Bisets as categories,

and tensor product of induced bimodules

Serge Bouc

Abstract : Bisets can be considered as categories. This note uses this point of view to give a simple proof of a Mackey-like formula expressing the tensor product of two induced bimodules.

AMS subject classification (2000) : 16D20, 20C20. Keywords : Biset, induced, bimodule, tensor product.

1. Introduction

Let R be a commutative ring, let G and H be finite groups, let X be a subgroup of H × G, and M be an RX-module. If m ∈ M and (h, g) ∈ X, set h · m · g^{−1} = (h, g) · m : this is a slight extension of the usual correspondence between R(H × G)-modules and (RH, RG)-bimodules.

The object of this note is to give a simple proof of the following result :

1.1. Theorem : Let R be a commutative ring, let G, H, and K be finite groups, let X be a subgroup of H × G and Y be a subgroup of K × H. Let M be an RX-module, and N be an RY -module. Then there is an isomorphism of (RK, RG)-bimodules

(Ind_{Y}^{K×H}N)⊗_{RH} (Ind_{X}^{H×G}M) ≅ ⊕_{t∈[p_{2}(Y)\H/p_{1}(X)]} Ind_{Y ∗^{(t,1)}X}^{K×G}(N ⊗_{k_{2}(Y)∩^{t}k_{1}(X)}^{(t,1)}M) ,

where the notation is as follows (cf. [1]) :

p_{1}(X) = {h ∈ H | ∃g ∈ G, (h, g) ∈ X}, k_{1}(X) = {h ∈ H | (h, 1) ∈ X}

p_{2}(Y) = {h ∈ H | ∃k ∈ K, (k, h) ∈ Y }, k_{2}(Y) = {h ∈ H | (1, h) ∈ Y }

Y ∗^{(t,1)}X = {(k, g) ∈ K × G | ∃h ∈ H, (k, h) ∈ Y, (h^{t}, g) ∈ X} .

The action of (k, g) ∈ Y ∗^{(t,1)}X on N ⊗_{k_{2}(Y)∩^{t}k_{1}(X)} ^{(t,1)}M is given by

k · (n ⊗ m) · g^{−1} = (k · n · h^{−1}) ⊗ (h^{t} · m · g^{−1}) ,

if h ∈ H is chosen such that (k, h) ∈ Y and (h^{t}, g) ∈ X.

2. Functors over bisets

Recall that when G and H are groups, an (H, G)-biset U is a set equipped with a left action of H and a right action of G which commute, i.e. such that (hu)g = h(ug) for any h ∈ H, u ∈ U, and g ∈ G.

2.1. Notation : Let G and H be groups. When U is an (H, G)-biset, let ⟨U⟩ denote the following category :

• The objects of ⟨U⟩ are the elements of U.

• If u, v ∈ U, then

Hom_{⟨U⟩}(u, v) = {(h, g) ∈ H × G | hu = vg} .

• If u, v, w ∈ U, the composition of the morphisms (h, g) : u → v and (h′, g′) : v → w is the morphism (h′h, g′g) : u → w.

• If u ∈ U, the identity morphism Id_{u} : u → u is the pair (1, 1) ∈ G × G.

Note that the category ⟨U⟩ is a groupoid (any morphism is an isomorphism), and that for any u ∈ U, the group

A(u) = Hom_{⟨U⟩}(u, u) = {(h, g) ∈ H × G | hu = ug}

is a subgroup of H × G.

A functor M from ⟨U⟩ to a category C consists of a collection of objects M(u) of C, for u ∈ U, together with morphisms

M(h, g) : M(u) → M(hug^{−1})

in the category C, for (h, g) ∈ H ×G, fulfilling the usual functorial conditions. In particular, for each u ∈ U, there is a group homomorphism A(u) → Aut_{C}M(u).

Functors from ⟨U⟩ to C are the objects of a category Fun(⟨U⟩, C), in which the morphisms are natural transformation of functors.

2.2. Notation : When C is a subcategory of the category Sets of sets, and M is a functor ⟨U⟩ → C, the image of m ∈ M(u) by the map M(h, g) : M(u) → M(hug^{−1}), for (h, g) ∈ H × G, will be denoted by hmg^{−1}.

In this case, a functor M : ⟨U⟩ → C is a collection of objects M(u) of C, for u ∈ U, together with morphisms m ↦ hmg^{−1} : M(u) → M(hug^{−1}) in C, for (h, g) ∈ H × G, such that h′(hmg^{−1})g^{′−1} = (h′h)m(g′g)^{−1} and 1m1 = m, for any (h, g), (h′, g′) in H × G, any u ∈ U, and any m ∈ M(u).

2.3. Example : Suppose that C = Sets. Then the disjoint union ⨆ M = ⨆_{u∈U} M(u) becomes an (H, G)-biset, and the map ⨆ M → U sending elements

of M(u) to u, for u ∈ U, is a map of (H, G)-bisets. Conversely, if π : S → U is a map of (H, G)-bisets, then the assignment u ↦ π^{−1}(u) is a functor from ⟨U⟩ to Sets.

In other words, a functor ⟨U⟩ → Sets is just an (H, G)-biset over U. More precisely, the category Fun(⟨U⟩, Sets) of such functors is equivalent to the category of (H, G)-bisets over U.

2.4. Example : Let R be a commutative ring. In the remainder of this note, the category C will be the category R-Mod of (left) R-modules. If M is functor from ⟨U⟩ to R-Mod, then for each u ∈ U, the R-module M(u) has a natural structure of RA(u)-module.

Conversely, let [H\U/G] be a set of representatives of (H, G)-orbits on U. Equivalently [H\U/G] is a set of representatives of isomorphism classes in the category ⟨U⟩. Since ⟨U⟩ is a groupoid, it is equivalent to its full subcategory [H\U/G]. In particular, this yields an equivalence of categories

(2.5)  Fun(⟨U⟩, R-Mod) ≅ ∏_{u∈[H\U/G]} RA(u)-Mod .

2.6. Remark : In the situation of Example 2.4, the direct sum

Σ(M) = ⊕_{u∈U} M(u)

has a natural structure of (RH, RG)-bimodule, i.e. using the usual group isomorphism (h, g) ↦ (h, g^{−1}) from H × G^{op} to H × G, of left R(H × G)- module.

Moreover, is is easy to see that there is an isomorphism of (RH, RG)- bimodules

Σ(M) ≅ ⊕_{u∈[H\U/G]} Ind_{A(u)}^{H×G}M(u) .

3. Product of bisets, and product of functors

Let G, H and K be groups. If U is an (H, G)-biset and V is a (K, H)-biset, recall that the product (or composition) of V and U is the set

V ×_{H} U = (V × U)/H ,

where the right action of H on (V × U) is defined by (v, u) · h = (vh, h^{−1}u), for v ∈ V, u ∈ U, and h ∈ H. The set V ×_{H} U is a (K, G)-biset for the following action

∀z ∈ K, ∀x ∈ G, ∀v ∈ V, ∀u ∈ U, z · (v,_{H} u) · x = (zv,_{H} ux) ,

where (v,_{H} u) denotes the H-orbit of (v, u).

3.1. Definition : Let G, H, and K be finite groups. Let U be a finite (H, G)-biset, and V be a finite (K, H)-biset. If M is a functor ⟨U⟩ → R-Mod and N is a functor ⟨V ⟩ → R-Mod, the tensor product N ⊗_{H} M is the functor ⟨V ×_{H} U⟩ → R-Mod defined by

(N ⊗_{H} M)(v,_{H} u) = ( ⊕_{h∈H} N(vh) ⊗_{R} M(h^{−1}u))/I_{v,u} ,

where I_{v,u} is the R-submodule generated by the elements of the form

[ny ⊗ y^{−1}m]_{hy} − [n ⊗ m]_{h} ,

where y ∈ H, and where [n⊗m]_{h} denotes the element n⊗m of the component indexed by h ∈ H in the direct sum, for n ∈ N(vh), and m ∈ M(h^{−1}u).

If (k, g) ∈ K × G, then by definition

k [n ⊗ m]_{h} g = [kn ⊗ mg]_{h} .

3.2. Remark : It follows from this definition that

(N ⊗_{H} M)((v,_{H} u)) ≅ N(v) ⊗_{RH_{v,u}} M(u) ,

where H_{v,u} is the set of elements h ∈ H such that vh = v and hu = u.

3.3. Lemma : There is an isomorphism of (RK, RG)-bimodules

Σ(N) ⊗_{RH} Σ(M) ≅ Σ(N ⊗_{H} M) ,

sending (from right to left) the element [n ⊗ m]_{h} to n ⊗_{RH} m.

Proof : To be more precise, the map α from

Σ(N ⊗_{H} M) = ⊕_{(v,_{H} u)∈V ×_{H}U} ( ⊕_{h∈H} N(vh) ⊗_{R} M(h^{−1}u))/I_{v,u}

sending the element [n ⊗ m]_{h} in the component indexed by (v,_{H} u) to the element n ⊗ m of the tensor product

Σ(N) ⊗_{RH} Σ(M) = ( ⊕_{v∈V} N(v)) ⊗_{RH} ( ⊕_{u∈U} M(u))

is well defined. To show that it is an isomorphism, define a map

β : Σ(N) ⊗_{RH} Σ(M) → Σ(N ⊗_{H} M)

in the following way : choose a set S of representatives of the classes (v,_{H} u). Now map the element n ⊗_{RH} m ∈ N(v) ⊗ M(u) ⊆ Σ(N) ⊗_{RH} Σ(M) to [n ⊗ m]_{h}, where h ∈ H is chosen such that (vh^{−1}, hu) ∈ S. Again, it is easy to see that this map is well defined, and that the maps α and β are mutual inverse isomorphisms of (RK, RG)-bimodules.

3.4. Corollary : Let G, H, and K be finite groups. Let X be a subgroup of H × G and Y be a subgroup of K × H. Let M be an RX-module, and N be an RY -module. Then there is an isomorphism of (RK, RG)-bimodules

(Ind_{Y}^{K×H}N)⊗_{RH} (Ind_{X}^{H×G}M) ≅ ⊕_{t∈[p_{2}(Y)\H/p_{1}(X)]} Ind_{Y ∗^{(t,1)}X}^{K×G}(N ⊗_{k_{2}(Y)∩^{t}k_{1}(X)}^{(t,1)}M) .

Proof : Set U = (H × G)/X. Then U is an (H, G)-biset by h · (t, s)X · g = (ht, g^{−1}s)X, and this biset is transitive. If u is the point X of U, then A(u) = X, and the equivalence of categories 2.5 reads

Fun(⟨U⟩, R-Mod) ≅ RX-Mod .

More precisely, for an RX-module M, this equivalence yields a functor M̃ : ⟨U⟩ → R-Mod in the following way : for any (h, g) ∈ H × G, set

M̃((h, g)X) = M .

Next, fix a set S of representatives of elements of U, i.e. X-cosets in H × G. For (t, s) ∈ S, and (h, g) ∈ H × G, define a map

M̃(h, g) : M̃((t, s)X) = M → M̃((ht, gs)X) = M

by M̃(h, g)(m) = (y, x)m, where (y, x) is the unique element of X such that (ht, gs)(y, x)^{−1} ∈ S.

Then it is easy to check that M̃ is indeed a functor, and that there is an isomorphism of (RH, RG)-bimodules

Σ(M̃) ≅ Ind_{X}^{H×G}M .

Similarly, set V = (K × H)/Y, and define a functor Ñ⟨V ⟩ → R-Mod, using the RY -module N. Then the corollary is a straightforward consequence of the lemma, applied to the functors M̃ and Ñ, using Remark 2.6 and Remark 3.2.

References

[1] S. Bouc. Foncteurs d’ensembles munis d’une double action. J. of Algebra, 183(0238):664–736, 1996.

Serge Bouc CNRS-LAMFA Université de Picardie 33 rue St Leu 80039 - Amiens Cedex 1 FRANCE

email :  serge.bouc@u-picardie.fr