : G-set → H-set.

• Let f : G → H be a group isomorphism, and let U be the set H, viewed as an (H, G)-biset by left multiplication, and by f followed by right multiplication. Then the functor U ∘_{G} − is the transport by f functor Iso_{G}^{H}(f) : G-set → H-set.

• One can show that any functor F : G-set → H-set which commutes with disjoint unions and preserves pull-back squares is isomorphic to a disjoint union of a finite number of functors which are composed of the five previous cases, i.e. of the form

Ind_{K}^{G} ∘ Inf_{K/M}^{K} ∘ Iso_{L/N}^{K/M}(f) ∘ Fix(N) ∘ Res_{L}^{H} ,

for some subgroups M \underline{◁} K ⊆ G and N \underline{◁} L ⊆ H, and some group isomorphism f : L/N → K/M.

2. Composition with bisets

2.1. Functors between categories of Mackey functors.

Proposition 2.1.1 : Let G and H be finite groups, and let U be a finite (G, H)-biset. If M is a Mackey functor for G over R, viewed as a bivariant functor G-set → R-Mod, denote by M ∘ U− the bivariant functor H-set → R-Mod obtained by precomposition of M with the functor U∘_{H} : H-set → G-set.

Then M ∘ U is a Mackey functor for H over R, and the correspondence M ↦ M ∘ U is an R-linear exact functor from Mack_{R}(G) to Mack_{R}(H).

This composition of Mackey functors with bisets can be extended to Green functors :

Proposition 2.1.2 : Let G and H be finite groups, and let U be a finite (G, H)-biset. If A is a Green functor for G over R, and if X and Y are finite H-sets, define a product ×^{U} : (A ∘ U)(X) × (A ∘ U)(Y) → (A ∘ U)(X × Y) by

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

where (\begin{matrix}u,x,y \\ ↓ \\ u,x,u,y \end{matrix}) is the map from U ∘_{H} (X × Y) to (U ∘_{H} X) × (U ∘_{H} Y)

sending (u, (x, y)) to ((u, x), (u, y). Let moreover ε_{A∘U} denote the element A^{∗}(p_{U/H})(ε_{A}) of A(U/H) = (A ∘ U)(•), where p_{U/H} is the unique map from U/H to •.

Then A ∘ U is a Green functor for H over R for the product ×^{U} and the identity element ε_{A∘U}.

Similarly, if M is an module over the Green functor A, then M ∘ U is a module over the Green functor A ∘ U :

Proposition 2.1.3 : Let G and H be finite groups, and let U be a finite (G, H)-b iset. Let A be a Green functor for G over R, and let M be an A-module. If X and Y are finite H-sets, define a product ×^{U} : (A ∘ U)(X) × (M ∘ U)(Y) → (M ∘ U)(X × Y) by

∀a ∈ (A ∘ U)(X), ∀m ∈ (M ∘ U)(Y), a ×^{U} b = M^{∗} (\begin{matrix}u,x,y \\ ↓ \\ u,x,u,y \end{matrix}) (a × m) .

Then M ∘ U is a module over the Green functor A ∘ U, for the product ×^{U}, and the correspondence M ↦ M ∘U is an R-linear exact functor from A-Mod to A ∘ U-Mod.

2.2. Examples.

• Let H be a subgroup of G, and let U be the set G, viewed as an (H, G)- biset by left and right multiplication. Then the functor M ↦ M ∘ U is the induction functor for Mackey functors. If M is a Mackey functor for H over R, this induced Mackey functor Ind_{H}^{G}M can be computed by

(Ind_{H}^{G}M)(X) = M(Res_{H}^{G}X)

for a finite G-set X.

• Let H be a subgroup of G, and let U be the set G, viewed as a (G, H)- biset by left and right multiplication. Then the functor M ↦ M ∘ U is the restriction functor for Mackey functors. If M is a Mackey functor for G over R, this restricted Mackey functor Res_{H}^{G}M can be computed by

(Res_{H}^{G}M)(X) = M(Ind_{H}^{G}X)

for a finite H-set X.

• Let N be a normal subgroup of G, and set H = G/N. Let U be the set H, viewed as a (G, H)-biset, by right multiplication, and by projection G → H followed by left multiplication. Then the functor M ↦ M ∘U is the deflation functor for Mackey functors. If M is a Mackey functor for G over R, this deflated Mackey functor Def_{H}^{G}M can be computed by

(Def_{H}^{G}M)(X) = M(Inf_{H}^{G}X)

for a finite H-set X.

• Let N be a normal subgroup of G, and set H = G/N. Let U be the set H, viewed as a (H, G)-biset, by left multiplication, and by projection G → H followed by right multiplication. Then the functor M ↦ M ∘ U is the inflation functor for Mackey functors. If M is a Mackey functor for H over R, this inflated Mackey functor Inf_{H}^{G}M can be computed by

(Inf_{H}^{G}M)(X) = M(X^{N})

for a finite G-set X.

3. Adjoint constructions.

3.1. Left and right adjoints.

Proposition 3.1.1 : [[2] Chapters 9 and 10]

1. Let G and H be finite groups, and let U be a finite (G, H)-biset. The

functor

M ↦ M ∘ U : Mack_{R}(G) → Mack_{R}(H)

admits a left adjoint L_{U} and a right adjoint R_{U}.

2. If G, H, and K are finite groups, if U is a finite (G, H)-biset and V is a finite (H, K)-biset, there are isomorphisms of functors

L_{U} ∘ L_{V} ≅ L_{U∘_{H}V} R_{U} ∘ R_{V} ≅ R_{U∘_{H}V} .

3. If A is a Green functor for H over R, then L_{U}(A) has a natural structure of Green functor for G over R. The correspondence A ↦ L_{U}(A) is a functor from Green_{R}(H) to Green_{R}(G).

4. If M is an A-module, then L_{U}(M) has a natural structure of L_{U}(A)- module, and the correspondence M ↦ L_{U}(M) is a functor from A-Mod to L_{U}(A)-Mod.

Remark 3.1.2 : The explicit description of the functors L_{U}(M) is rather complicated in the general case (see [2] Chapter 9). However, a partial description can be obtained by the following argument, which also gives a proof for the existence of the functors L_{U} and R_{U} : recall that the Mackey algebra μ_{R}(G) can be identified with RB(Ω_{G}×Ω_{G}), where Ω_{G} = ⊔_{K⊆G} G/K. If M is a

Mackey functor for G over R, then M(Ω_{G}) is a μ_{R}(G)-module. Set similarly Ω_{H} = ⊔_{L⊆H} H/L, and identify μ_{R}(H) with RB(Ω_{H} × Ω_{H}).

Now consider

RB_{U} = RB(Ω_{G} × (U ∘_{H} Ω_{H})) .

This is a (μ_{R}(G), μ_{R}(H))-bimodule, for the actions extending linearly the

following products : suppose that (X, (a, b)) is a G-set over Ω_{G} × Ω_{G}, that

(Y, (c, d)) is an H-set over Ω_{H} × Ω_{H}, and that (Z, (e, f)) is a G-set over Ω_{G} × (U ∘_{H} Ω_{H}). Build the following diagram

\xymatrix{ &  &  & E \ar[dl]_{k} \ar[dr]^{l} \\  &  & C \ar[dl]_{g} \ar[dr]^{h} &  & D \ar[dl]_{i} \ar[dr]^{j} \\  & X \ar[dl]_{a} \ar[dr]^{b} &  & Z \ar[dl]_{e} \ar[dr]^{f} &  & U ∘_{H} Y \ar[dl]_{U∘_{H}c} \ar[dr]^{U∘_{H}d} \\ Ω_{G} &  & Ω_{G} &  & U ∘_{H} Ω_{H} &  & U ∘_{H} Ω_{H}}

where all the squares are pull-back squares. Then the left and right actions on RB_{U} are defined by

(X, (a, b)).(Z, (e, f)).(Y, (c, d)) = (E, (agk, (U ∘_{H} d)jl)) .

It is easy to this from this definition that there is an isomorphism of left μ_{R}(G)-modules

RB_{U} ≅ RB_{U∘_{H}Ω_{H}}(Ω_{G}) .

In particular RB_{U} is projective and finitely generated as μ_{R}(G)-module. Moreover, one can show that if M is a Mackey functor for G over R, then the natural isomorphism of R-modules

(M ∘ U)(Ω_{H}) = M(U ∘_{H} Ω_{H}) ≅ Hom_{μ_{R}(G)}(RB_{U}, M(Ω_{G}))

is an isomorphism or μ_{R}(H)-modules. Thus if N is a Mackey functor for H over R, this gives by standard arguments

L_{U}(N)(Ω_{G}) ≅ RB_{U} ⊗_{μ_{R}(H)} N(Ω_{H}) .

But since RB_{U} is projective and finitely generated as a left μ_{R}(G)-module, one has also that

(M ∘ U)(Ω_{H}) ≅ RB_{U}^{♯} ⊗_{μ_{R}(G)} M(Ω_{G}) ,

where

RB_{U}^{♯} = Hom_{μ_{R}(G)}(RB_{U}, μ_{R}(G))
= Hom_{μ_{R}(G)}(RB_{U}, RB_{Ω_{G}}(Ω_{G}))
= (RB_{Ω_{G}} ∘ U)(Ω_{H}) ≅ RB((U ∘_{H} Ω_{H}) × Ω_{G})
≅ RB_{U} .

In other words RB_{U}^{♯} can be identified to RB_{U}, viewed as a (μ_{R}(H), μ_{R}(G))- bimodule for the action defined by

∀a ∈ μ_{R}(H), ∀b ∈ μ_{R}(G), ∀c ∈ RB_{U}, a.c.b = σ_{G}(b)cσ_{H}(a) ,

where σ_{G} (resp. σ_{H}) is the anti-automorphism of μ_{R}(G) (resp.μ_{R}(H)). Now it follows that if N is a Mackey functor for H over R,

R_{U}(N)(Ω_{G}) ≅ Hom_{μ_{R}(H)}(RB_{U}^{♯}, N(Ω_{H})) .

3.2. Examples.

• Let H be a subgroup of G. Then the functors

Mack_{R}(G) ∋ M ↦ Res_{H}^{G}M ∈ Mack_{R}(H)

Mack_{R}(H) ∋ N ↦ Ind_{H}^{G}N ∈ Mack_{R}(G)

are mutual left and right adjoint functors ([4] Prop. 4.2).

• Let N \underline{◁} G, and H = G/N. Then then inflation functor Inf_{H}^{G} : Mack_{R}(H) → Mack_{R}(G) has a left adjoint M ↦ M^{N}, and a right adjoint M ↦ M_{N}, which can be computed as follows : if K/N is a subgroup of H = G/N, then

M^{N}(K/N) = M(K)/ ∑_{N⊈L⊆K} t_{L}^{K}M(L) M_{N}(K/N) = ⋂_{N⊈L⊆K} Ker r_{L}^{K} .

In the same situation, the left and right adjoints to the deflation functor Def_{H}^{G} are described in Section 9.9.3 of [2].

4. Biset functors

4.1. Bisets as morphisms. The previous sections suggest the following idea : if G and H are finite group, then a finite (G, H)-biset is a kind of generalized morphism from H to G. At least it gives a way in many situations to transport to the group G known structures associated to H. This idea can be formalized as follows :

Definition 4.1.1 : Let C_{R} denote the following category :

• The objects of C_{R} are finite groups.

• If H and G are finite groups, then

Hom_{C_{R}}(H, G) = RB(G × H^{op})

is the Burnside group of (G, H)-bisets, with coefficients in R.

• Composition of morphisms is obtained by linearity from the product (U, V) ↦ U ×_{H} V.

• The identity morphism of the group G is the biset G itself, for left and right multiplication.

Let F_{R} denote the category of R-linear functors from C_{R} to R-Mod.

Remark 4.1.2 : It is often interesting to consider proper subcategories of C_{R} (consisting for example only of finite p-groups, for some prime p), which need not be full subcategories (it may be appropriate to consider only those bisets which are free on the right, or on the left, or both, in building the Hom-sets).

In particular if one allows only left and right free bisets as morphisms, the objects of F_{R} are called global Mackey functors : this is because left and right free bisets can be viewed as disjoint unions of bisets composed of restrictions, group isomorphisms, and inductions.

It is also interesting to replace the composition product ×_{H} by the composition product ∘_{H}. This gives examples of categories with the same objects, the same morphisms, but with different composition product. Such categories may have completely different properties.

4.2. Examples.

• Let R = ℤ. Let R_{ℚ}(G) denote the Grothendieck group of the category of (finite dimensional) ℚG-modules. If H is another finite group, if L is a finite dimensional ℚH-module, and if U is a finite (G, H)-biset, then ℚU ⊗_{ℚH} L is a finite dimensional ℚG-module.

This construction gives a linear map I_{U} : R_{ℚ}(H) → R_{ℚ}(G). If U′ is another finite (G, H)-biset, then clearly I_{U⊔U′} = I_{U} + I_{U′}. Moreover, if K is another finite group, and if V is a finite (H, K)-biset, it is clear that I_{U} ∘ I_{V} = I_{U×_{H}V}. Hence the correspondence G ↦ R_{ℚ}(G) can be viewed as an object of F_{ℤ}.

• Let R = ℤ, and let k be a field of characteristic p > 0. Let R_{k}(G) be the Grothendieck group of the category of the category of (finite dimensional) kG-modules, for relations given by short exact sequences. The previous example cannot be extended to this situation, because the functor L ↦ kU ⊗_{kH} L does not preserve short exact sequences in general. It does indeed if the module kU is flat as a right kH-module, which happens for example if U is free on the right. This shows the interest of restricting the set of morphisms in the category C_{R}.

• Let p be a prime number, and let k = 𝔽_{p} be the field with p elements. If G is a finite p-group, let D_{k}(G) denote the Dade group of endo-permutation kG-modules. One can show (see [3]) that the correspondence G ↦ D_{k}(G) is a functor on the full subcategory of C_{ℤ} consisting of finite p-groups. This shows the interest of restricting the class of objects in the category C_{R}.

• Let F (G) = K_{0}(μ_{R}(G)) be the Grothendieck group of finitely generated μ_{R}(G)-modules. If H is a finite group and U is a finite (G, H)-biset, then the functor L_{U} is left adjoint to an exact functor. Hence it maps projective μ_{R}(H)-modules to projective μ_{R}(G)-modules. This defines a linear map F (U) : F (H) → F (G). If K is another finite group and if V is a finite (H, K)-biset, it is clear that F (U) ∘ F (V) = F (U ∘_{H} V). In other words, we get a functor on the category C_{ℤ}, equipped with the product ∘_{H} instead of ×_{H}.

4.3. Simple biset functors. The category F_{R} is an abelian category. One can try to classify and describe its simple objects.

Lemma 4.3.1 : There is a direct sum decomposition

End_{C_{R}}(G) = A_{G} ⊕ I_{G}

where I_{G} is a two sided ideal of End_{C_{R}}(G), and A_{G} is a subalgebra isomorphic to the algebra over R of the group Out(G) of outer automorphisms of G.

In particular, any ROut(G)-module can be viewed as a module for the algebra End_{C_{R}}(G).

Notation 4.3.2 : Let H be an object of C_{R}, and let V be an ROut(H)- module. If G is an object of C_{R}, set

L_{H,V} (G) = Hom_{C_{R}}(H, G) ⊗_{End_{C_{R}}(H)} V ,

where the right End_{C_{R}}(H)-module structure on Hom_{C_{R}}(H, G) is given by composition of morphisms in C_{R}.

If f : G → G′ is a morphism in C_{R}, define a map L_{H,V} (f) : L_{H,V} (G) → L_{H,V} (G′) by composition with f on the left.

Clearly L_{H,V} is an R-linear functor from C_{R} to R-Mod, i.e. an object of F_{R}.

Proposition 4.3.3 : 1. Let H be an object of C_{R}, and let V be a simple ROut(H)-module. Then L_{H,V} admits a unique maximal proper subfunctor J_{H,V}. The quotient S_{H,V} = L_{H,V} /J_{H,V} is a simple object of F_{R}, and S_{H,V} (K) = 0 for any object K of C_{R} with |K| < |H|.

2. If S is a simple object of F_{R}, let H be an object of C_{R} of minimal order such that S(H) ≠ 0. Then S(H) is a simple ROut(H)-module, and S ≅ S_{H,V}.

3. If H (resp. K) is an object of C_{R}, and if V (resp. W) is a simple ROut(H)-module (resp. a simple ROut(K)-module), then the functors S_{H,V} and S_{K,W} are isomorphic if and only if there is a group isomorphism φ : H → K such that φ(V) = W.

Remark 4.3.4 : This proposition gives a nice classification of the simple objects in F_{R}. Unfortunately, this does not give any easy way to compute the evaluations S_{H,V} (G) of a simple functor at the finite group G. For example, suppose that R = k is a field, that C_{R} consists of all finite groups, that all finite bisets are allowed as morphisms, and that the composition product is ×_{H}.

Let H be a finite group, and let V = k be the trivial kOut(H)-module. If G is a finite group, let B_{H}(G) denote the k-vector space with basis the set of conjugacy classes of pairs (K, L) of subgroups of G, with L \underline{◁} K and K/L ≅ H. Define a bilinear form on B_{H}(G), with values in k, by

⟨(K, L) | (K′, L′)⟩ = |{x ∈ K\G/K′ | K.^{x}L′ = L.^{x}K′ K ∩^{x}L′ = L∩^{x}K′}| .

Then one can show that S_{H,k}(G) ≃ B_{H}(G)/Rad⟨ | ⟩. It is far from obvious however even to compute in general the k-dimension of this space.

References

[1] Serge Bouc. Construction de foncteurs entre catégories de G-ensembles. J. of Algebra, 183(0239):737–825, 1996.

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

[3] Serge Bouc and Jacques Thévenaz. The group of endo-permutation modules. Invent. Math., 139:275–349, 2000.

[4] 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.