Monomial G-posets and their Lefschetz invariants

Serge Bouc and Hatice Mutlu

Abstract Let G be a finite group, and C be an abelian group. We introduce the notions of C-monomial G-sets and C-monomial G-posets, and state some of their categorical properties. This gives in particular a new description of the C-monomial Burnside ring B_{C}(G). We also introduce Lefschetz invariants of C-monomial G-posets, which are elements of B_{C}(G). These invariants allow for a definition of a generalized tensor induction multiplicative map T_{U,λ} : B_{C}(G) → B_{C}(H) associated to any C-monomial (G, H)-biset (U, λ), which in turn gives a group homomorphism B_{C}(G)^{×} → B_{C}(H)^{×} between the unit groups of C-monomial Burnside rings. AMS Classification: 06A11, 19A22, 20J15 Keywords: Burnside ring, monomial, tensor induction, Lefschetz invariant

1 Introduction

Let G be a finite group, and C be an abelian group. In this work, we first introduce the notion of C-monomial G-set: this is a pair (X, 𝔩) consisting of a finite G-set X, together with a functor from the transporter category X̂ of X, to the the groupoid •_{C} with one object and automorphism group C. The C-monomial G-sets form a category _{C}MG-set, and we show that it is equivalent to the category _{C}F G-set of C-fibred G-sets considered by Barker ([1]). In particular, the C-monomial Burnside ring B_{C}(G) introduced by Dress ([5]) is isomorphic to the Grothendieck ring of the category _{C}MG-set.

We extend these definitions to the notion of C-monomial G-poset: this is a pair (X, 𝔩) consisting of a finite G-poset X, and a functor 𝔩 from the transporter category X̂ to •_{C}. We associate to each such pair (X, 𝔩) a Lefschetz invariant Λ_{(X,𝔩)} lying in B_{C}(G). We show that any element of B_{C}(G) is equal to the Lefschetz invariant of some (non unique) C-monomial G-poset.

We also introduce the category _{C}MG-poset of C-monomial G-posets, and show that there are natural functors of induction Ind_{H}^{G} : _{C}MH-poset → _{C}MG-poset and of restriction Res_{H}^{G} : _{C}MG-poset → _{C}MH-poset, whenever H is a subgroup of G. These functors are compatible with the construction of Lefschetz invariants.

We extend several classical properties of the Lefschetz invariants of G-posets to Lefschetz invariants of C-monomial G-posets (the classical case being the case where C is trivial).

We next turn to the construction of generalized tensor induction functors

T_{U,λ} : _{C}MG-poset → _{C}MH-poset

associated, for arbitrary finite groups G and H, to any C-monomial (G, H)-biset (U, λ). We show that these functors induce well defined tensor induction maps

T_{U,λ} : B_{C}(G) → B_{C}(H),

which are not additive in general, but multiplicative and preserve identity elements. In particular, we get induced group homomorphisms between the corresponding unit groups of monomial Burnside rings, similar to those obtained by Carman ([4]) for other usual representation rings.

We show moreover that under an additional assumption, these tensor induction functors and their associated tensor induction maps are well behaved for composition. This yields to a (partial) fibred biset functor structure on the group of units of the monomial Burnside ring.

2 The monomial Burnside ring

Let G be a finite group and C be an abelian group which is noted multiplicatively. We denote by G-set the category of finite G-sets (with G-equivariant maps as morphisms), and B(G) the usual Burnside ring of G, i.e. the Grothendieck ring of G-set for relations given by disjoint union decompositions of finite G-sets.

2.1 The category of C-fibred G-sets

A C-fibred G-set is defined to be a C-free (C × G)-set with finitely many C-orbits. Let CF G-set denote the category of C-fibred G-sets where morphisms are (C×G)-equivariant maps. The coproduct of C-fibred G-sets X, Y is their coproduct X ⊔ Y as sets, with the obvious (C × G)-action. If X and Y are C-fibred G-sets, there is a C-action on X × Y defined by c(x, y) = (cx, c^{−1}y) for any c ∈ C and (x, y) ∈ X × Y. The C-orbit of an element (x, y) of X × Y is denoted by x ⊗ y and the set of C-orbits is denoted by X ⊗ Y. Moreover C × G acts on X ⊗ Y by

(c, g)(x ⊗ y) = cgx ⊗ gy

for any (c, g) ∈ C × G and x ⊗ y ∈ X ⊗ Y. One checks easily that X ⊗ Y is again a C-fibred G-set, called the tensor product of X and Y.

We denote the isomorphism class of a C-fibred G-set X by [X]. The C-monomial Burnside ring B_{C}(G), introduced by Dress ([5]), is defined as the Grothendieck group of the category of C-fibred G-sets, for relations given by [X] + [Y] = [X ⊔ Y]. The ring structure of B_{C}(G) is induced by [X] · [Y] = [X ⊗ Y]. The identity element is the set C with trivial G-action and the zero element is the empty set. If C is trivial we recover the ordinary Burnside ring of the group G.

Given a C-fibred G-set X, we denote the set of C-orbits on X by C\X. The group G acts on C\X, and X is (C × G)-transitive if and only if C\X is G-transitive. If C\X is transitive as a G-set it is isomorphic to G/U for some U ≤ G. There exists a group homomorphism μ : U → C such that if U is the stabilizer of the orbit Cx, then ax = μ(a)x for all a ∈ U. Since the stabilizer (C × G)_{x} of x in C × G is equal to

(C × G)_{x} = {(μ(a)^{−1}, a) | a ∈ U},

the C-fibred G-set X is determined up to isomorphism by the subgroup U and μ.

Conversely, let U be a subgroup of G, and μ : U → C be a group homomorphism. Then we set U_{μ} = {(μ(a)^{−1}, a) | a ∈ U}, and denote by [U, μ]_{G} the C-fibred G-set (C × G)/U_{μ}. The pair (U, μ) is called a C-subcharacter of G. We denote the set of C-subcharacters by ch(G). The group G acts on ch(G) by conjugation. The G-set ch(G) is a poset with the relation ≤ defined by

(U, μ) ≤ (V, ν) ⇔ U ≤ V and Res_{U}^{V} ν = μ

for any (U, μ) and (V, ν) in ch(G).

As an abelian group we have

B_{C}(G) = ⨁_{(U,μ)∈_{G} ch(G)} ℤ[U, μ]_{G}

where (V, ν) runs over G-representatives of the C-subcharacters of G, details can be seen in [1].

2.2 The category of C-monomial G-sets

Let G be a finite group and C be an abelian group. Given a G-set X, we consider its transporter category X̂ whose objects are the elements of X and given x, y in X the set of morphisms from x to y is

HomX̂(x, y) = {g ∈ G | gx = y}.

Let •_{C} denote the category with one object where morphisms are the elements of C and composition is multiplication in C. Now we define C-monomial G-sets as follows.

Definition 1. A C-monomial G-set is a pair (X, 𝔩) consisting of a finite G-set X and a functor 𝔩 : X̂ → •_{C}.

In otherwords, for each x, y ∈ X and g ∈ G such that gx = y, we have an element 𝔩(g, x, y) of C, with the property that 𝔩(h, y, z)𝔩(g, x, y) = 𝔩(hg, x, z) if h ∈ G and hy = z, and 𝔩(1, x, x) = 1 for any x ∈ X.

Let (X, 𝔩) and (Y, 𝔪) be C-monomial G-sets. If f : X → Y is a map of G-sets, we slightly abuse notation and also denote by f : X̂ → Ŷ the obvious functor induced by f. Now a map (f, λ) : (X, 𝔩) → (Y, 𝔪) of C-monomial G-sets is a pair consisting of a map f : X → Y of G-sets and a natural transformation λ : 𝔩 → 𝔪 ∘ f. We denote by CMG-set the category whose objects are C-monomial G-sets, morphisms are the maps of C-monomial G-sets, and composition is the obvious one.

Let (X, 𝔩) and (X′, 𝔩′) be C-monomial G-sets. We define the disjoint union of Cmonomial G-sets as (X, 𝔩) ⊔ (X′, 𝔩′) = (X ⊔ X′, 𝔩 ⊔ 𝔩′) where X ⊔ X′ is the disjoint union of G-sets and

𝔩 ⊔ 𝔩′ : { \atop X ⊔ X′} → •_{C}

is the functor such that

(𝔩 ⊔ 𝔩′)(g, z_{1}, z_{2}) = \begin{cases}𝔩(g, z_{1}, z_{2}) & z_{1}, z_{2} ∈ X \\ 𝔩′(g, z_{1}, z_{2}) & z_{1}, z_{2} ∈ X′ \end{cases}

for any z_{1}, z_{2} ∈ X ⊔ X′ such that gz_{1} = z_{2} for some g ∈ G.

The product of C-monomial G-sets (X, 𝔩), (X′, 𝔩′) is defined to be (X × X′, 𝔩 × 𝔩′) where X × X′ is the product of G-sets and 𝔩 × 𝔩′ : { \atop X × Y} → •_{C} is the functor defined by

(𝔩 × 𝔩′)(g, (x, x′), (y, y′)) = 𝔩(g, x, y)𝔩′(g, x′, y′)

for g ∈ G and (x, x′), (y, y′) ∈ X × X′ such that g(x, x′) = (y, y′).

Our goal is to show that the categories _{C}MG-set and _{C}F G-set are equivalent. For this, we define a functor F : _{C}MG-set → _{C}F G-set as follows: given a C-monomial G-set (X, 𝔩), we set

F (X, 𝔩) = C ×_{𝔩} X,

which is the direct product C×X endowed with the (C×G)-action defined by (k, g)(c, x) = (kc𝔩(g, x, gx), gx) for any (k, g) ∈ C × G and (c, x) ∈ C × X.

Given a map (f, λ) : (X, 𝔩) → (Y, 𝔪) of C-monomial G-sets, we define

F (f, λ) : C ×_{𝔩} X → C ×_{𝔪} Y

by F (f, λ)(c, x) = (cλ_{x}, f(x)) for any (c, x) ∈ C ×_{𝔩} X. Then F (f, λ) is a (C × G)-map: indeed, given (k, g) ∈ C × G and (c, x) ∈ C × X, we have

(k, g)F (f, λ)(c, x) = (k, g)(cλ_{x}, f(x)) = (kcλ_{x}𝔪(g, f(x), f(gx)), f(gx))
= (kcλ_{gx}𝔩(g, x, gx), f(gx)) = F (f, λ)(kc𝔩(g, x, gx), gx)
= F (f, λ)((k, g)(c, x)).

It is clear that F : _{C}MG-set → _{C}F G-set is a functor.

Lemma 2. Let C be an abelian group and G be a finite group. Then the above functor F : _{C}MG-set → _{C}F G-set is an equivalence of categories.

Proof. We prove that F is fully faithful and essentially surjective. First we show that F is essentially surjective. Given a C-fibred G-set X, let C\X be the set of C-orbits. Clearly C\X is a G-set. We define a functor 𝔩 : { \atop C\X} → •_{C}. Let Cx, Cy ∈ C\X such that Cgx = Cy for some g ∈ G. Then there exists a unique c ∈ C such that gx = cy. We set 𝔩(g, Cx, Cy) = c. We have F (C\X, 𝔩) = C ×_{𝔩} (C\X). Now choose a set [C\X] of G-representatives of the G-action on C\X. Then for any x ∈ X, there exits a unique Cσ_{x} ∈ [C\X] such that x ∈ Cσ_{x}. Since X is C-free, there exists a unique c_{x} ∈ C such that x = c_{x}σ_{x}. We define a (C×G)-map f : X → C×_{𝔩}(C\X) such that f(x) = (c_{x}, Cσ_{x}). Then

(c, g)f(x) = (c, g)(c_{x}, Cσ_{x}) = (c_{x}c𝔩(g, Cσ_{x}, Cgσ_{x}), Cgσ_{x}) = (c_{x}c, Cgσ_{x})
= (c_{cgx}, Cgσ_{x}) = f((c, g)x).

So f is a (C × G)-map and clearly an isomorphism. Thus, F is essentially surjective. Let (X, 𝔩) and (Y, 𝔪) be C-monomial G-sets. We need to show that the map

\overline{F} : Hom((X, 𝔩), (Y, 𝔪)) → Hom(F (X, 𝔩), F (Y, 𝔪))

induced by F is surjective and injective. Let φ : C ×_{𝔩} X → C ×_{𝔪} Y be a (C × G)- map. Given (1, x) ∈ C ×_{𝔩} X, let φ(1, x) = (c_{x}, z_{x}) for (c_{x}, z_{x}) ∈ C × Y. Since φ is a (C × G)-map, we get

φ(c, x) = (cc_{x}, z_{x})

and

φ(1, gx) = (c_{x}𝔪(g, z_{x}, gz_{x})𝔩^{−1}(g, x, gx), gz_{x})

for any c ∈ C and g ∈ G. We define a map

(f, λ) : (X, 𝔩) → (Y, 𝔪)

such that f : X → Y is defined by f(x) = z_{x} and λ : 𝔩 → 𝔪 ∘ f is defined by λ_{x} = c_{x} for any x ∈ X. Clearly, f is a G-set map. Let x ∈ X and g ∈ G. Then

𝔪(g, f(x), f(gx))λ_{x} = 𝔪(g, f(x), f(gx))c_{x} = c_{x}𝔪(g, f(x), f(gx))𝔩^{−1}(g, x, gx)𝔩(g, x, gx)

= 𝔩(g, x, gx)c_{gx} = 𝔩(g, x, gx)λ_{gx}.

So λ : 𝔩 → 𝔪 ∘ f is a natural transformation and (f, λ) is a map of C-monomial G-sets. Thus, \overline{F} (f, λ) = φ and \overline{F} is surjective. The injectivity is clear, so F is fully faithful.

Proposition 3. Let G be a finite group. Then B_{C}(G) is isomorphic to the Grothendieck ring of the category _{C}MG-set, for relations given by decomposition into disjoint unions of C-monomial G-sets and multiplication induced by product of C-monomial G-sets.

Proof. We let B_{C}^{1}(G) denote the Grothendieck ring of the category _{C}MG-set. The equivalence

F : _{C}MG-set → _{C}F G-set

induces a bijection

F̂ : B_{C}^{1}(G) → B_{C}(G)

such that

F̂([(X, 𝔩)]) = [C ×_{𝔩} X]

for any C-monomial G-set (X, 𝔩). Now we show that F̂ is a ring homomorphism. Let (X_{1}, 𝔩_{1}) and (X_{2}, 𝔩_{2}) be C-monomial G-sets. Then

F̂([(X_{1}, 𝔩_{1})] + [(X_{1}, 𝔩_{1})]) = F̂([(X_{1}, 𝔩_{1}) ⊔ (X_{1}, 𝔩_{1})]) = F̂([(X_{1} ⊔ X_{2}, 𝔩_{1} ⊔ 𝔩_{2})])

= [C ×_{𝔩_{1}⊔𝔩_{2}} (X_{1} ⊔ X_{2})] = [(X_{1}, 𝔩_{1}) ⊔ (X_{2}, 𝔩_{2})] = [C ×_{𝔩_{1}} X_{1}] + [C ×_{𝔩_{2}} X_{2}].

For multiplicativity of F̂ we define a map

f : C ×_{𝔩_{1}×𝔩_{2}} (X_{1} × X_{2}) → (C ×_{𝔩_{1}} X_{1}) ×_{C} (C ×_{𝔩_{2}} X_{2})

such that f(c, (x_{1}, x_{2})) = (c, x_{1}) ×_{C} (1, x_{2}). Let (k, g) ∈ C × G and (c, (x_{1}, x_{2})) ∈ C ×_{𝔩_{1}×𝔩_{2}} (X_{1} × X_{2}). Then

(k, g)f(c, (x_{1}, x_{2})) = (k, g)((c, x_{1}) ×_{C} (1, x_{2})) = ((k, g)(c, x_{1}) ×_{C} (1, g)(1, x_{2}))
= (kc𝔩_{1}(g, x_{1}, gx_{1}), gx_{1}) ×_{C} (𝔩_{2}(g, x_{2}, gx_{2}), gx_{2})
= (kc𝔩_{1}(g, x_{1}, gx_{1})𝔩_{2}(g, x_{2}, gx_{2}), gx_{1}) ×_{C} (1, gx_{2})
= f(kc𝔩_{1}(g, x_{1}, gx_{1})𝔩_{2}(g, x_{2}, gx_{2}), g(x_{1}, x_{2}))
= f((k, g)(c, (x_{1}, x_{2}))).

So f is a (C × G)-map and obviously, f is a (C × G)-isomorphism. Using f we get

F̂([X_{1}, 𝔩_{1}]·[X_{2}, 𝔩_{2}]) = F̂([X_{1}×X_{2}, 𝔩_{1}×𝔩_{2}]) = [C×_{𝔩_{2}×𝔩_{2}}(X_{1}×X_{2})] = [(C×_{𝔩_{1}}X_{1})×_{C}(C×_{𝔩_{2}}X_{2})].

Thus, the desired result follows.

Remark 4. Let (X, 𝔩) be a C-monomial G-set. For all x ∈ X, we get a character 𝔩_{x} : G_{x} → C defined by 𝔩_{x}(g) = 𝔩(g, x, x) for g ∈ G_{x}. On the other hand given a subgroup U of G and a group homomorphism μ : U → C we get a C-monomial G-set (G/U, μ̂) where and μ̂ : { \atop G/U} → •_{C} is the functor such that given gU, kU ∈ G/U if hgU = kU for some g ∈ G then μ̂(h, gU, kU) = μ(k^{−1}hg). Moreover, [U, μ]_{G} and [G/U, μ̂] represents the same element in B_{C}(G).

2.3 The Lefschetz invariant attached to a monomial G-poset

A G-poset X is a partially ordered set (X, ≤) with a compatible G-action (that is gx ≤ gy whenever g ∈ G and x ≤ y in X). A map of G-posets is a G-equivariant map of posets. We denote by G-poset the category of finite G-posets obtained in this way.

There is an obvious functor ι_{G} : G-set → G-poset sending each finite G-set to the set X ordered by the equality relation, and each G-equivariant map to itself.

The Lefschetz invariant attached to a finite G-poset, which is an element of the Burnside ring of G has been introduced in [7] by Thévenaz. We will define similarly a Lefschetz invariant attached to a C-monomial G-poset as an element of the C-monomial Burnside ring of G.

2.3.1 The category of C-monomial G-posets

Given a G-poset X, we consider the category X̂ whose objects are the elements of X and given x, y in X the set of morphisms from x to y is

HomX̂(x, y) = {g ∈ G | gx ≤ y}.

Now we define a C-monomial G-poset as follows.

Definition 5. A C-monomial G-poset is a pair (X, 𝔩) consisting of a G-poset X and a functor 𝔩 : X̂ → •_{C}.

In otherwords, for each x, y ∈ X and g ∈ G such that gx ≤ y, we have an element 𝔩(g, x, y) of C, with the property that 𝔩(h, y, z)𝔩(g, x, y) = 𝔩(hg, x, z) if h ∈ G and hy ≤ z, and 𝔩(1, x, x) = 1 for any x ∈ X.

Let (X, 𝔩) and (Y, 𝔪) be C-monomial G-posets. A map of C-monomial G-posets from (X, 𝔩) to (Y, 𝔪) is a pair (f, λ) : (X, 𝔩) → (Y, 𝔪), where f : X → Y is a map of G-posets and λ : 𝔩 → 𝔪∘f is a natural transformation. We denote the category of C-monomial Gposets by _{C}MG-poset. Product and disjoint union of C-monomial G-posets are defined as for C-monomial G-sets. When C is the trivial group, we will identify the category CMG-poset with G-poset.

Remark 6. If (X, 𝔩) is a C-monomial G-poset, then for any x ∈ X we get a character 𝔩_{x} : G_{x} → C defined by 𝔩_{x}(g) = 𝔩(g, x, x). Moreover, if x ≤ y, then

res_{G_{x}∩G_{y}}^{G_{x}}𝔩_{x} = res_{G_{x}∩G_{y}}^{G_{y}}𝔩_{y}

because we have the following commutative diagram:

\xymatrix{𝔩(x) \ar[r]^{𝔩(1,x,y)} \ar[d]_{𝔩(g,x,x)} & 𝔩(y) \ar[d]^{𝔩(g,y,y)} \\ 𝔩(x) \ar[r]_{𝔩(1,x,y)} & 𝔩(y).}

Let H be a subgroup of G and (X, 𝔩) be a C-monomial H-set. We let G ×_{H} X to be the quotient of G × X by the action of H. The set G ×_{H} X is a G-set via the action g(u,_{H} x) = (gu,_{H} x), for any g ∈ G, and (u,_{H} x) ∈ G ×_{H} X. We define an order relation ≤ on G ×_{H} X as

∀(u,_{H} x), (v,_{H} y) ∈ G ×_{H} X, (u,_{H} x) ≤ (v,_{H} y) ⇔ ∃h ∈ H, u = vh, x ≤ h^{−1}y.

Since we have

G ×_{H} X = ⨆_{g∈G/H} g ×_{H} X,

it’s enough to consider the chains of type (u,_{H} x_{0}) < ... < (u,_{H} x_{n}) in G ×_{H} X for some u ∈ G and a chain x_{0} < ... < x_{n} in X for some n ∈ ℕ.

Let (u,_{H} x), (u,_{H} y) ∈ G ×_{H} X and g ∈ G such that g(u,_{H} x) ≤ (u,_{H} y). Then there exists h ∈ H such that gu = uh and hx ≤ y. We define the induced C-monomial G-poset Ind_{H}^{G}(X, 𝔩) of (X, 𝔩) as the pair (G×_{H} X, G×_{H} 𝔩) where G×_{H} 𝔩 : { \atop G ×_{H} X} → •_{C} is defined by

(G ×_{H} 𝔩)(g, (u,_{H} x), (u,_{H} y)) = 𝔩(h, x, y).

Now show that (G ×_{H} X, G ×_{H} 𝔩) is a C-monomial G-poset.

Let (u,_{H} x), (u,_{H} y), (u,_{H} z) ∈ G ×_{H} X such that

g(u,_{H} x) ≤ (u,_{H} y)

and

g′(u,_{H} y) ≤ (u,_{H} z)

for some g, g′ ∈ G. Then there exist some h, h′ ∈ H such that

gu = uh, g′u = uh′, hx ≤ y, h′y ≤ z.

Then t = h′h ∈ H. Moreover g′gu = uh′h = ut and tx = h′hx ≤ z. Now we get

(G ×_{H} 𝔩)(g′g, (u,_{H} x), (u,_{H} z)) = 𝔩(t, x, z) = 𝔩(h′h, x, z) = 𝔩(h′, y, z)𝔩(h, x, y)
= (G ×_{H} 𝔩)(g′, (u,_{H} x), (u,_{H} y))(G ×_{H} 𝔩)(g, (u,_{H} y), (u,_{H} z)).

We also have (G ×_{H} 𝔩)(1, (u,_{H} x), (u,_{H} x)) = 1 for any (u,_{H} x) ∈ G ×_{H} X. Thus G ×_{H} 𝔩 is a functor. So Ind_{H}^{G}(X, 𝔩) is a C-monomial G-poset.

Given a C-monomial G-poset (Y, 𝔪), the restriction Res_{H}^{G}(Y, 𝔪) of (Y, 𝔪) is the pair ( Res_{H}^{G}Y, res_{H}^{G}𝔪) where Res_{H}^{G}Y is the restriction of the G-poset Y to H-poset and res_{H}^{G}𝔪

is the restriction of the functor 𝔪 from Ŷ to { \atop Res_{H}^{G}Y}.

Proposition 7. Let G be a finite group.

1. If Y is a finite G-poset, denote by 1_{Y} : Ŷ → •_{C} the trivial functor defined by 1_{Y} (g, x, y) = 1 for any g ∈ G and x, y ∈ Y such that gx ≤ y. Then the assignment Y ↦ (Y, 1_{Y}) is a functor τ_{G} from G-poset to _{C}MG-poset.

2. Let H be a subgroup of G. The assignment (X, 𝔩) ↦ Ind_{H}^{G}(X, 𝔩) is a functor Ind_{H}^{G} : _{C}MH-poset → _{C}MG-poset, and the assignment (Y, 𝔪) ↦ Res_{H}^{G}(Y, 𝔪) is a functor Res_{H}^{G} : _{C}MG-poset → _{C}MH-poset.

3. Moreover the diagrams

\xymatrix{H-poset \ar[r]^{^{Ind^{G}^{H}}} \ar[d]_{τ_{H}} & G-poset \ar[d]_{τ_{G}} & and & G-poset \ar[r]^{^{Res^{G}^{H}}} \ar[d]_{τ_{G}} & H-poset \ar[d]_{τ_{H}} \\ _{C}MH-poset \ar[r]^{_{Ind_{G}_{H}}} & _{C}MG-poset &  & _{C}MG-poset \ar[r]^{_{Res_{G}_{H}}} & _{C}MH-poset}

of categories and functors are commutative.

Proof. 1. Let f : X → Y be a map of G-posets. We set

τ_{G}(f) = (f, 1_{f}) : (X, 1_{X}) → (Y, 1_{Y}),

where 1_{f} : 1_{X} → 1_{Y} ∘ f is defined by 1_{f}_{x} = 1 for any x ∈ X. Obviously (f, 1_{f}) is a map of C-monomial G-posets and τ_{G} is a functor.

2. Let (f, λ) : (X, 𝔩) → (Y, 𝔪) be a map of C-monomial H-posets. We set the pair

Ind_{H}^{G}(f, λ) = (G ×_{H} f, G ×_{H} λ) : (G ×_{H} X, G ×_{H} 𝔩) → (G ×_{H} Y, G ×_{H} 𝔪)

where

G ×_{H} f : G ×_{H} X → G ×_{H} Y

is defined by (G ×_{H} f)(u,_{H} x) = (u,_{H} f(x)) and

G ×_{H} λ : G ×_{H} 𝔩 → (G ×_{H} 𝔪) ∘ (G ×_{H} f)

is defined by (G ×_{H} λ)_{(u,_{H} x)} = λ_{x} for any (u,_{H} x) ∈ G ×_{H} X. It’s clear that G ×_{H} f is a map of C-monomial G-posets. Now we show that G ×_{H} λ is a natural transformation. Let (u,_{H} x), (u,_{H} y) ∈ G ×_{H} X such that g(u,_{H} x) ≤ (u,_{H} y) for some g ∈ G. Then gu = uh and hx ≤ y for some h ∈ H. Since λ : 𝔩 → 𝔪 ∘ f is a natural transformation, we get

(G ×_{H} 𝔪)(g, (u,_{H} f(x)), (u,_{H} f(y)))(G ×_{H} λ)_{(u,_{H} x)} = 𝔪(h, f(x), f(y))λ_{x}
= λ_{y}𝔩(h, x, y) = (G ×_{H} λ)_{(u,_{H} y)}(G ×_{H} 𝔩)(g, (u,_{H} x), (u,_{H} y)).

Now consider ( id_{X}, id_{𝔩}) : (X, 𝔩) → (X, 𝔩) where id_{X} : X → X is the identity map on the H-set X and id_{𝔩} : 𝔩 → 𝔩 ∘ id_{X} is the identity transformation. Then we get

Ind_{H}^{G}( id_{X}, id_{𝔩}) = ( id_{G×_{H}X}, id_{G×_{H}𝔩}).

Now let (f, λ) : (X, 𝔩) → (Y, 𝔪) and (t, β) : (Y, 𝔪) → (Z, 𝔯) be the maps of Cmonomial H-posets. We obviously have

(G ×_{H} t) ∘ (G ×_{H} f) = G ×_{H} (t ∘ f)

and

(G ×_{H} β) ∘ (G ×_{H} λ) = G ×_{H} (β ∘ λ).

Thus,

Ind_{H}^{G}(t, β) ∘ Ind_{H}^{G}(f, λ) = Ind_{H}^{G}((t, β) ∘ (f, λ)).

So Ind_{H}^{G} : _{C}MH-poset → _{C}MG-poset is a functor.

Now let (f, λ) : (X, 𝔩) → (Y, 𝔪) be a map of C-monomial G-posets. We set the pair

Res_{H}^{G}(f, λ) = (f|_{H}, λ|_{H}) : ( Res_{H}^{G}X, res_{H}^{G}𝔩) → ( Res_{H}^{G}Y, res_{H}^{G}𝔪)

where f|_{H} : Res_{H}^{G}X → Res_{H}^{G}Y is defined as the restriction of map of G-posets f to map of H-posets and λ|_{H} : res_{H}^{G}𝔩 → res_{H}^{G}𝔪 ∘ f|_{H} is defined as the restriction of λ. Clearly, we get that Res_{H}^{G} : _{C}MG-poset → _{C}MH-poset is a functor.

3. Let X be an H-poset. Commmutativity of the first diagram follows from

τ_{G} ∘ Ind_{H}^{G}(X) = τ_{G}(G×_{H} X) = (G×_{H} X, 1_{G×_{H}X}) = Ind_{H}^{G}(X, 1_{X}) = Ind_{H}^{G} ∘τ_{H}(X).

Now let Y be a G-poset. Commutativity of the second diagram follows from

τ_{H} ∘ Res_{H}^{G}(Y) = τ_{H}( Res_{H}^{G}Y) = ( Res_{H}^{G}Y, 1_{Res_{H}^{G}Y})
= ( Res_{H}^{G}Y, res_{H}^{G}1_{Y}) = Res_{H}^{G}(Y, 1_{Y}) = Res_{H}^{G} ∘ τ_{H}(Y).

Proposition 8. Let G be a finite group and H be subgroup of G. Then the functor Ind_{H}^{G} : _{C}MH-poset → _{C}MG-poset is left adjoint to the functor (Y, 𝔪) ↦ Res_{H}^{G}(Y, 𝔪).

Proof. We prove that for any C-monomial H-poset (X, 𝔩) and any C-monomial G-poset (Y, 𝔪) we have a bijection

Hom_{_{C}MG}( Ind_{H}^{G}(X, 𝔩), (Y, 𝔪)) ≅ Hom_{_{C}MH}((X, 𝔩), Res_{H}^{G}(Y, 𝔪))

natural in (X, 𝔩) and (Y, 𝔪).

We define

φ : Hom_{_{C}MG}( Ind_{H}^{G}(X, 𝔩), (Y, 𝔪)) → Hom_{_{C}MH}((X, 𝔩), Res_{H}^{G}(Y, 𝔪))

where

φ : (f, λ) ↦ (φ(f), φ(λ))

such that

φ(f) : X → Res_{H}^{G}(Y)

defined by φ(f)(x) = f(1,_{H} x) and

φ(λ) : 𝔩 → res𝔪 ∘ φ(f)

defined by φ(λ)_{x} = λ_{(1,_{H} x)} for any x ∈ X. Obviously, φ(f) is a map of H-posets. We need to show that

φ(λ) : 𝔩 → res𝔪 ∘ φ(f)

is a natural transformation. Let x, y ∈ X such that gx ≤ y for some g ∈ G. Then

𝔪(h,φ(f)(x), φ(f)(y))φ(λ)_{x} = 𝔪(h, f(1,_{H} x), f(1,_{H} y))λ_{(1,_{H} x)}
= λ_{(1,_{H} y)}𝔩(h, x, y) = φ(λ)_{y}(G ×_{H} 𝔩)(h, (1,_{H} x), (1,_{H} y)).

We define an inverse map to φ as

θ : Hom_{_{C}MH}((X, 𝔩), Res_{H}^{G}(Y, 𝔪)) → Hom_{_{C}MG}( Ind_{H}^{G}(X, 𝔩), (Y, 𝔪))

where

θ : (ψ, β) ↦ (θ(ψ), θ(β))

such that

θ(ψ) : G ×_{H} X → Y

defined as θ(ψ)(u,_{H} x) = uψ(x) and

θ(β) : G ×_{H} 𝔩 → 𝔪 ∘ θ(ψ)

defined as

θ(β)_{(u,_{H} x)} = 𝔪(u, ψ(x), uψ(x))β_{x}

for any (u,_{H} x) ∈ G ×_{H} X. Obviously, the map θ(ψ) is a map of G-posets. We need to show that θ(β) is a natural transformation. Let (u,_{H} x), (u,_{H} y) ∈ G ×_{H} X such that g(u,_{H} x) ≤ (u,_{H} y) for some g ∈ G. Then there exists some h ∈ H such that gu = uh and hx ≤ y. Now, we have

𝔪(g, θ(ψ)(u,_{H} x), θ(ψ)(u,_{H} y))θ(β)_{(u,_{H} x)} = 𝔪(g, uψ(x), uψ(y))𝔪(u, ψ(x), uψ(x))β_{x}
= 𝔪(u, ψ(y), uψ(y))𝔪(h, ψ(x), ψ(y))β_{x} = 𝔪(u, ψ(y), uψ(y))β_{y}𝔩(h, x, y)
= θ(β)_{(u,_{H} y)}(G ×_{H} 𝔩)(h, (u_{H}x), (u,_{H} y)).

Clearly, φ and θ are mutual inverse maps, and natural in (X, 𝔩) and (Y, 𝔪).

2.3.2 The Lefschetz invariant attached to a C-monomial G-poset

Let (X, 𝔩) be a C-monomial G-poset. The Lefschetz invariant Λ_{(X,𝔩)} of (X, 𝔩) is the element of B_{C}(G) defined by

Λ_{(X,𝔩)} = ∑_{x_{0}<...<x_{n}∈_{G}X}(−1)^{n}[G_{x_{0},...,x_{n}}, Res_{G_{x_{0},...,xn}}^{G_{x_{0}}}(𝔩_{x_{0}})]_{G}

where x_{0} < ... < x_{n} runs over G-representatives of the chains in X. The group G_{x_{0},...,x_{n}} is the stabilizer of the set {x_{0}, ..., x_{n}}, that is G_{x_{0},...,x_{n}} = ∩_{i=0}^{n}G_{x_{i}}. Here Res_{G_{x_{0},...,xn}}^{G_{x_{0}}}(𝔩_{x_{0}}) denotes the restriction of the character 𝔩_{x_{0}} introduced in Remark 4. Observe that if x_{0} < ... < x_{n} is a chain in X for some n ∈ ℕ, by Remark 6 we have

Res_{G_{x_{0},...,xn}}^{G_{x_{0}}}𝔩_{x_{0}} = Res_{G_{x_{0},...,xn}}^{G_{x_{i}}}𝔩_{x_{i}}

for any 0 ≤ i ≤ n.

Let (X, 𝔩) be a C-monomial G-poset. Given n ∈ ℕ, let Sd_{n}(X) denote the set of chains in X with order n + 1. Obviously, the set Sd_{n}(X) is a G-set. Then ( Sd_{n}(X), 𝔩_{n}) is a C-monomial G-set where 𝔩_{n} : { \atop Sd_{n}(X)} → •_{C} is the functor defined by

𝔩_{n}(g, x_{0} < ... < x_{n}, y_{0} < ... < y_{n}) = 𝔩(g, x_{0}, y_{0})

for any x_{0} < ... < x_{n}, and y_{0} < ... < y_{n} in Sd_{n}(X) such that

g(x_{0} < ... < x_{n}) = y_{0} < ... < y_{n}

for some g ∈ G.

Remark 9. Given a C-monomial G-poset (X, 𝔩), we have the following isomorphism of monomial G-sets:

( Sd_{n}(X), 𝔩_{n}) ≅ ⨆_{x_{0}<...<x_{n}∈_{G} Sd_{n}(X)} (G/G_{x_{0},...,x_{n}}, { \atop Res_{G_{x_{0},...,xn}}^{G_{x_{0}}}(𝔩_{x_{0}})})

for any n ∈ ℕ.

Proof. Let [G/ Sd_{n}(X)] be a set of representative of the G-action on Sd_{n}(X). Let x = x_{0} < ... < x_{n} be a chain in Sd_{n}(X) then there exist some g_{x} ∈ G and a unique σ_{x} ∈ [G/ Sd_{n}(X)] such that x = g_{x}σ_{x} where σ_{x} = σ_{x_{0}} < ... < σ_{x_{n}}. We define

(f, λ) : ( Sd_{n}(X), 𝔩_{n}) → ⨆_{x_{0}<...<x_{n}∈_{G} Sd_{n}(X)} (G/G_{x_{0},...,x_{n}}, { \atop Res_{G_{x_{0},...,xn}}^{G_{x_{0}}}(𝔩_{x_{0}})})

where f(x) = g_{x}G_{σ_{x}} ∈ G/G_{σ_{x}} and λ_{x} = 𝔩(g_{x}^{−1}, g_{x}σ_{x_{0}}, σ_{x_{0}}). Obviously,

f : Sd_{n}(X) → ⨆_{x_{0}<...<x_{n}∈_{G} Sd_{n}(X)} G/G_{x_{0},...,x_{n}}

is an isomorphism of G-sets. We show that

λ : 𝔩_{n} → ⨆_{x_{0}<...<x_{n}∈_{G}Sd_{n}(X)} { \atop Res_{G_{x_{0},...,xn}}^{G_{x_{0}}}(𝔩_{x_{0}})} ∘ f

is a natural transformation. Let x = x_{0} < ... < x_{n}, and y = y_{0} < ... < y_{n} be sequences in Sd_{n}(X) such that gx = y for some g ∈ G. There exist a unique σ_{x}, σ_{y} ∈ [G/ Sd_{n}(X)] such that x = g_{x}σ_{x} and y = g_{y}σ_{y} for some g_{x} and g_{y} in G. Then x_{0} = g_{x}σ_{x_{0}} and y_{0} = g_{y}σ_{y_{0}} so y_{0} = gx_{0} = gg_{x}σ_{x_{0}}. Thus, by uniqueness σ_{x_{0}} = σ_{y_{0}} and so g_{y}^{−1}gg_{x} ∈ G_{σ_{x_{0}}}.

Then setting r = { \atop Res_{G_{x}}^{G_{x_{0}}}(𝔩_{x_{0}})}(g, f(x), f(y))λ_{x}, we have that

r = 𝔩̂_{x_{0}}(g, g_{x}G_{σ_{x}}, g_{y}G_{σ_{y}})𝔩(g_{x}^{−1}, g_{x}σ_{x_{0}}, σ_{x_{0}}) = 𝔩_{x_{0}}(g_{y}^{−1}gg_{x})𝔩(g_{x}^{−1}, g_{x}σ_{x_{0}}, σ_{x_{0}})
= 𝔩(g_{y}^{−1}gg_{x}, x_{0}, x_{0})𝔩(g_{x}^{−1}, g_{x}σ_{x_{0}}, σ_{x_{0}})
= 𝔩(g_{x}, σ_{x_{0}}, g_{x}σ_{x_{0}})𝔩(g, g_{x}σ_{x_{0}}, gg_{x}σ_{x_{0}})𝔩(g_{y}^{−1}, gg_{x}σ_{x_{0}}, σ_{x_{0}})𝔩(g_{x}^{−1}, g_{x}σ_{x_{0}}, σ_{x_{0}})
= 𝔩(g, x_{0}, y_{0})𝔩(g_{y}^{−1}, g_{y}σ_{y_{0}}, σ_{y_{0}})
= 𝔩_{n}(g, x, y)λ_{y}.

By Remark 9, the Lefschetz invariant of a C-monomial G-set (X, 𝔩) can be written as

Λ_{(X,𝔩)} = ∑_{x_{0}<...<x_{n}∈_{G}X}(−1)^{n}[G_{x_{0},...x_{n}}, Res_{G_{x_{0},...,xn}}^{G_{x_{0}}}(𝔩_{x_{0}})]_{G} = ∑_{n∈ℕ}(−1)^{n}( Sd_{n}(X), 𝔩_{n}).

It follows that Λ_{X} = Λ_{τ_{G}(X)}, where Λ_{X} the Lefschetz invariant of the G-poset X introduced in [2].

We define similarly the reduced Lefschetz invariant of (X, 𝔩)

Λ̃_{(X,𝔩)} = Λ_{(X,𝔩)} − [G, 1_{G}]_{G}

where 1_{G} is the trivial character of G.

Lemma 10. Let G be a finite group and C be an abelian group.

1. Let (X, 𝔩) be a C-monomial G-set, viewed a a C-monomial G-poset ordered by the equality relation on X. Then Λ_{(X,𝔩)} = [C ×_{𝔩} X] in B_{C}(G).

2. Let (X, 𝔩) and (Y, 𝔪) be C-monomial G-posets. Then Λ_{(X⊔Y,𝔯)} = Λ_{(X,𝔩)} + Λ_{(Y,𝔪)} in B_{C}(G).

3. Given C-monomial G-posets (X, 𝔩) and (Y, 𝔪), we have Λ_{(X×Y,𝔩×𝔪)} = Λ_{(X,𝔩)}Λ_{(Y,𝔪)} in B_{C}(G).

Proof. 1. and 2. are clear.

3. In the following proof using the inclusion

B_{C}(G) ↪ ℚ ⊗_{ℤ} B_{C}(G)

we identify the elements of B_{C}(G) with their image in ℚ ⊗_{ℤ} B_{C}(G). We start with rearranging the chains in X × Y as in the proof of Lemma 11.2.9 in [2]. Let n ∈ ℕ. Given a chain z = z_{0} < ... < z_{n} in X × Y projection of z on X is denoted by z_{X} and on Y is denoted by z_{Y}. Then z_{X} is a chain in X with order i + 1 for some i ≤ n and z_{Y} is a chain in Y with order j + 1 for some j ≤ n such that i + j = n. Let \underline{s}_{i} be the chain s_{0} < ... < s_{i} and \underline{t}_{j} be the chain t_{0} < ... < t_{j}. Now

Λ_{(X×Y,𝔩×𝔪)} = ∑_{\begin{matrix}n∈ℕ, \\ z∈_{G} Sd_{n}(X×Y) \end{matrix}}(−1)^{n}[G_{z}, Res_{G_{z}}^{G_{z_{0}}}(𝔩_{z_{0}})]_{G} = ∑_{\begin{matrix}n∈ℕ, \\ z∈ Sd_{n}(X×Y) \end{matrix}}(−1)^{n} \frac{|G_{z}|}{|G|}[G_{z}, Res_{G_{z}}^{G_{z_{0}}}(𝔩_{z_{0}})]_{G}

= ∑_{\begin{matrix}i,j∈ℕ \\ \underline{s}_{i}∈X \\ \underline{t}_{j}∈Y \end{matrix}} Γ_{\underline{s}_{i},\underline{t}_{j}}

where

Γ_{\underline{s}_{i},\underline{t}_{j}} = ∑_{\begin{matrix}n∈ℕ \\ z∈ Sd_{n}(X×Y):z_{X}=\underline{s}_{i}, z_{Y} =\underline{t}_{j} \end{matrix}} (−1)^{n} \frac{|G_{\underline{s}_{i}} ∩ G_{\underline{t}_{j}}|}{|G|}[G_{\underline{s}_{i}} ∩ G_{\underline{t}_{j}}, Res_{G_{\underline{s}_{i}}}^{G_{s_{0}}}(𝔩_{s_{0}}) Res_{G_{\underline{t}_{j}}}^{G_{t_{0}}}(𝔪_{t_{0}})]_{G}

= \frac{|G_{\underline{s}_{i}} ∩ G_{\underline{t}_{j}}|}{|G|}[G_{\underline{s}_{i}} ∩ G_{\underline{t}_{j}}, Res_{G_{\underline{s}_{i}}}^{G_{s_{0}}}(𝔩_{s_{0}}) Res_{G_{\underline{t}_{j}}}^{G_{t_{0}}}(𝔪_{t_{0}})]_{G} ∑_{\begin{matrix}n∈ℕ \\ z∈ Sd_{n}(X×Y):z_{X}=\underline{s}_{i}, z_{Y} =\underline{t}_{j} \end{matrix}} (−1)^{n}

= \frac{|G_{\underline{s}_{i}} ∩ G_{\underline{t}_{j}}|}{|G|}[G_{\underline{s}_{i}} ∩ G_{\underline{t}_{j}}, Res_{G_{\underline{s}_{i}}}^{G_{s_{0}}}(𝔩_{s_{0}}) Res_{G_{\underline{t}_{j}}}^{G_{t_{0}}}(𝔪_{t_{0}})]_{G}(−1)^{i+j}.

Now,

Λ_{(X×Y,𝔩×𝔪)} = ∑_{\begin{matrix}i,j∈ℕ \\ \underline{s}_{i}∈X \\ \underline{t}_{j}∈Y \end{matrix}}(−1)^{i+j} \frac{|G_{\underline{s}_{i}} ∩ G_{\underline{t}_{j}}|}{|G|}[G_{\underline{s}_{i}} ∩ G_{\underline{t}_{j}}, Res_{G_{\underline{s}_{i}}}^{G_{s_{0}}}(𝔩_{s_{0}}) Res_{G_{\underline{t}_{j}}}^{G_{t_{0}}}(𝔪_{t_{0}})]_{G}.

On the other hand

Λ_{(X,𝔩)}Λ_{(Y,𝔪)} = ∑_{\begin{matrix}i∈ℕ \\ \underline{s}_{i}∈_{G}X \end{matrix}}(−1)[G_{\underline{s}_{i}}, Res_{G_{\underline{s}_{i}}}^{G_{s_{0}}}(𝔩_{s_{0}})] ∑_{\begin{matrix}j∈ℕ \\ \underline{t}_{j}∈_{G}Y \end{matrix}} (−1)^{j}[G_{\underline{t}_{j}}, Res_{G_{\underline{t}_{j}}}^{G_{t_{0}}}(𝔪_{t_{0}})]_{G}

= ∑_{\begin{matrix}i,j∈ℕ \\ \underline{s}_{i}∈X \\ \underline{t}_{j}∈Y \\ G_{i}gG_{\underline{t}_{j}} ⊆G \end{matrix}} (−1)^{i+j} \frac{|G_{\underline{s}_{i}}||G_{\underline{t}_{j}}|}{|G|^{2}}[G_{\underline{s}_{i}} ∩^{g}G_{\underline{t}_{j}}, Res_{G_{\underline{s}_{i}}}^{G_{s_{0}}}(𝔩_{s_{0}}) Res_{^{g}G_{\underline{t}_{j}}}^{^{g}G_{t_{0}}}(^{g}𝔪_{t_{0}})]_{G}

= ∑_{\begin{matrix}i,j∈ℕ \\ \underline{s}_{i}∈X \\ \underline{t}_{j}∈Y \\ g∈G \end{matrix}}(−1)^{i+j} \frac{|G_{\underline{s}_{i}}||G_{\underline{t}_{j}}|}{|G|^{2}|G_{\underline{s}_{i}}gG_{\underline{t}_{j}}|}[G_{\underline{s}_{i}} ∩^{g}G_{\underline{t}_{j}}, Res_{G_{\underline{s}_{i}}}^{G_{s_{0}}}(𝔩_{s_{0}}) Res_{^{g}G_{\underline{t}_{j}}}^{^{g}G_{t_{0}}}(^{g}𝔪_{t_{0}})]_{G}

= ∑_{\begin{matrix}i,j∈ℕ \\ \underline{s}_{i}∈X \\ \underline{t}_{j}∈Y \\ g∈G \end{matrix}}(−1)^{i+j} \frac{|G_{\underline{s}_{i}} ∩^{g}G_{\underline{t}_{j}}|}{|G|^{2}}[G_{\underline{s}_{i}} ∩^{g}G_{\underline{t}_{j}}, Res_{G_{\underline{s}_{i}}}^{G_{s_{0}}}(𝔩_{s_{0}}) Res_{^{g}G_{\underline{t}_{j}}}^{^{g}G_{t_{0}}}(^{g}𝔪_{t_{0}})]_{G}

= ∑_{\begin{matrix}i,j∈ℕ \\ \underline{s}_{i}∈X \\ \underline{t}_{j}∈Y \\ g∈G \end{matrix}}(−1)^{i+j} \frac{|G_{\underline{s}_{i}} ∩ G_{g\underline{t}_{j}}|}{|G|^{2}}[G_{\underline{s}_{i}} ∩ G_{g\underline{t}_{j}}, Res_{G_{\underline{s}_{i}}}^{G_{s_{0}}}(𝔩_{s_{0}}) Res_{G_{g\underline{t}_{j}}}^{G_{gt_{0}}}(𝔪_{gt_{0}})]_{G}

= ∑_{\begin{matrix}i,j∈ℕ \\ \underline{s}_{i}∈X \\ \underline{t}_{j}∈Y \end{matrix}}(−1)^{i+j} \frac{|G_{\underline{s}_{i}} ∩ G_{\underline{t}_{j}}|}{|G|}[G_{\underline{s}_{i}} ∩ G_{\underline{t}_{j}}, Res_{G_{\underline{s}_{i}}}^{G_{s_{0}}}(𝔩_{s_{0}}) Res_{G_{\underline{t}_{j}}}^{G_{t_{0}}}(𝔪_{t_{0}})]_{G}.

Thus, Λ_{(X×Y,𝔩×𝔪)} = Λ_{(X,𝔩)}Λ_{(Y,𝔪)}. The first assertion of Lemma 10 tells us that every positive element of B_{C}(G) is

in of the form Λ_{(X,𝔩)} for some C-monomial G-poset (X, 𝔩). Now consider the poset X = {a, b, c, d, e} with the ordering {a ≤ c, a ≤ d, a ≤ e, b ≤ c, b ≤ d, b ≤ e}. Consider trivial G-action on X. Then Λ_{τ_{G}(X)} = −1_{B_{C}(G)}. So as a consequence of Lemma 10 we get the following corollary.

Corollary 11. Any element of the monomial Burnside ring can be expressed as the Lefschetz invariant of some (non unique) monomial G-poset.

Proposition 12. Let H be a subgroup of G. Given a C-monomial H-poset (X, 𝔩), we have

Ind_{H}^{G}(Λ_{(X,𝔩)}) = Λ_{Ind_{H}^{G}(X,𝔩)}.

Proof. Since

Ind_{H}^{G}(Λ_{(X,𝔩)}) = ∑_{n∈ℕ}(−1)^{n} Ind_{H}^{G}( Sd_{n}(X), 𝔩_{n}),

we need to show that there exists a C-monomial G-set isomorphism between

(G ×_{H} Sd_{n}(X), G ×_{H} 𝔩_{n})

and

( Sd_{n}(G ×_{H} X), (G ×_{H} 𝔩)_{n})

for any n ∈ ℕ.

We define

(f_{n}, id) : (G ×_{H} Sd_{n}(X), G ×_{H} 𝔩_{n}) → ( Sd_{n}(G ×_{H} X), (G ×_{H} 𝔩)_{n})

where

f_{n} : G ×_{H} Sd_{n}(X) → Sd_{n}(G ×_{H} X)

such that

f_{n}(u,_{H} x_{0} < ... < x_{n}) = ((u,_{H} x_{0}) < ... < (u,_{H} x_{n}))

for any chain (u,_{H} x_{0} < ... < x_{n}) in G ×_{H} Sd_{n}(X).

Let (u_{0},_{H} x_{0}) < ... < (u_{n},_{H} x_{n}) be a chain in Sd_{n}(G ×_{H} X). There exist some h_{i} ∈ H such that u_{i}h_{i} = u_{i+1} and h_{i}^{−1}x_{i} < x_{i+1} for all 0 ≤ i ≤ n − 1. Then

f_{n}(u_{0},_{H} x_{0} < h_{0}x_{1} < ... < h_{0}...h_{n−1}x_{n}) = (u_{0},_{H} x_{0}) < ... < (u_{n},_{H} x_{n}).

Obviously, f_{n} is a map of G-sets and injective.

Now, we show that G ×_{H} 𝔩_{n} = (G ×_{H} 𝔩)_{n} ∘ f_{n}. We consider an element k ∈ G, and chains (u,_{H} x_{0} < ... < x_{n}) in G ×_{H} Sd_{n}(X) such that

k(u,_{H} x_{0} < ... < x_{n}) = (v,_{H} y_{0} < ... < y_{n}).

There exists some h ∈ H such that ku = vh and hx_{i} = y_{i} for all 0 ≤ i ≤ n. Then

(G ×_{H} 𝔩)_{n}(k, f_{n}(u,_{H} x_{0} < ... < x_{n}), f_{n}(v,_{H} y_{0} < ... < y_{n}))
= (G ×_{H} 𝔩)_{n}(k, (u,_{H} x_{0}) < ... < (u,_{H} x_{n}), (v,_{H} y_{0}) < ... < (v,_{H} y_{n}))
= 𝔩_{n}(h, x_{0} < ... < x_{n}, y_{0} < ... < y_{n})
= (G ×_{H} 𝔩_{n})(k, (u,_{H} x_{0} < ... < x_{n}), (v,_{H} y_{0} < ... < y_{n})).

Let (X, 𝔩) be a G-poset and let x ∈ X. Then the pairs (]x, ·[_{X}, 𝔩_{>x}) and (]·, x[_{X}, 𝔩^{<x}) are C-monomial G_{x}-posets where

]x, ·[_{X}= {y ∈ X | x < y}, ]·, x[_{X}= {y ∈ X | y < x}

which are G_{x}-posets and 𝔩_{>x} : { \atop ]x, ·[_{X}} → •_{C} and 𝔩^{<x} : { \atop ]·, x[_{X}} → •_{C} are the restrictions of the functor 𝔩.

Lemma 13. Let (X, 𝔩) be a monomial G-poset. We have

Λ_{(X,𝔩)} = − ∑_{x∈[G/X]}Ind_{G_{x}}^{G}([G_{x}, 𝔩_{x}]_{G_{x}} · Λ̃_{]x,·[_{X}}).

Proof.

Λ_{(X,𝔩)} = ∑_{x_{0}<...<x_{n}∈_{G}X} (−1)^{n}[G_{x_{0},...,x_{n}}, Res_{G_{x_{0},...,xn}}^{G_{x_{0}}}(𝔩_{x_{0}})]_{G}

= ∑_{x_{0}∈_{G}X} ∑_{x_{1}<...<x_{n}∈_{G}X:x_{0}<x_{1}} (−1)^{n}[G_{x_{0},...,x_{n}}, Res_{G_{x_{0},...,xn}}^{G_{x_{0}}}(𝔩_{x_{0}})]_{G}

= ∑_{x_{0}∈_{G}X} Ind_{G_{x_{0}}}^{G} ∑_{x_{1}<...<x_{n}∈_{Gx_{0}} ]x_{0},·[_{X}} (−1)^{n}[G_{x_{0},...,x_{n}}, Res_{G_{x_{0},...,xn}}^{G_{x_{0}}}(𝔩_{x_{0}})]_{G_{x_{0}}}

= ∑_{x_{0}∈_{G}X}Ind_{G_{x_{0}}}^{G}[G_{x_{0}}, 𝔩_{x_{0}}]_{G_{x_{0}}} ∑_{x_{1}<...<x_{n}∈_{_{Gx_{0}}_{]x_{0},·[_{X}}}}(−1)^{n}[G_{x_{0},...,x_{n}}, 1_{G_{x_{0},..,xn}}]_{G_{x_{0}}}

= − ∑_{x∈_{G}X}Ind_{G_{x}}^{G}([G_{x}, 𝔩_{x}]_{G_{x}} · Λ̃_{]x,·[_{X}}).

Remark 14. We can define the opposite of a C-monomial G-poset (X, 𝔩) as follows. We consider the pair (X^{op}, 𝔩^{op}) where X^{op} is the opposite G-poset with the order ≤^{op} defined by

∀x, y ∈ X, g ∈ G, gx ≤^{op} y ⇔ y ≤ gx

and 𝔩^{op} : X̂^{op} → •_{C} is defined by

𝔩^{op}(g, x, y) = 𝔩^{−1}(g^{−1}, y, x)

for any x, y ∈ X^{op} and g ∈ G such that gx ≤^{op} y. Obviously, the pair (X^{op}, 𝔩^{op}) is a C-monomial G-poset. Moreover the assignment (X, 𝔩) ↦ (X^{op}, 𝔩^{op}) is a functor CMG-poset → _{C}MG-poset: if (f, λ) : (X, 𝔩) → (Y, 𝔪) is a map of C-monomial Gposets, then f : X^{op} → Y ^{op} is a map of G-posets and for any gx ≤^{op} x′, we get the commutative diagram

\xymatrix{𝔩(x) \ar[r]^{λ_{x}} \ar[d]_{𝔩^{op}(g,x,x′)} & 𝔪 ∘ f(x) \ar[d]^{𝔪^{op}(g,f(x),f(x′))} \\ 𝔩(x′) \ar[r]_{λ_{x′}} & 𝔪 ∘ f(x′).}

Observe that (𝔩^{op})_{x}(g) = 𝔩^{−1}(g^{−1}, x, x) = 𝔩(g, x, x) = 𝔩_{x}(g), for any x ∈ X and g ∈ G_{x}. It follows that Λ_{(X,𝔩)} = Λ_{(X^{op},𝔩^{op})}.

Let (f, λ) : (X, 𝔩) → (Y, 𝔪) be a map of C-monomial G-posets. Given y ∈ Y, following [3] we set

f^{y} = {x ∈ X | f(x) ≤ y}, f_{y} = {x ∈ X | f(x) ≥ y}

which are both G_{y}-posets. We denote by (f^{y}, 𝔩_{|f^{y}}) the C-monomial G_{y}-poset where 𝔩_{|f^{y}} : f̂^{y} → •_{C} is the restriction of the functor 𝔩. Similarly, we denote by (f_{y}, 𝔩_{|f_{y}}) to be C-monomial G_{y}-poset where 𝔩_{|f_{y}} : f̂_{y} → •_{C} is the restriction of the functor 𝔩. Example 15. Let (f, λ) : (X, 𝔩) → (Y, 𝔪) be a map of C-monomial G-posets. We define a G-poset X ∗_{f,λ} Y with underlying G-set X ⊔ Y as follows: for z, z′ ∈ X ⊔ Y, we set

z ≤ z′ ⇔ \begin{cases}z, z′ ∈ X & and z ≤ z′ ∈ X \\ z, z′ ∈ Y & and z ≤ z′ ∈ Y \\ z ∈ X, z′ ∈ Y & and f(z) ≤ z′ ∈ Y \end{cases} .

We define the functor 𝔩 ∗_{f,λ} 𝔪 : { \atop X ⊔ Y} → •_{C} by

(𝔩 ∗_{f,λ} 𝔪)(g, z, z′) = \begin{cases}𝔩(g, z, z′) & if z, z′ ∈ X \\ 𝔪(g, z, z′) & if z, z′ ∈ Y \\ 𝔪(g, f(z), z′)λ_{z} & if z ∈ X, z′ ∈ Y. \end{cases} .

for any z, z′ ∈ X ∗_{f,λ} Y and g ∈ G such that gz ≤ z′.

Now let z_{1}, z_{2}, z_{3} ∈ X ∗_{f,λ} Y and g, g′ ∈ G such that gz_{1} ≤ z_{2} and g′z_{2} ≤ z_{3}. We aim to show that

(𝔩 ∗_{f,λ} 𝔪)(g′g, z_{1}, z_{3}) = (𝔩 ∗_{f,λ} 𝔪)(g′, z_{2}, z_{3})(𝔩 ∗_{f,λ} 𝔪)(g, z_{1}, z_{2}).

We have four cases to consider:

• z_{1}, z_{2}, z_{3} ∈ X

• z_{1}, z_{2} ∈ X and z_{3} ∈ Y

• z_{1} ∈ X and z_{2}, z_{3} ∈ Y

• z_{1}, z_{2}, z_{3} ∈ Y. In the first case we get

(𝔩 ∗_{f,λ} 𝔪)(g′g, z_{1}, z_{3}) = 𝔩(g′g, z_{1}, z_{3}) = 𝔩(g′, z_{2}, z_{3})𝔩(g, z_{1}, z_{2})

= (𝔩 ∗_{f,λ} 𝔪)(g′, z_{2}, z_{3})(𝔩 ∗_{f,λ} 𝔪)(g, z_{1}, z_{2}).

In the second case, using the naturality of λ we get

(𝔩 ∗_{f,λ} 𝔪)(g′g, z_{1}, z_{3}) = 𝔪(g′g, f(z_{1}), z_{3})λ_{z_{1}} = 𝔪(g′, f(z_{2}), z_{3})𝔪(g, f(z_{1}), f(z_{2}))λ_{z_{1}}

= 𝔩(g, z_{1}, z_{2})𝔪(g′, f(z_{2}), z_{3})λ_{z_{2}} = (𝔩 ∗_{f,λ} 𝔪)(g′, z_{2}, z_{3})(𝔩 ∗_{f,λ} 𝔪)(g, z_{1}, z_{2}).

In the third case, we get

(𝔩 ∗_{f,λ} 𝔪)(g′g, z_{1}, z_{3}) = 𝔪(g′g, f(z_{1}), z_{3})λ_{z_{1}} = 𝔪(g′, f(z_{2}), z_{3})𝔪(g, f(z_{1}), f(z_{2}))λ_{z_{1}}

= (𝔩 ∗_{f,λ} 𝔪)(g, z_{1}, z_{2})(𝔩 ∗_{f,λ} 𝔪)(g′, z_{2}, z_{3}).

In the fourth case

(𝔩 ∗_{f,λ} 𝔪)(g′g, z_{1}, z_{3}) = 𝔪(g′g, z_{1}, z_{3}) = 𝔪(g′, z_{2}, z_{3})𝔪(g, z_{1}, z_{2})
= (𝔩 ∗_{f,λ} 𝔪)(g′, z_{2}, z_{3})(𝔩 ∗_{f,λ} 𝔪)(g, z_{1}, z_{2}).

Let z ∈ X ∗_{f,λ} Y then obviously we have (𝔩 ∗_{f,λ} 𝔪)(1, z, z) = 1. Thus, (X ∗_{f,λ} Y, 𝔩 ∗_{f,λ} 𝔪) is a C-monomial G-poset.

Lemma 16. Let (f, λ) : (X, 𝔩) → (Y, 𝔪) be a map of C-monomial G-posets. Then Λ_{(X∗_{f,λ}Y,𝔩∗_{f,λ}𝔪)} = Λ_{(Y,𝔪)}.

Proof. 1. Let z ∈ Z = X ∗_{f,λ} Y. If z ∈ X consider the map g : ]z, ·[_{Z}→ [f(z), ·[_{Y} defined by

g(t) = \begin{cases}f(t) & if t ∈ X \\ t & if t ∈ Y \end{cases} .

Let g′ : [f(z), ·[ → ]z, ·[ defined by g′(s) = s. Then g and g′ are maps of G_{z}-posets such that g ∘ g′ = Id and Id ≤ g′ ∘ g. So if z ∈ X using [[3], Lemma 4.2.4 and Proposition 4.2.5], we get Λ̃_{]z,·[} = Λ̃_{[f(z),·[} = 0. Thus,

Λ_{(X∗_{f,λ}Y,𝔩∗_{f,λ}𝔪)} = − ∑_{z∈[G\X∗_{f,λ}Y]} Ind_{G_{z}}^{G}([G_{z}, 𝔩_{z}]_{G_{z}} · Λ̃_{]z,·[})

= − ∑_{y∈[G\Y]}Ind_{G_{y}}^{G}([G_{y}, 𝔩_{y}]_{G_{y}} · Λ̃_{]y,·[}) = Λ_{(Y,𝔪)}.

As a consequence, we give an analogue of Proposition 4.2.7. in [3], which in turn was inspired by a much deeper theorem of Quillen in [6].

Proposition 17. Let (f, λ) : (X, 𝔩) → (Y, 𝔪) be a map of C-monomial G-posets. Then in B_{C}(G)

Λ̃_{(Y,𝔪)} = Λ̃_{(X,𝔩)} +∑_{y∈G\Y}Ind_{G_{y}}^{G}(Λ̃_{f^{y}} Λ̃_{(]y,·[_{Y} ,𝔪^{>y})}).

Λ̃_{(Y,𝔪)} = Λ̃_{(X,𝔩)} +∑_{y∈G\Y}Ind_{G_{y}}^{G}(Λ̃_{f_{y}} Λ̃_{(]·,y[_{Y} ,𝔪_{<y})}).

Proof. We follow the proof of Proposition 4.2.7 in [3]. For any n ∈ ℕ, any chain z = z_{0} < ... < z_{n} ∈ Sd_{n}(X ∗_{f,λ} Y) can be of two types, depending on z_{n} ∈ X or z_{n} ∈ Y. For a sequence z of the first type we get

Res_{G_{z_{0},...,zn}}^{G_{zn}}(𝔩 ∗_{f,λ} 𝔪)_{n}_{z_{n}} = Res_{G_{z_{0},...,zn}}^{G_{zn}}𝔩_{n}.

Now a sequence z of the second type has a smallest element y = z_{i} in Y, thus, we can write the sequence as

x_{0} < ... < x_{i−1} < y < y_{0} < ... < y_{n−i−1}

such that x_{0} < ... < x_{i−1} is in Sd_{i−1}(f^{y}), and y_{0} < ... < y_{n−i−1} is in Sd_{n−i−1}(]y, ·[_{Y}). We get

Res_{G_{z_{0},...,zn}}^{G_{zn}}(𝔩 ∗_{f,λ} 𝔪)_{n}_{z_{n}} = Res_{G_{z_{0},...,zn}}^{G_{y}}(𝔪).

Let \underline{x}_{i−1} denote the chain x_{0} < ... < x_{i−1} and \underline{y}_{n−i−1} denote the chain y_{0} < ... < y_{n−i−1}. Then, by Lemma 10 and Lemma 16 we get

Λ_{(Y,𝔪)} = Λ_{(X∗_{f,λ}Y,𝔩∗_{f,λ}𝔪)} = ∑_{n∈ℕ}(−1)^{n}( Sd_{n}(X ∗_{f,λ} Y), (𝔩 ∗_{f,λ} 𝔪)_{n})

= ∑_{\begin{matrix}n∈ℕ \\ z_{0}<...<z_{n}∈ Sd_{n}(𝔩∗_{f,λ}𝔪) \end{matrix}} (−1)^{n}[G_{z_{0},...,z_{n}}, Res_{G_{z_{0},...,zn}}^{G_{zn}}(𝔩 ∗_{f,λ} 𝔪)_{n}]_{G}

= ∑_{n∈ℕ}(−1)^{n}( Sd_{n}(X), 𝔩_{n})

+ ∑_{y∈[G\Y]}Ind_{G_{y}}^{G} ∑_{i=0}^{n} ∑_{\begin{matrix}\underline{x}_{i−1}∈ Sd_{i−1}(f^{y}) \\ \underline{y}_{n−i−1}∈ Sd_{n−i−1}(]y,·[_{Gy}) \end{matrix}} [G_{\underline{x}_{i−1},y,\underline{y}_{n−i−1}}, Res_{G_{\underline{x}_{i−1},y,\underline{y}_{n−i−1}}}^{G_{y}}𝔪_{y}]_{G_{y}}

= Λ_{(X,𝔩)} + ∑_{y∈[G\Y]}Ind_{G_{y}}^{G}(Λ̃_{(f^{y},1_{f^{y}})}Λ̃_{(]y,·[_{Y} ,𝔪_{>y})}).

For the second assertion we consider the opposite map

(f, λ) : (X^{op}, 𝔩^{op}) → (Y ^{op}, 𝔪^{op})

Since we have Λ_{(X,𝔩)} = Λ_{(X^{op},𝔩^{op})} by Remark 14, the result follows.

Corollary 18. Let (f, λ) : (X, 𝔩) → (Y, 𝔪) be a map of C-monomial G-posets. If Λ_{f^{y}} = 0 for all y ∈ Y (resp. if Λ_{f_{y}} = 0 for all y ∈ Y), then Λ_{X,𝔩} = Λ_{Y,𝔪}.

Remark 19. The assumption of this corollary is fulfilled in particular if f : X̂ → Ŷ admits a right adjoint g, in other words if there exists a map of posets g : Y → X such that f(x) ≤ y ⇔ x ≤ g(y) for any x ∈ X and y ∈ Y, i.e. equivalently if f ∘ g(y) ≤ y and g ∘ f(x) ≤ x for any x ∈ X and any y ∈ Y.

Now we set some notation. Given a C-monomial G-set (X, 𝔩), we can rewrite its Lefschetz invariant as

Λ_{(X,𝔩)} = ∑_{x_{0}<...<x_{n}∈_{G}X} (−1)^{n}[G_{x_{0},...x_{n}}, Res_{G_{x_{0},...,xn}}^{G_{x_{0}}}(𝔩_{x_{0}})]_{G}

= ∑_{(V,ν)∈_{G} ch(G)} γ_{V,ν}^{X,𝔩}[V, ν]_{G}

where

γ_{V,ν}^{X,𝔩} = ∑_{\begin{matrix}x_{0}<...<x_{n}∈_{G}X \\ (G_{x_{0},...,xn}, Res_{Gx_{0},...,x_{n}}^{Gx_{0}} 𝔩_{x_{0}})=_{G}(V,ν) \end{matrix}} (−1)^{n} = \frac{1}{|N_{G}(V, ν) : V |} ∑_{\begin{matrix}x_{0}<...<x_{n}∈X \\ (G_{x_{0},...,xn}, Res_{Gx_{0},...,x_{n}}^{Gx_{0}} 𝔩_{x_{0}})=(V,ν) \end{matrix}} (−1)^{n}.

Given a C-monomial G-poset (X, 𝔩) we let the set (X, 𝔩)^{U,μ} to be

(X, 𝔩)^{U,μ} = {x ∈ X^{U} | Res_{U}^{G_{x}}𝔩_{x} = μ}

where (U, μ) is a subcharacter of G. Then given a C-subcharacter (U, μ) ∈ ch(G) we have

χ((X, 𝔩)^{U,μ}) = ∑_{\begin{matrix}n∈ℕ \\ x_{0}<...<x_{n}∈X^{U} \\ Res_{Gx_{0},...,x_{n}}^{Gx_{0}} 𝔩_{x_{0}}=μ \end{matrix}} (−1)^{n} = ∑_{\begin{matrix}(V,ν)∈ ch(G) \\ U⊆V \\ Res_{U}^{V} ν=μ \end{matrix}}m_{V,ν}^{X,𝔩}

where

m_{V,ν}^{X,𝔩} = ∑_{\begin{matrix}n∈ℕ \\ x_{0}<...<x_{n}∈X \\ (G_{x_{0},...,xn}, Res_{Gx_{0},...,x_{n}}^{Gx_{0}} 𝔩_{x_{0}})=(V,ν) \end{matrix}} (−1)^{n}.

Now |N_{G}(V, ν) : V |m_{V,ν}^{X,𝔩} = γ_{V,ν}^{X,𝔩}. Using this fact we prove the following lemma.

Lemma 20. Let (X, 𝔩) and (Y, 𝔪) be C-monomial G-posets then Λ_{(X,𝔩)} = Λ_{(Y,𝔪)} if and only if χ((X, 𝔩)^{U,μ}) = χ((Y, 𝔪)^{U,μ}) for every C-subcharacter (U, μ) of G.

Proof. Assume Λ_{(X,𝔩)} = Λ_{(Y,𝔪)}. Then

∑_{(V,ν)∈_{G} ch(G)} γ_{V,ν}^{X,𝔩}[V, ν]_{G} = ∑_{(V,ν)∈_{G} ch(G)} γ_{V,ν}^{Y,𝔪}[V, ν]_{G}

∑_{(V,ν)∈_{G} ch(G)} (γ_{V,ν}^{X,𝔩} − γ_{V,ν}^{Y,𝔪})[V, ν]_{G} = 0.

So γ_{V,ν}^{X,𝔩} = γ_{V,ν}^{Y,𝔪} and then m_{V,nu}^{X,𝔩} = m_{V,nu}^{Y,𝔪} for every C-subcharacter (V, ν) of G. We get

∑_{(U,μ)≤(V,ν)∈_{G} ch(G)} m_{V,ν}^{X,𝔩} = ∑_{(U,μ)≤(V,ν)∈_{G} ch(G)} m_{V,ν}^{Y,𝔪}.

Thus, χ((X, 𝔩)^{U,μ}) = χ((Y, 𝔪)^{U,μ}) for every C-subcharacter (U, μ) of G.

Conversely, assume that χ((X, 𝔩)^{U,μ}) = χ((Y, 𝔪)^{U,μ}) for every C-subcharacter (U, μ) of G. Then

∑_{(U,μ)≤(V,ν)∈ ch(G)} m_{V,ν}^{X,𝔩} = ∑_{(U,μ)≤(V,ν)∈ ch(G)} m_{V,ν}^{Y,𝔪},

∑_{(U,μ)≤(V,ν)∈ ch(G)} (m_{V,ν}^{X,𝔩} − m_{V,ν}^{Y,𝔪}) = 0.

Let z be the matrix with the coefficients

z(U, μ; V, ν) = ⌊(U, μ) ≤ (V, ν)⌋ = \begin{cases}1 & if (U, μ) ≤ (V, ν) \\ 0 & otherwise \end{cases}

for any C-subcharacters (U, μ), (V, ν). If we list the C-subcharacters in non-decreasing order of size of the subgroups, the matrix z is upper triangular with nonzero diagonal coefficients. Thus, z is nonsingular and so m_{V,ν}^{X,𝔩} = m_{V,ν}^{Y,𝔪}. This implies γ_{V,ν}^{X,𝔩} = γ_{V,ν}^{Y,𝔪}. We get

Λ_{(X,𝔩)} = ∑_{(V,ν)∈_{G} ch(G)} γ_{V,ν}^{X,𝔩}[V, ν]_{G} = ∑_{(V,ν)∈_{G} ch(G)} γ_{V,ν}^{Y,𝔪}[V, ν]_{G} = Λ_{(Y,𝔪)}.

This proves the lemma.

3 Generalized tensor induction

Let G and H be finite groups. A set U is a (G, H)-biset if U is a left G-set and right H-set such that the G-action and the H-action commute. Any (G, H)-biset U is a left (G × H)-set with the following action:

∀u ∈ U, (g, h) ∈ G × H (g, h) · u = guh^{−1}.

A C-monomial (G×H)-set (U, λ) will be called a C-monomial (G, H)-biset, and usually denoted by U_{λ} for simplicity.

Now let U_{λ} be a C-monomial (G × H)-set and u, u′ ∈ U. Then the set of morphisms from u to u′ in Û is

HomÛ(u, u′) = {(g, h) ∈ G × H | gu = u′h}.

If (g, h) ∈ HomÛ(u, u′), we denote the image of (g, h) under λ by λ(g, h, u, u′).

Let U_{λ} be a C-monomial (G, H)-biset and V_{ρ} be a C-monomial (H, K)-biset. Consider the set

U_{λ} ∘ V_{ρ} = {(u, v) ∈ U × V | ∀h ∈ H_{u} ∩ H_{v}, λ(1, h, u, u)ρ(h, 1, v, v) = 1}.

The set U_{λ} ∘ V_{ρ} is an H-set with the action

∀(u, v) ∈ U_{λ} ∘ V_{ρ}, ∀h ∈ H, h(u, v) = (uh^{−1}, hv).

Indeed, the condition that we impose on U_{λ} ∘ V_{ρ} amounts to saying that given (u, v) ∈ U_{λ} ∘ V_{ρ}, the linear character ξ_{u,v} : h ↦ λ(1, h, u, u)ρ(h, 1, v, v) of H_{u} ∩ H_{v} is trivial. Moreover we have ξ_{ux,x^{−1}v}(h) = ξ_{u,v}(xhx^{−1}) = 1 for x ∈ H and h ∈ H_{ux} ∩ H_{x^{−1}v}, i.e. xhx^{−1} ∈ H_{u} ∩ H_{v}.

We let U_{λ} ∘_{H} V_{ρ} denote the set of H-orbits on U_{λ} ∘ V_{ρ} and (u,_{H} v) denote the H-orbit containing (u, v). The set U_{λ} ∘_{H} V_{ρ} is (G, K)-biset with the action

(u,_{H} v) ∈ U_{λ} ∘_{H} V_{ρ}, (g, k) ∈ G × K, g(u,_{H} v)k = (gu,_{H} vk).

We obtain a C-monomial (G, K)-biset (U_{λ} ∘_{H} V_{ρ}, λ×ρ), where λ×ρ is defined as follows: if (u,_{H} v) and (u′,_{H} v′) ∈ U_{λ} ∘_{H} V_{ρ} and (g, k) ∈ G × K are such that g(u,_{H} v) = (u′,_{H} v′)k, then there exists h ∈ H such that gu = u′h and hv = v′k. This element h need not be unique, but it is well defined up to multiplication on the right by an element of H_{u} ∩ H_{v}. We set

(λ × ρ)(g, k, (u,_{H} v), (u′,_{H} v′) = λ(g, h, u, u′)ρ(h, k, v, v′),

which does not depend on the choice of h, by the defining property of U_{λ} ∘ V_{ρ}. Note that U_{λ} ∘_{H} V_{ρ} = U ×_{H} V when V is a left free (H, K)-biset, or when λ and ρ are both equal to the trivial functor.

Given a C-monomial G-poset (X, 𝔩), we let t_{U,λ}(X, 𝔩) be the set of G-equivariant maps f : U → X such that

𝔩(g, f(u), f(u)) = λ(g, 1, u, u)

for all u ∈ U and g ∈ G_{u}. Then t_{U,λ}(X, 𝔩) is an H-poset with the action (hf)(u) = f(uh), for any h ∈ H, for any f ∈ t_{U,λ}(X, 𝔩), for any u ∈ U. The order ≤ is given as follows:

∀f, f′ ∈ t_{U,λ}(X, 𝔩), f ≤ f′ ⇔ ∀u ∈ U, f(u) ≤ f′(u) in X.

Now we define a functor 𝔏_{U,λ} : { \atop t_{U,λ}(X, 𝔩)} → •_{C}. Let f, f′ ∈ t_{U,λ}(X, 𝔩) and h ∈ H

such that hf ≤ f′. We choose a set [G\U] of representatives of G-orbits of U. Then for all u ∈ U there exist some g_{h,u} ∈ G and a unique σ_{h}(u) ∈ [G\U] such that

uh = g_{h,u}σ_{h}(u).

Since hf ≤ f′, we get g_{h,u}f(σ_{h}(u)) ≤ f′(u), and we set

𝔏_{U,λ}(h, f, f′) = ∏_{u∈[G\U]} 𝔩(g_{h,u}, f(σ_{h}(u)), f′(u))λ^{−1}(g_{h,u}, h, σ_{h}(u), u).

Now we show that this definition does not depend on the choice of g_{h,u}. Assume that there exist g_{h,u}, g′_{h,u} ∈ G such that

uh = g_{h,u}σ_{h}(u) = g′_{h,u}σ_{h}(u).

So there exists w ∈ G_{σ_{h}(u)} such that g_{h,u} = g′_{h,u}w. We get

𝔩(w, f(σ_{h}(u)), f(σ_{h}(u))) = λ(w, 1, σ_{h}(u), σ_{h}(u)).

Furthermore, we get the following commutative diagram:

\xymatrix{σ_{h}(u) \ar[rr]^{w} \ar[dr]_{g_{h,u}} &  & σ_{h}(u) \ar[dl]^{g′_{h,u}} \\  & u}

Thus,

𝔏_{U,λ}(h, f, f′) = ∏_{u∈[G\U]} 𝔩(g_{h,u}, f(σ_{h}(u)), f′(u))λ^{−1}(g_{h,u}, h, σ_{h}(u), u)

= ∏_{u∈[G\U]} 𝔩(g′_{h,u}w, f(σ_{h}(u)), f′(u))λ^{−1}(g′_{h,u}w, h, σ_{h}(u), u)

= ∏_{u∈[G\U]} 𝔩(g′_{h,u}, f(σ_{h}(u)), f′(u))λ^{−1}(g′_{h,u}, h, σ_{h}(u), u).

Definition 21. The above construction T_{U,λ} : (X, 𝔩) ↦ (t_{U,λ}(X, 𝔩), 𝔏_{U,λ}) is called the generalized tensor induction for C-monomial G-posets, associated to (U, λ).

Lemma 22. Let G and K be finite groups and U be a (G, K)-biset. Then there exists a bijection between the sets {(u, t) | u ∈ [G\U/K], t ∈ [(K ∩ G^{u})\K]} and [G\U].

Proof. Let u ∈ [G\U/K] and t ∈ [(K ∩ G^{u})\K] then there exist some g_{t,u} ∈ G and a unique σ_{t}(u) ∈ [G\U] such that

ut = g_{t,u}σ_{t}(u).

We define ψ : {(u, t) | u ∈ [G\U/K], t ∈ [(K ∩ G^{u})\K]} → [G\U] by ψ(u, t) = σ_{t}(u).

Lemma 23. Let G and H be finite groups, (U, λ) be a monomial (G, H)-biset and (X, 𝔩) be a C-monomial G-poset.

1. (t_{U,λ}(X, 𝔩), 𝔏_{U,λ}) is a C-monomial H-poset.

2. (t_{U,λ}(X, 𝔩), 𝔏_{U,λ}) does not depend on the choice of representative set [G\U], up to isomorphism.

Proof. 1. We show that 𝔏_{U,λ} : { \atop t_{U,λ}(X, 𝔩)} → •_{C} is a functor. Let h, h′ ∈ H and f, f′, f′′ ∈ t_{U,λ}(X, 𝔩) such that hf ≤ f′ and h′f′ ≤ f′′. Let u ∈ [G\U]. Then there exist some g_{h,u}, g_{h′,u}, g_{h′h,u} in G and unique elements σ_{h}(u), σ_{h′}(u), σ_{h′h}(u) in [G\U] such that

uh = g_{h,u}σ_{h}(u), uh′ = g_{h′,u}σ_{h′}(u), uh′h = g_{h′h,u}σ_{h′h}(u).

Also there exist some g_{h,σ_{h′}(u)} ∈ G and a unique σ_{h}(σ_{h′}(u)) ∈ [G\U] such that

σ_{h′}(u)h = g_{h,σ_{h′}(u)}σ_{h}(σ_{h′}(u)).

Now we get

uh′h = g_{h′,u}g_{h,σ_{h′}(u)}σ_{h}(σ_{h′}(u))

and

σ_{h′h}(u) = σ_{h}(σ_{h′}(u)).

Then there exists w ∈ G_{σ_{h′h}(u)} such that

g_{h′h,u} = g_{h′,u}g_{h,σ_{h′}(u)}w.

We have the following commutative diagram:

\xymatrix{σ_{h′h}(u) \ar[rr]^{w} \ar[dr]_{g_{h′h,u}} &  & σ_{h′h}(u) \ar[dl]^{g_{h′,u}g_{h,σ_{h′} (u)}} \\  & uh′h}

On the other hand since w ∈ G_{σ_{h′h}(u)}, we get

𝔩(w, f(σ_{h′h}(u)), f(σ_{h′h}(u))) = λ(w, 1, σ_{h′h}(u), σ_{h′h}(u)).

Thus, setting L = 𝔏_{U,λ}(h′h, f, f′′), we have

L = ∏_{u∈[G\U]} 𝔩(g_{h′h,u}, f(σ_{h′h}(u)), f′′(u))λ^{−1}(g_{h′h,u}, h′h, σ_{h′h}(u), u)

= ∏_{u∈[G\U]} 𝔩(g_{h′,u}g_{h,σ_{h′}(u)}w, f(σ_{h′h}(u)), f′′(u))λ^{−1}(g_{h′,u}g_{h,σ_{h′}(u)}w, h′h, σ_{h′h}(u), u)

= ∏_{u∈[G\U]} 𝔩(g_{h′,u}g_{h,σ_{h′}(u)}, f(σ_{h′h}(u)), f′′(u))λ^{−1}(g_{h′,u}g_{h,σ_{h′}(u)}, h′h, σ_{h′h}(u), u)

= 𝔏(h′, f′, f′′)𝔏(h, f, f′).

Moreover, given f ∈ T_{U,λ}(X, 𝔩) we have

𝔏(1, f, f) = ∏_{u∈[G\U]} 𝔩(1, f(u), f(u))λ^{−1}(1, 1, u, u) = 1.

Thus, 𝔏_{U,λ} : { \atop t_{U,λ}(X, 𝔩)} → •_{C} is a functor.

2. Let h ∈ H and f, f′ ∈ t_{U,λ}(X, 𝔩) such that hf ≤ f′. Let S = [G\U] and let S′ be the another choice of representatives. If u′ ∈ S′ then there exist some a_{u} ∈ G, and a unique u ∈ S such that u′ = a_{u}u. Then there exist some g_{h,a_{u}u}, g_{h,u} ∈ G, a unique σ′_{h}(a_{u}u) ∈ S′, and a unique σ_{h}(u) ∈ S such that

a_{u}uh = g_{h,a_{u}u}σ′_{h}(a_{u}u)

and

uh = g_{h,u}σ_{h}(u).

Then

a_{u}uh = a_{u}g_{h,u}σ_{h}(u) = a_{u}g_{h,u}a_{σ_{h}(u)}^{−1}a_{σ_{h}(u)}σ_{h}(u).

So σ′_{h}(a_{u}u) = a_{σ_{h}(u)}σ_{h}(u). Note that a_{σ_{h}(u)}σ_{h}(u) ∈ S′. We get the following commutative diagram:

\xymatrix{a_{σ_{h}(u)}f(σ_{h}(u)) \ar[r]^{a_{u}g_{h,u}a_{σ_{h}(u)}^{−1}} \ar[d]_{a_{σ_{h}(u)}^{−1}} & a_{u}f′(u) \\ f(σ_{h}(u)) \ar[r]_{g_{h,u}} & f(u). \ar[u]_{a_{u}}}

Thus, setting L = 𝔏′_{U,λ}(h, f, f′), we have

L= ∏_{a_{u}u∈S′}𝔩(a_{u}g_{h,u}a_{σ_{h}(u)}^{−1}, f(a_{σ_{h}(u)}σ_{h}(u)), f′(a_{u}u))λ^{−1}(a_{u}g_{h,u}a_{σ_{h}(u)}^{−1}, h, a_{σ_{h}(u)}σ_{h}(u), a_{u}u)

= 𝔏′_{U,λ}(h, f, f′) = 𝔏_{U,λ}(h, f, f′)α_{f′}α_{f}^{−1}

where

α_{f′} = ∏_{u∈S} 𝔩(a_{u}, f′(u), a_{u}f′(u))λ^{−1}(a_{u}, 1, u, a_{u}u)

and

α_{f}^{−1} = ∏_{u∈S} 𝔩(a_{u}^{−1}, a_{u}f(u), f(u))λ^{−1}(a_{u}^{−1}, 1, a_{u}u, u).

Proposition 24. Let G and H be finite groups and (U, λ) be a C-monomial (G, H)-biset.

1. Let (X, 𝔩), (X′, 𝔩′) be C-monomial G-posets then T_{U,λ}((X, 𝔩) × (X′, 𝔩′)) ≅ T_{U,λ}(X, 𝔩) × T_{U,λ}(X′, 𝔩′).

2. T_{U,λ} : _{C}MG-poset → _{C}MH-poset is a functor. Proof. 1. is clear.

2. Let (φ, β) : (X, 𝔩) → (Y, 𝔪) be a map of C-monomial G-posets. We define a map

of C-monomial G-posets (T_{U,λ}(φ), T_{U,λ}(β)) : (t_{U,λ}(X, 𝔩), 𝔏) → (t_{U,λ}(Y, 𝔪), 𝔐) where

T_{U,λ}(φ) : t_{U,λ}(X, 𝔩) → t_{U,λ}(Y, 𝔪)

such that T_{U,λ}(φ)(f) = φ ∘ f and

T_{U,λ}(β) : 𝔏(f) → 𝔐 ∘ T_{U,λ}(φ)(f)

such that

T_{U,λ}(β) = ∏_{u∈[G\U]} β_{f(u)}

for any f ∈ t_{U,λ}(X, 𝔩). Clearly, φ ∘ f : U → X → Y is a map of G-posets. Since given g ∈ G_{u} and u ∈ U the map β : 𝔩 → 𝔪 ∘ φ is natural, we have the following commutative diagram:

\xymatrix{𝔩(f(u)) \ar[r]^{β_{f(u)}} \ar[d]_{𝔩(g,f(u),f(u))} & 𝔪 ∘ φ(f(u)) \ar[d]^{𝔪(g,φ∘f(u),φ∘f(u))} \\ 𝔩(f(u)) \ar[r]_{β_{f(u)}} & 𝔪 ∘ φ(f(u)).}

So

β_{f(u)}𝔩(g, f(u), f(u)) = 𝔪(g, φ(f(u)), φ(f(u)))β_{f(u)}.

Since g ∈ G_{f(u)}, we have

𝔩(g, f(u), f(u)) = λ(g, 1, u, u).

Then we get

𝔪(g, φ(f(u)), φ(f(u))) = λ(g, 1, u, u).

Thus, φ ∘ f ∈ t_{U,λ}(Y, 𝔪).

Now we show that

T_{U,λ}(β) : 𝔏 → 𝔐 ∘ T_{U,λ}(φ)

is a natural transformation. Let f, f′ ∈ t_{U,λ}(X, 𝔩) and h ∈ H such that hf ≤ f′. We show that the following diagram is commutative:

\xymatrix{𝔏(f) \ar[r]^{T_{U,λ}(β)_{f}} \ar[d]_{𝔏(h,f,f′)} & 𝔐 ∘ T_{U,λ}(φ)(f) \ar[d]^{𝔐(h,φ∘f,φ∘f′)} \\ 𝔏(f′) \ar[r]_{T_{U,λ}(β)_{f′}} & 𝔐 ∘ T_{U,λ}(φ)(f′).}

Let u ∈ [G\U]. Then there exist some g_{h,u} ∈ G and a unique σ_{h}(u) ∈ [G\U] such that

uh = g_{h,u}σ_{h}(u).

Since β : 𝔩 → 𝔪 ∘ φ is a natural transformation, we obtain the following commutative diagram :

\xymatrix{𝔩(f(σ_{h}(u))) \ar[r]^{β_{f(σ_{h}(u))}} \ar[d]_{𝔩(g_{h,u}, f(σ_{h}(u)), f′(u))} & 𝔪 ∘ φ(f(σ_{h}(u))) \ar[d]^{𝔪(g_{h,u}, φ(f(σ_{h}(u))), φ(f′(u)))} \\ 𝔩(f′(u)) \ar[r]_{β_{f′(u)}} & 𝔪 ∘ φ(f′(u)).}

Using the commutativity of the above diagram, and setting T = T_{U,λ}(β)_{f′} ∘𝔏(h, f, f′), we get

T = T_{U,λ}(β)_{f′}( ∏_{u∈[G\U]} 𝔩(g_{h,u}, f(σ_{h}(u)), f′(u))λ^{−1}(g_{h,u}, h, σ_{h}(u), u))

= ∏_{u∈[G\U]} β_{f′(u)}𝔩(g_{h,u}, f(σ_{h}(u)), f′(u))λ^{−1}(g_{h,u}, h, σ_{h}(u), u)

= ∏_{u∈[G\U]} 𝔪(g_{h,u}, φ(f(σ_{h}(u))), φ(f′(u)))λ^{−1}(g_{h,u}, h, σ_{h}(u), u)β_{f(σ_{h}(u))}

= 𝔐(h, φ ∘ f, φ ∘ f′)β_{f}.

So T_{U,λ}(β) : 𝔏 → 𝔐 ∘ T_{U,λ}(φ) is a natural transformation. Thus,

(T_{U,λ}(φ), T_{U,λ}(β)) : (t_{U,λ}(X, 𝔩), 𝔏) → (t_{U,λ}(Y, 𝔪), 𝔐)

is a map of C-monomial G-posets.

Lemma 25. Let G, H and K be finite groups. If U is a (G, H)-biset and V is a left free (H, K)-biset, then the map (u, v) ∈ U × V ↦ (u,_{H} v) ∈ U ×_{H} V restricts to a bijection π : [G\U] × [H\V] → [G\(U ×_{H} V)], where brackets denote sets of representatives of orbits.

Proof. For (u, v) ∈ U × V, there exists v_{0} ∈ [G\V] and h ∈ H such that v = hv_{0}. Then there exists u_{0} ∈ [G\U] and g ∈ G such that uh = gu_{0}. Then (u,_{H} v) = g(u_{0},_{H} v_{0}). Hence π is surjective. Now if (u_{0}, v_{0}) and (u_{1}, v_{1}) are pairs in [G\U] × [H\V] which lie in the same G-orbit, there exists g ∈ G and h ∈ H such that (gu_{0}, v_{0}) = (u_{1}h^{−1}, hv_{1}). Hence hv_{1} = v_{0}, so v_{0} = v_{1} = hv_{1}, and h = 1 since H act freely on V. Then gu_{0} = u_{1}, so u_{0} = u_{1}, and π is injective.

Proposition 26. Let G, H and K be finite groups.

1. Let (•, 1) be the C-monomial G-poset where • is G-poset with one element and 1 : • → •_{C} is the functor such that 1(g, •, •) = 1. Then T_{U,λ}(•, 1) = (•, 1).

2. Let (∅, z) be the empty C-monomial (G, H)-poset. Then T_{∅,z} is the constant functor with value (•, 1).

3. Let (U, λ) and (U′, λ′) be C-monomial (G, H)-bisets and let (X, 𝔩) be a C-monomial

G-poset then

T_{U⊔U′,λ⊔λ′}(X, 𝔩) = T_{U,λ}(X, 𝔩)T_{U′,λ′}(X, 𝔩).

4. Let id_{G} stand for the identity (G, G)-biset. Then T_{id_{G},1}(X, 𝔩) = (X, 𝔩) for any C-monomial G-poset (X, 𝔩).

5. Let (V, ρ) be a C-monomial left free (H, K)-biset, and (U, λ) be a C-monomial

(H, G)-biset. Then

T_{V,ρ} ∘ T_{U,λ} = T_{U×_{H}V,λ×ρ}.

Proof. 1., 2., 3. and 4. are clear.

5. Note that since V is left free, we have U_{λ} ∘_{H} V_{ρ} ≅ _{G}U ×_{H} V_{K}. Let (X, 𝔩) be a C-monomial G-poset. We need to show that

(t_{V,ρ}(t_{U,λ}(X, 𝔩), 𝔏_{U,λ}), 𝔏_{V,ρ} ∘ 𝔏_{U,λ}) = (t_{U×_{H}V,λ×ρ}(X, 𝔩), 𝔏_{U×_{H}V,λ×ρ}).

We define a K-poset map φ : t_{V,ρ}(t_{U,λ}(X, 𝔩), 𝔏_{U,λ}) → t_{U×_{H}V,λ×ρ}(X, 𝔩) such that

φ(f)(u,_{H}v) = f(v)(u)

for any f ∈ t_{V,ρ}(t_{U,λ}(X, 𝔩), 𝔏_{U,λ}) and (u,_{H}v) ∈ U ×_{H} V. It’s clear that the map φ(f) is a map of G-posets.

Let g ∈ G_{(u,_{H} v)}. Note that since V is H-free, we have g ∈ G_{u}. Then

𝔩(g, φ(f)(u,_{H} v), φ(f)(u,_{H} v)) = 𝔩(g, f(v)(u), f(v)(u))
= λ(g, 1, u, u)ρ(1, 1, v, v) = (λ × ρ)(g, 1, (u,_{H}v), (u,_{H}v)).

and so φ(f) ∈ t_{U×_{H}V,λ×ρ}(X, 𝔩).

Now we define a map

θ : t_{U×_{H}V,λ×ρ}(X, 𝔩) → t_{V,ρ}(t_{U,λ}(X, 𝔩), 𝔏_{U,λ})

such that θ(t)(v)(u) = t(u,_{H}v) for any t ∈ t_{U×_{H}V,λ×ρ}(X, 𝔩), u ∈ U and v ∈ V. We show that θ(t) ∈ t_{V,ρ}(t_{U,λ}(X, 𝔩), 𝔏_{U,λ}). Indeed, the map θ(t) is clearly a map of H-sets and moreover, since V is H-free, we have H_{v} = 1 for any v ∈ V. Then

𝔏_{U,λ}(1, θ(t)(v), θ(t)(v)) = 1 = ρ(1, 1, v, v).

Clearly, θ(t)(v) is a map of G-sets. Let g ∈ G_{u}. Then g ∈ G_{(u,_{H} v)}, and we get

𝔩(g, θ(t)(v)(u), θ(t)(v)(u)) = 𝔩(g, t(u,_{H}v), t(u,_{H}v))
= λ(g, 1, u, u)ρ(1, 1, v, v) = λ(g, 1, u, u).

So θ(t) ∈ t_{V,ρ}(t_{U,λ}(X, 𝔩), 𝔏_{U,λ}).

Now we show that 𝔏_{V,ρ}∘𝔏_{U,λ} = 𝔏_{U×_{H}V,λ×ρ}. Let k ∈ K and f, f′ ∈ t_{V,ρ}(t_{U,λ}(X, 𝔩), 𝔏_{U,λ}) such that kf ≤ f′. Let v ∈ [H\V]. Then there exist a unique σ_{k}(v) ∈ [H\V] and some h_{k,v} ∈ H such that

vk = h_{k,v}σ_{k}(v).

Let u ∈ [G\U]. Then there exist a unique σ_{h_{k,v}}(u) ∈ [G\U] and some g_{h_{k,v},u} ∈ G such that

uh_{k,v} = g_{h_{k,v},u}σ_{h_{k,v}}(u).

Then

(u,_{H}v) = (uh_{k,v}h_{k,v}^{−1},_{H} v) = (uh_{k,v},_{H} h_{k,v}^{−1}v)
= (g_{h_{k,v},u}σ_{h_{k,v}}(u),_{H} σ_{k}(v)k^{−1}) = g_{h_{k,v},u}(σ_{h_{k,v}}(u),_{H} σ_{k}(v))k^{−1}.

We get

(u,_{H}v)k = g_{h_{k,v},u}(σ_{h_{k,v}}(u),_{H} σ_{k}(v)).

Then

(σ_{h_{k,v}}(u),_{H} σ_{k}(v)) = σ_{k}(u,_{H}v)

and

g_{h_{k,v},u} = g_{k,(u,_{H} v)}w

for some w ∈ G_{σ_{k}(u,_{H} v)}. We get the following commutative diagram:

\xymatrix{σ_{k}(u,_{H}v) \ar[rr]^{w} \ar[dr]_{g_{h_{k,v},u}} &  & σ_{k}(u,_{H}v) \ar[dl]^{g_{k,(u,_{H} v)}} \\  & (u,_{H}v)k}

Using the commutativity of the above diagram and Lemma 25 we get

𝔏_{V,ρ} ∘ 𝔏_{U,λ}(k, f, f′) = ∏_{v∈[H\V]} 𝔏_{U,λ}(h_{k,v}, f(σ_{k}(v)), f′(v))ρ^{−1}(h_{k,v}, k, σ_{k}(v), v)

= ∏_{\begin{matrix}u∈[G\U] \\ v∈[H\V] \end{matrix}}𝔩(g_{h_{k,v},u}, f(σ_{k}(v))(σ_{h_{k,v}}(u)), f′(v)(u))λ^{−1}(g_{h_{k,v},u}, h_{k,v}, σ_{h_{k,v}}(u), u)ρ^{−1}(h_{k,v}, k, σ_{k}(v), v)

= ∏_{(u,_{H} v)∈[G\(U×_{H}V)]} 𝔩(g_{h_{k,v},u}, f(σ_{h_{k,v}}(u),_{H} σ_{k}(v)), f′(u,_{H}v))(λ×ρ)^{−1}(g_{h_{k,v},u}, k, (σ_{h_{k,v}}(u),_{H}σ_{k}(v)), (u,_{H}v))

= ∏_{(u,_{H} v)∈[G\(U×_{H}V)]} 𝔩(g_{k,(u,_{H} v)}w, f(σ_{k}(u,_{H}v)), f′(u,_{H}v))(λ×ρ)^{−1}(g_{k,(u,_{H} v)}w, k, (σ_{h_{k,v}}(u),_{H}σ_{k}(v)), (u,_{H}v))

= ∏_{(u,_{H} v)∈[G\(U×_{H}V)]} 𝔩( g_{k,(u,_{H} v)}, f(σ_{k}(u,_{H}v)), f′(u,_{H}v))(λ×ρ)^{−1}(g_{k,(u,_{H} v)}, k, (σ_{h_{k,v}}(u),_{H}σ_{k}(v)), (u,_{H}v))

= 𝔏_{U×_{H}V} (X, 𝔩).

Remark 27. The following example shows that the assumption that V is left free seems to be necessary for Assertion 5. Suppose that H = N ⋊ K is a semidirect product of a normal subgroup N with K. Let G be the group K, viewed as a subgroup of H. Let moreover U be the set H, viewed as a (G, H)-biset by left and right multiplication, and let V be the set K, acted on by K on the right by multiplication, and by H on the left by projection to K = H/N, followed by multiplication in K. Let moreover λ and ρ be equal to the trivial functor on Û and V̂, respectively.

Then U_{λ} ∘_{H} V_{ρ} = U ×_{H} V, as λ and ρ are both trivial. Moreover U ×_{H} V = G×_{H} K is equal to the identity (K, K)-biset (this makes sense since G = K), so T_{U×_{H}V,λ×ρ} = T_{id_{K},1} is the identity functor, by Assertion 4.

On the other hand G\U = K\(NK) ≅ N, and H\V has cardinality 1. So in the computation of the functor L_{U,1} appearing in T_{U,1}(X, 𝔩), we have a product of values of 𝔩, indexed by N. So the composition T_{V,1} ∘ T_{U,1} cannot act in general as the identity on (X, 𝔩), if N is non trivial. Hence T_{V,λ} ∘ T_{U,ρ} ≠ T_{U×_{H}V,λ×ρ} in this situation.

Remark 28. Let G and H be finite groups, and U be a (finite) (G, H)-biset. Then one can check that the diagram

\xymatrix{G-poset \ar[r]^{τ_{G}} \ar[d]^{T_{U}} & _{C}MG-poset \ar[d]^{T_{U,1_{U}}} \\ H-poset \ar[r]^{τ_{H}} & _{C}MH-poset}

of categories and functors is commutative, up to isomorphism, where the functor T_{U} on the left is the usual generalized tensor induction functor for G-posets.

Lemma 29. Let G and H be finite groups, and let (U, λ) be a C-monomial (G, H)-biset. Then there exists a unique map

T_{U,λ} : B_{C}(G) → B_{C}(H)

such that T_{U,λ}(Λ_{(X,𝔩)}) = Λ_{T_{U,λ}(X,𝔩)} for any finite C-monomial G-poset (X, 𝔩).

Proof. We show that if (X, 𝔩) and (Y, 𝔪) are finite C-monomial G-posets such that if Λ_{(X,𝔩)} = Λ_{(Y,𝔪)} in B_{C}(G), then Λ_{T_{U,λ}(X,𝔩)} = Λ_{T_{U,λ}(Y,𝔪)} in B_{C}(H). So it’s enough to show

that χ(T_{U,λ}((X, 𝔩))^{K,θ}) = χ(T_{U,λ}((Y, 𝔪))^{K,θ}), by Lemma 20 for any (K, θ) of ch(G).

Let u ∈ [G\U/K], k ∈ K and t ∈ [K ∩ G^{u}\K] then there exist a unique σ_{k}(ut) ∈ [G\U] and some g_{k,ut} ∈ G such that

utk = g_{k,ut}σ_{k}(ut).

Also there exist some c_{k,t} ∈ K ∩ G^{u} and a unique τ_{k}(t) ∈ [K ∩ G^{u}\K] such that

tk = c_{k,t}τ_{k}(t).

Since c_{k,t} ∈ K ∩ G^{u}, there exists γ_{k,t,u} ∈ G such that

uc_{k,t} = γ_{k,t,u}u.

Now

utk = uc_{k,t}τ_{k}(t) = γ_{k,t,u}uτ_{k}(t) = g_{k,ut}σ_{k}(ut).

So σ_{k}(ut) = uτ_{k}(t) and there exists w ∈ G_{σ_{k}(ut)} such that g_{k,ut} = γ_{k,t,u}w. We get the following commutative diagram:

\xymatrix{σ_{k}(ut) \ar[rr]^{w} \ar[dr]_{g_{k,ut}} &  & σ_{k}(ut) \ar[dl]^{γ_{k,t,u}} \\  & utk}

Now let f ∈ t_{U,λ}(X, 𝔩)^{K,θ} and k ∈ K such that kf = f. Note that since f is K-fixed, we have

f(utk) = f(ut) = f(u) = γ_{k,t,u}f(uτ_{k}(t)) = γ_{k,t,u}f(u),

so γ_{k,t,u} ∈ G_{f(u)}. Hence

f(σ_{k}(ut)) = f(uτ_{k}(t)) = γ_{k,t,u}f(utk) = γ_{k,t,u}f(u) = f(u).

Let γ_{k,u} = ∏_{t∈[K∩G^{u}\K]} γ_{k,t,u} and ϕ_{u}(k) = ∏_{t∈[K∩G^{u}\K]} λ^{−1}(γ_{k,t,u}, k, σ_{k}(ut), ut). Then

𝔏(k, f, f) = ∏_{u∈[G\U]} 𝔩(g_{k,u}, f(σ_{k}(u)), f(u))λ^{−1}(g_{k,u}, k, σ_{k}(u), u)

= ∏_{\begin{matrix}u∈[G\U/K] \\ t∈[K∩G^{u}\K] \end{matrix}} 𝔩(g_{k,ut}, f(σ_{k}(ut)), f(ut))λ^{−1}(g_{k,ut}, k, σ_{k}(ut), ut)

= ∏_{\begin{matrix}u∈[G\U/K] \\ t∈[K∩G^{u}\K] \end{matrix}} 𝔩(γ_{k,t,u}w, f(σ_{k}(ut)), f(ut))λ^{−1}(γ_{k,t,u}w, k, σ_{k}(ut), ut)

= ∏_{\begin{matrix}u∈[G\U/K] \\ t∈[K∩G^{u}\K] \end{matrix}} 𝔩(γ_{k,t,u}, f(σ_{k}(ut)), f(ut))λ^{−1}(γ_{k,t,u}, k, σ_{k}(ut), ut)

= ∏_{u∈[G\U/K]} 𝔩_{f(u)}(γ_{k,u})ϕ_{u}(k)

= θ(k).

Let Ξ be the family of the sets ξ = {ξ_{u}}_{u∈[G\U/K]} where ξ_{u} : ^{u}K → C is a character such that res_{^{u}K}^{G_{f(u)}}(𝔩_{f(u)}) = ξ_{u} and

θ(k) = ∏_{u∈[G\U/K]}ξ_{u}(γ_{k,u})ϕ_{u}(k)

for all k ∈ K and u ∈ [G\U/K].

We claim that

T_{U,λ}(X, 𝔩)^{K,θ} = ⨆_{ξ∈Ξ} ∏_{u∈[G\U/K]}(X, 𝔩)^{^{u}K,ξ_{u}}.

Let f ∈ T_{U,λ}(X, 𝔩)^{K,θ}, then f(guk) = gf(u) for all g ∈ G, u ∈ U, and k ∈ K. So to determine f, it’s enough to know f(u) for u ∈ [G\U/K]. Let f(u) = x_{u}. Then res_{^{u}K}^{G_{xu}}𝔩_{x_{u}} ∈ {ξ_{u}}_{u∈[G\U/K]} for some {ξ_{u}}_{u∈[G\U/K]} ∈ ξ.

Conversely, let us choose x_{u} ∈ X for any u ∈ [G\U/K]. Let v ∈ V then v = guk for some g ∈ G, for some k ∈ K and a unique u ∈ [G\U/K]. We set f(v) = gx_{u}. Now f is well defined if and only if gx_{u} = x_{u} whenever g ∈ ^{u}K or equivalently x_{u} ∈ X^{^{u}K}. We want that f ∈ T_{U,λ}(X, 𝔩)^{K,θ}. If x_{u} ∈ (X, 𝔩)^{^{u}K,ξ_{u}} then res_{^{u}K}^{G_{xu}}(𝔩_{x_{u}}) = ξ_{u} and

θ(k) = ∏_{u∈[G\U/K]}ξ_{u}(γ_{k,u})ϕ_{u}(k).

So f ∈ T_{U,λ}(X, 𝔩)^{K,θ}.

Now using [[2], Lemma 11.2.9] we get

χ(T_{U,λ}(X, 𝔩)^{K,θ}) = ∑_{ξ∈Ξ} ∏_{u∈[G\U/K]} χ((X, 𝔩)^{^{u}K,ξ_{u}}).

Thus, if Λ_{(X,𝔩)} = Λ_{(Y,𝔪)} then Λ_{T_{U,λ}(X,𝔩)} = Λ_{T_{U,λ}(Y,𝔪)}. So we can define a map

T_{U,λ} : B_{C}(G) → B_{C}(H)

such that T_{U,λ}(a) = Λ_{T_{U,λ}(X,𝔩)} where (X, 𝔩) is a C-monomial G-poset such that a = Λ_{(X,𝔩)}, as in Corollary 11.

Proposition 30. Let G and H be finite groups, and let (U, λ) be a C-monomial (G, H)- biset.

1. T_{U,λ}([G, 1_{G}]_{G}) = [H, 1_{H}]_{H}.

2. T_{U,λ}(ab) = T_{U,λ}(a)T_{U,λ}(b), for any a, b ∈ B_{C}(G). In particular, the restriction of T_{U,λ} to B_{C}(G)^{×} is a group homomorphism

T_{U,λ}^{×} : B_{C}(G)^{×} → B_{C}(H)^{×}.

Proof. 1. Consider the C-monomial G-poset (•, 1) then clearly Λ_{(•,1)} = [H, 1_{H}]_{H}. So

using the first assertion of Proposition 26 we get

T_{U,λ}([G, 1]_{G}) = Λ_{T_{U,λ}(•,1)} = Λ_{(•,1)} = [H, 1_{H}]_{H}.

2. Let a, b ∈ B_{C}(G) then by Corollary 11 there exist C-monomial G-posets (X, 𝔩)

and (Y, 𝔪) such that Λ_{(X,𝔩)} = a and Λ_{(Y,𝔪)} = b. Then

T_{U,λ}(ab) = T_{U,λ}(Λ_{(X,𝔩)}Λ_{(Y,𝔪)}) = T_{U,λ}(Λ_{X×Y,𝔩×𝔪}) = Λ_{T_{U,λ}(X×Y,𝔩×𝔪)}

= Λ_{T_{U,λ}(X,𝔩)}Λ_{T_{U,λ}(Y,𝔪)} = T_{U,λ}(a)T_{U,λ}(b)

Proposition 31. Let G, H, and K be finite groups.

1. Let id_{G} stand for the identity (G, G)-biset. Then T_{id_{G},1_{G}} is the identity map of B_{C}(G).

2. Let (U, λ) and (U′, λ′) be C-monomial (G, H)-bisets. Then for any a ∈ B_{C}(G) T_{U⊔U′,λ⊔λ′}(a) = T_{U,λ}(a)T_{U′,λ′}(a).

3. Let (U, λ) be a C-monomial (G, H)-biset and let (V, ρ) be a monomial left free

(H, K)-biset then

T_{V,ρ} ∘ T_{U,λ} = T_{U×_{H}V,λ×ρ}.

Proof. Let a ∈ B_{C}(G) then by Corollary 11 there exists a C-monomial G-poset (X, 𝔩) such that a = Λ_{(X,𝔩)}.

1. Using the third assertion of Proposition 26, we get T_{id_{G},1_{G}}(a) = T_{id_{G},1_{G}}(Λ_{(X,𝔩)}) = Λ_{T_{id_{G},1_{G}}(X,𝔩)} = Λ_{(X,𝔩)} = a

2. Using the second assertion of Proposition 26, we get

T_{U⊔U′,λ⊔λ′}(a) = T_{U⊔U′,λ⊔λ′}(Λ_{(X,𝔩)}) = Λ_{T_{U⊔U′,λ⊔λ′}(X,𝔩)}
= Λ_{T_{U,λ}(X,𝔩)×T_{U′,λ′}(X,𝔩)} = Λ_{T_{U,λ}(X,𝔩)}Λ_{T_{U′,λ′}(X,𝔩)}
= T_{U,λ}(Λ_{(X,𝔩)})T_{U′,λ′}(Λ_{(X,𝔩)}) = T_{U,λ}(a)T_{U′,λ′}(a).

3. Using the fourth assertion of Proposition 26, we get

T_{V,ρ} ∘ T_{U,λ}(a) = T_{V,ρ} ∘ T_{U,λ}(Λ_{(X,𝔩)}) = Λ_{T_{V,ρ}∘T_{U,λ}(X,𝔩)}
= Λ_{T_{U×_{H} V,λ×ρ}(X,𝔩)} = T_{U×_{H}V,λ×ρ}(Λ_{(X,𝔩)}) = T_{U×_{H}V,λ×ρ}(a).

Corollary 32. Let G and H be finite groups. The map (U, λ) ↦ T_{U,λ}^{×} of Proposition 30 extends to a bilinear map

B_{C}(G, H) × B_{C}(G)^{×} → B_{C}(H)^{×}.

Proof. This follows from Assertion 2 of Proposition 26 and Assertion 2 of Proposition 31, and from the fact that the map T_{(U,λ)}^{×} depends only on the isomorphism class of (U, λ).

Remark 33. It follows from Remark 28 that if U is a finite (G, H)-biset, the square

\xymatrix{B(G) \ar[r]^{t_{G}} \ar[d]^{T_{U}} & B_{C}(G) \ar[d]^{T_{U,1_{U}}} \\ B(H) \ar[r]^{t_{H}} & B_{C}(H)}

of groups and multiplicative maps, is commutative, where T_{U} on the left is the usual generalized tensor induction map for Burnside rings, and the horizontal maps t_{G} and t_{H} are the ring homomorphisms induced by the functors τ_{G} and τ_{H}.

Acknowledgement

The second author was granted the Fellowship Program for Abroad Studies 2214-A by the Scientific and Technological Research Council of Turkey (Tübitak). The second author also wishes to thank LAMFA for their hospitality during the visit.

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Serge Bouc CNRS-LAMFA, Université de Picardie - Jules Verne 33 rue St Leu 80039 - Amiens - France email: serge.bouc@u-picardie.fr

Hatice Mutlu Department of Mathematics, Bilkent University 06800 - Bilkent, Ankara - Turkey email: hatice.mutlu@bilkent.edu.tr