Crossed Burnside rings and Bouc’s construction of Green functors

Fumihito Oda

Department of Liberal Arts, Toyama National College of Technology, Toyama 939-8630, Japan

                             E-mail: oda@toyama-nct.ac.jp

Abstract

We use a formula for primitive idempotents of the crossed Burnside ring given by F. Oda and T. Yoshida, and a result from the theory of Green functors obtained by S. Bouc to prove an induction theorem for the Drinfel’d double for a finite group over the complex field. We obtain Artin’s induction theorem for group algebras as a corollary of the theorem.

MSC (2000) :16E40, 19A22. Keywords : Crossed G-sets, Crossed Burnside rings, Drinfel’d double, Artin induction theorem.

1 Introduction

We study the Grothendieck (character) ring of the Drinfel’d double of a finite group G over the complex field ℂ. Witherspoon studied the representation ring of the Drinfel’d double of a group algebra in positive characteristic [Wi96]. In particular, she gave a decomposition of the representation ring as a direct sum of ideals involving Green rings of subgroups by using Thévenaz’s twin functor construction for Green functors [Th88]. Dress introduced the construction of a Mackey functor M_{Γ} from a Mackey functor M by simply setting M_{Γ}(X) := M(X ×Γ) for all finite G-sets X when Γ is a finite G-set [Dr73]. We call this construction for Mackey functors the Dress construction. Bouc introduced the Dress construction for a Green functor ([Bo03a] Theorem 5.1): If A is a Green functor for G over a commutative ring O, and Γ is a crossed G-monoid, then the Mackey functor A_{Γ} obtained by the Dress construction has a natural structure of a Green functor, and its evaluation A_{Γ}(G) is an O-algebra. Bouc’s construction involves as special cases the construction of the crossed Burnside ring obtained from the Burnside ring Green functor, the Hochschild cohomology ring of G obtained from the group cohomology Green functor, and the Grothendieck ring of the Drinfel’d double of G obtained from the Grothendieck ring Green functor for a group algebra. We also point out that Bouc’s construction is discussed in [Wi04]. In this paper, we obtain an induction theorem for the Drinfel’d double for G by using a formula for the primitive idempotents of the crossed Burnside ring [OY01], Bouc’s construction, and some properties of Witherspoon’s Green functor R(D_{G}(∗)). The theorem implies Artin’s induction theorem for a group algebra over ℂ. This is a new proof of Artin induction theorem.

The paper is organized as follows: Section 2 recalls a summary of basic results on crossed G-sets and the crossed Burnside ring. Section 3 describes the relationship between the crossed Burnside ring Green functor and the Grothendieck ring Green functor for the Drinfel’d double. Section 4 exposes a kind of induction theorem for ℂ-representations of the Drinfel’d double of G. We shall note that the result is not exactly an induction theorem for the ℂ-representation of the Drinfel’d double of G, because the evaluation R_{ℂ}(D_{G}(H)) of the Green functor R_{ℂ}(D_{G}(∗)) at a subgroup H of G is not isomorphic to the ℂ-representation ring R_{ℂ}(D(H)) of the Drinfel’d double of H.

We refer the reader to [Bo97], [Bo00], [TW95] or [We00] for standard definitions and results regarding Burnside rings and Green functors, and to [Bo03a], [Bo03b], [OY01], and [OY04] for basic results about crossed G-sets and crossed Burnside rings.

2 Crossed G-sets

(2.1) Notation. Let G be a finite group. If H is a subgroup of G, and g ∈ G, the conjugate subgroup gHg^{−1} of G is denoted by ^{g}H. The normalizer of H in G is denoted by N_{G}(H). The centralizer of H (resp. g ∈ G) in G is denoted by C_{G}(H) (resp. C_{G}(g)). A set of representatives in G of G/H is denoted by [G/H]. If X is a G-set, the stabilizer in G of element x of X is denoted by G_{x}. If X and Y are G-sets, the intersection G_{x} ∩ G_{y} of stabilizers in G of element (x, y) of X × Y is denoted by G_{x,y}. The set of orbits of H on X is denoted by H\X, and [H\X] denotes a set of representatives in X of H\X.

(2.2) Crossed Burnside rings. Let G be a finite group. In [Bo03a], Bouc defined a crossed G-monoid as follows. A crossed G-monoid (Γ, φ) is a pair consisting of a finite monoid Γ with a left action of G by monoid automorphisms (denoted by (g, γ) ↦ gγ or (g, γ) ↦ ^{g}γ, for g ∈ G and γ ∈ Γ), and a map of G-monoids φ from Γ to the G-set G^{c} with G-action defined by conjugation (i.e. a map φ which is both a map of monoids and a map of G-sets). In this paper, since we use only the trivial crossed G-monoid (Γ, φ) = (G^{c}, id_{G^{c}}), we denote by Γ or G^{c} a crossed G-monoid. A crossed G-set (X, α) over a crossed G-monoid Γ, is a pair consisting of a finite G-set X, together with a map α of G-sets from X to Γ. A morphism of crossed G-sets from (X, α) to (Y, β) is a G-map f from X to Y such that β ∘ f = α. Crossed G-sets over Γ and crossed G-maps make a category G-xset/Γ. The tensor product of crossed G-sets (X, α) and (Y, β) is defined by (X × Y, α.β), where X × Y is the direct product of X and Y, with diagonal G-action, and α.β is the map from X × Y to G^{c} defined by α.β(x, y) = α(x)β(y). We denote by XΩ(G, Γ) the Grothendieck ring of the category G-xset/Γ with respect to disjoint union and tensor product. We call it the crossed Burnside ring. The crossed Burnside ring Gxset/1^{c} over the crossed 1^{c}-monoid is the ordinary Burnside ring B(G). Since any crossed G-set is a disjoint union of transitive crossed G-sets (see 2.12 of [OY01]), G-xset/Γ has the following free ℤ-basis as an abelian group:

{(G/D)_{s} | D ∈ [G \ S(G)], s ∈ [G \ C_{Γ}(D)]}.

If Γ is a normal subgroup of G or an abelian group, then a formula for the primitive idempotents of 𝕂XΩ(G, Γ) over a splitting field 𝕂 of characteristic 0 has been given by Oda and Yoshida (see Lemma (5.5) of [OY01]).

(2.3) Theorem. [OY01] Let 𝕂 be a field of characteristic 0 which is a splitting field for all subgroups of G.

(1) For H ≤ G and an irreducible 𝕂-character θ of C_{Γ}(H), we put

e_{H,θ} = \frac{θ(1)}{|N_{G}(H)||C_{Γ}(H)|} ∑_{D≤H} ∑_{s∈C_{Γ}(H)} |D|μ(D, H)θ̃(s^{−1})(G/D)_{s},

where θ̃ is the sum of all distinct N_{G}(H)-conjugates of θ. Then

{e_{H,θ} | H ∈ [G \ S(G)], θ ∈ [N_{G}(H) \ Irr_{𝕂}(C_{Γ}(H))]}

is a set of orthogonal idempotents of the crossed Burnside ring 𝕂XΩ(G, Γ) over 𝕂 such that

(G/G)_{1_{G}} = 1_{𝕂XΩ(G,Γ)} = ∑_{H,θ} e_{H,θ}.

Moreover, the idempotents e_{H,θ} are all primitive and conversely any primitive idempotent of 𝕂XΩ(G, Γ) has this form.

A formula for the primitive idempotents of OXΩ(G, G^{c}) over a p-local ring O has been given by Bouc [Bo03b].

3 Bouc’s constructions of Green functors

(3.1) Burnside Green functors. We recall the crossed Burnside ring Green functor XΩ(∗, G^{c}) in terms of subgroups of G (see 4.1 of [OY04]). Let S(H) be the family of all subgroups of H ≤ G and C_{G}(D) the centralizer of D ≤ H. Then the assignment

H(≤ G) ⟼ XΩ(H, G^{c}) = ⟨(H/D)_{s} | D ∈ [H \ S(H)] s ∈ [H \ C_{G}(D)]⟩_{ℤ}

gives a Green functor for G over ℤ equipped with

ind_{L}^{H} : XΩ(L, G^{c}) ⟶ XΩ(H, G^{c}) : (L/D)_{s} ⟼ (H/D)_{s},
res_{L}^{H} : XΩ(H, G^{c}) ⟶ XΩ(L, G^{c}) : (H/D)_{s} ⟼ ∑_{g∈[L\H/D]}(L/L ∩^{g}D)_{^{g}s},
con_{H,g} : XΩ(H, G^{c}) ⟶ XΩ(^{g}H, G^{c}) : (H/D)_{s} ⟼ (^{g}H/^{g}D)_{^{g}s},

where D ≤ L ≤ H ≤ G and g ∈ G. In order to note the Green functor structure of XΩ(∗, G^{c}), we shall discuss briefly an equivalence between the category G-set↓_{(G/H×G^{c})} of finite G-sets over the G-set G/H × G^{c} (see 2.4 of [Bo97]) and the category H-set↓_{G^{c}} of finite H-sets over the H-set G^{c} with the H-action defined by conjugation. Let Ω be the Burnside Green functor for G over ℤ in terms of G-sets. By Proposition 2.4.2 of [Bo97], Ω_{G^{c}}(G/H) = Ω((G/H) × G^{c}) is isomorphic to the Grothendieck group of G-set↓_{(G/H×G^{c})}, with relations given by decomposition into disjoint union. It is easy to see that the G-sets

[K, s] : G/K → G/H × G^{c} : gK ↦ (gH,^{g}s)

over G/H × G^{c}, for K ∈ [H\S(H)] and s ∈ [H\C_{G}(K)], form a basis of Ω(G/H × G^{c}) over ℤ. We denote by (G/K, [K, s]) an element of the basis of Ω(G/H × G^{c}). Theorem 5.1 of [Bo03a] shows that Ω_{G^{c}} is a Green functor. If (G/K, [K, s]) and (G/L, [L, t]) are elements of the basis of Ω(G/H × G^{c}), then we have the following commutative diagram

[figure: ⨆_{x∈K\H/L} G/K ∩^{x}L; ⊔_{x}π_{K∩^{x}L}^{K}×(π_{K^{x}∩L}^{L}c_{K∩^{x}L},x); G/K × G/L; [K,s]×[L,t]; (G/H × G^{c}) × (G/H × G^{c})]
P.B.
[figure: G/H × G^{c}; ⊔_{x}[K∩^{x}L,s·^{x}t]; δ_{G/H}×Id_{G^{c}}; G/H × G/H × G^{c}; id; G/H × G/H × G^{c},; f]

where the map f from G/H × G^{c} × G/H × G^{c} to G/H × G/H × G^{c} maps (xK, γ_{1}, yL, γ_{2}) to (xK, yL, γ_{1}γ_{2}) (see section 5 of [Bo03a]). The left square is a pullback square. Theorem 5.1 of [Bo03a] shows that the product of (G/K, [K, s]) and (G/L, [L, t]) on Ω(G/H × G^{c}) is given by

(G/K, [K, s]) · (G/L, [L, t]) = ∑_{x∈[K\H/L]}(G/K ∩^{x}L, [K ∩^{x}L, s ·^{x}t]).  (3.1.1)

We have a functor F mapping G-set↓_{(G/H×G^{c})} to H-set↓_{G^{c}} defined for a transitive G-set [K, s] : G/K → G/H × G^{c} over G/H × G^{c} by

F : (G/K, [K, s]) ↦ ⟨K, s⟩ : [K, s]^{−1}({H} × G^{c}) → G^{c}

as in Lemma 2.4.1 of [Bo97]. We also denote by [K, s] the H-map H/K → G^{c} defined by gK ↦ ^{g}s. It is clear that the H-sets

[K, s] : H/K → G^{c} : gK ↦ ^{g}s

over the H-set G^{c}, for K ∈ [H\S(H)] and s ∈ [H\C_{G}(K)], form a basis of Ω↓_{H}^{G}(G^{c}) over ℤ, where Ω↓_{H}^{G} is a Green functor for H given by the restricion to H of G. We denote by (H/K, [K, s]) this element of the basis of Ω↓_{H}^{G}(G^{c}). It is easy to see that F gives an equivalence of categories from G-set↓_{(G/H×G^{c})} to H-set↓_{G^{c}} for any subgroup H of G. The inverse equivalence is given by the induction functor from H-set↓_{G^{c}} to G-set↓_{(G/H×G^{c}).} The images of (3.1.1) under F are

(H/K, [K, s]) · (H/L, [L, t]) = ∑_{x∈[K\H/L]}(H/K ∩^{x}L, [K ∩^{x}L, s ·^{x}t]).

in H-set↓_{G^{c}}. The Grothendieck group of H-set↓_{G^{c}} is isomorphic to XΩ(H, G^{c}). We can define a product

(H/K)_{s} · (H/L)_{t} = ∑_{x∈[K\H/L]}(H/K ∩^{x}L)_{s·^{x}t}

for any two elements (H/K)_{s} and (H/L)_{t} of the basis of XΩ(H, G^{c}). It is clear that the element (H/H)_{1_{G}} for the identity element 1_{G} of G is the identity element of XΩ(H, G^{c}). This gives a unitary ring structure to XΩ(H, G^{c}) for a subgroup H of G.

(3.2) Witherspoon’s Green functor. Witherspoon introduced a Green functor R_{ℂ}(D_{G}(∗)) for G over ℤ (see [Wi96] Section 5). For each subgroup H of G, there is a subalgebra

D_{G}(H) = ∑_{g∈G, h∈H} ℂϕ_{g}h

of the Drinfel’d (quantum) double D(G) of ℂG [Dr86], where ϕ_{g} is an element of the basis {ϕ_{g}}_{g∈G} of the dual space (ℂG)^{∗} = Hom_{ℂ}(ℂG, ℂ). Note that D_{G}(G) = D(G) and R(D(G)) is the representation ring of D(G) or equivalently the Grothendieck ring of Hopf bimodules for the Hopf algebra ℂG ([Ro95], [Bo03a], [OY04]). Let R_{ℂ}(D_{G}(H)) be the Grothendieck (representation) ring of D_{G}(H) for subgroup H of G. Then the assignment

H ⟼ R_{ℂ}(D_{G}(H))

where H ≤ G gives a Green functor for G over ℤ with operations given by

Dres_{L}^{H}  : R_{ℂ}(D_{G}(H)) ⟶ R_{ℂ}(D_{G}(L)) : U ⟼ U ↓_{D_{G}(L)},
Dind_{L}^{H} : R_{ℂ}(D_{G}(L)) ⟶ R_{ℂ}(D_{G}(H)) : V ⟼ D_{G}(H) ⊗_{D_{G}(L)} V,
Dconj_{H,g} : R_{ℂ}(D_{G}(H)) ⟶ R_{ℂ}(D_{G}(^{g}H)) : U ⟼ ^{g}U = gD_{G}(H) ⊗_{D_{G}(H)} U,

where U ↓_{D_{G}(L)} is a D_{G}(L)-module by restriction of the action from D_{G}(H) to D_{G}(L), L ≤ H ≤ G and g ∈ G. We use the equivalence of the category of H-vector bundles on G^{c} with the category of D_{G}(H)-modules (see [Wi96] Section 2).

(3.3) A morphism of Green functors. Let Ω be the Burnside ring Green functor for G over ℤ (see [Bo97] 2.4.2):

• If X is a finite G-set, then Ω(X) is the Grothendieck ring of the category of finite G-sets over X, where the relations are given by decomposition into disjoint union and product of G-sets.

• If X → X′ is a G-map, then Ω_{∗}(f) : Ω(X) → Ω(X′) is defined by Ω_{∗}(f)((Y, ϕ)) =

(Y, fϕ) for any G-set (Y, ϕ) = Y → X^{ϕ} over X.

• If X′ → X is a G-map, then Ω^{∗}(f) : Ω(X) → Ω(X′) is defined by Ω^{∗}(f)((Y, ϕ)) = (Y′, ϕ′), where (Y′, ϕ′) is the pull-back of (Y, ϕ) along f, obtained by filling the cartesian square

[figure: Y′; a; Y; X′; ϕ′; f; X.; ϕ]

Suppose that R_{ℂ} is the ℂ-representation (character ring) Green functor for G over ℤ, defined on subgroups of G. Then setting R_{ℂ}(G/H) = R_{ℂ}(H) leads by linearity to a definition of G-equivariant ℂ-vector bundles R_{ℂ}(X) on a G-set X (see [Wi96] Section 2) by using Remark 2.3 of [Bo03a]:

• If X is a finite G-set, then R_{ℂ}(X) is the Grothendieck ring of the category of Gequivariant ℂ-vector bundles on the G-set X, for relations given by decomposition into direct sum of vector bundles, the ring structure being induced by the tensor product of vector bundles: one can set

R_{ℂ}(X) = (⨁_{x∈X} R_{ℂ}(G_{x}))^{G}

where the exponent denotes fixed points under the natural action of G on ⊕_{x∈X}R_{ℂ}(G_{x}) by permutation of the components, and G_{x} is the stabilizer of x in G.

• If f : X → X′ is a G-map, then R_{ℂ}_{∗}(f) : R_{ℂ}(X) → R_{ℂ}(X′) is defined by

R_{ℂ}_{∗}(f)(u)_{y} = ∑_{x∈[G_{y}\f^{−1}(y)]} t_{G_{x}}^{G_{y}}(u_{x}),

where t_{G_{x}}^{G_{y}} is the induction map from R_{ℂ}(G_{x}) to R_{ℂ}(G_{y}), u ∈ R_{ℂ}(X), and y ∈ X′.

• If f : X′ → X is a G-map, then R_{ℂ}^{∗}(f) : R_{ℂ}(X) → R_{ℂ}(X′) is defined by

R_{ℂ}^{∗}(f)(v)_{x} = r_{G_{f(x)}}^{G_{x}}(v_{f(x)}),

where r_{G_{f(x)}}^{G_{x}} is the restriction map from R_{ℂ}(G_{x}) to R_{ℂ}(G_{f(x)}), v ∈ R_{ℂ}(X), and x ∈ X′.

• The product of the elements a ∈ R_{ℂ}(X) and b ∈ R_{ℂ}(Y) is defined by

(a × b)_{x,y} = r_{G_{(x,y)}}^{G_{x}}(a_{x}) · r_{G_{(x,y)}}^{G_{y}}(b_{y}).

If X is a finite G-set, denote the natural morphism

θ : Ω → R_{ℂ}

of Green functors defined by the maps θ(X) : Ω(X) → R_{ℂ}(X) by

(Y, φ) = (φ : Y → X) ⟼ {ℂ[φ^{−1}(x)]}_{x∈X},

where ℂ[φ^{−1}(x)] is the permutation module associated to the G_{x}-set φ^{−1}(x).

The following theorem is essential in the proof of Theorem 4.1 of this paper.

(3.4) Theorem (Bouc [Bo03a] 5.1). Let A be a Green functor for G over a commutative ring O, Γ a crossed G-monoid, and ε_{A} an element of A(•) such that for any G-set X and for any a ∈ A(X)

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

denoting by p_{x} (resp. q_{x}) the bijective projection from X × • (resp. from • × X) to X (see 1.2.1 of [Bo03a]). Then the functor A_{Γ} is a Green functor for G over O, with unit ε_{A_{Γ}},

where ε_{A_{Γ}} is the element A_{∗}( \begin{matrix}• \\ ↓ \\ 1_{G} \end{matrix} )(ε_{A}) of A(Γ) = A_{Γ}(•). Moreover the correspondence

A ↦ A_{Γ} is an endo-functor of the category of Green functors for G over O.

(3.5) Lemma. Let Ω be the Burnside ring Green functor and G^{c} the crossed G-monoid. Then there is an isomorphism of Green functors

XΩ(∗, G^{c}) ≅ Ω_{G^{c}}.

Proof: Let H be a subgroup of G. We set Ω_{G^{c}}(H) = Ω_{G^{c}}(G/H)(= Ω((G/H) × G^{c})). By Bouc’s Theorem (3.4), we have

Ω((G/H) × G^{c}) = (  ⨁_{(aH,g)∈G/H×G^{c}} Ω(G_{aH,g}))^{G} ≅ ⨁_{(H,g)∈[G\(G/H×G^{c})]} Ω(G_{H,g}),

where G_{aH,g} = G_{aH} ∩ G_{g} = ^{a}H ∩ C_{G}(g) = C_{^{a}H}(g). The map

f_{H} : XΩ(H, G^{c}) ⟶ ( ⨁_{(aH,g)∈G/H×G^{c}} Ω(G_{aH,g}))^{G}

is defined for (H/K)_{s} ∈ XΩ(H, G^{c}) by

(H/K)_{s} ⟼ ([K, s]^{−1}(aH, g))_{(aH,g)∈G/H×G^{c}} ,

where the map [K, s] : G/K → G/H × G^{c} is the G-map defined by xK ↦ (xH,^{x}s). It is well known that the natural projection p_{L}^{H} : G/L → G/H gives the induction Ω_{G^{c}}_{∗}(p_{L}^{H}) and the restriction Ω_{G^{c}}^{∗}(p_{L}^{H}). Moreover,

γ_{x,H} : G/H → G/^{x}H : gx^{−1}^{x}H = gHx^{−1}

gives the conjugation Ω_{G^{c}}_{∗}(γ_{x,H}). It is clear that the collection of homomorphisms of ℤ-modules f_{H} : XΩ(H, G^{c}) → Ω_{G^{c}}(H) makes a morphism f from the Mackey functor XΩ(∗, G^{c}) to the Mackey functor Ω_{G^{c}}, which commutes with restriction, induction and conjugation. Since f_{H} is the inverse of the homomorphism from Ω_{G^{c}}(H) to XΩ(H, G^{c}) obtained by the functor F of (3.1) and we defined the Green functor structure on XΩ(∗, G^{c}) using Green functor structure on Ω_{G^{c}}, f_{H} is an isomorphism of rings for H.

Finally we have to check that the morphism f_{H} preserves the identity element. Recall from Theorem 5.1 of [Bo03a] that the identity element ε_{Ω_{G^{c}}} of the Green functor Ω_{G^{c}} is

the element Ω_{∗}( \begin{matrix}• \\ ↓ \\ 1_{G} \end{matrix} )(ε_{Ω}) of Ω(G^{c}) = Ω_{G^{c}}(•). The identity element ε_{Ω} of the Burnside

Green functor Ω is the identity element • of the Burnside ring Ω(G), so Ω_{∗}( \begin{matrix}• \\ ↓ \\ 1_{G} \end{matrix} )(ε_{Ω}) is

the composition • → • → G^{c} : the identity element of the crossed Burnside ring Ω(G^{c}) (see 3.1 of [OY01] or 2.1 of [Bo03a]). Moreover, the identity element (ε_{Ω_{G^{c}}})_{G/H} of the ring Ω_{G^{c}}(G/H) is obtained by the formula (see 2.3 of [Bo97]) (ε_{Ω_{G^{c}}})_{G/H} = Ω_{G^{c}}^{∗}(π_{H}^{G})(ε_{Ω_{G^{c}}}), where π_{H}^{G} is the natural projection G/H → G/G = •. The pull-back diagram shows that (ε_{Ω_{G^{c}}})_{G/H} is the G-set [H, 1_{G}] : G/H → G/H × G^{c} over G/H × G^{c}. By the definition of f_{H},

[figure: G/H; •; G/H × G^{c}; π_{H}^{G} × id_{G^{c}}; • × G^{c}]

f_{H}((H/H)_{1_{G}}) = ([H, 1_{G}]^{−1}(aH, α))_{(aH,α)∈G/H×G^{c}} .

We have that [H, 1_{G}]^{−1}(aH, α) is the coset aH for α = 1_{G} and the empty set ϕ for all other elements, so f_{H} preserves the identity element, and this completes the proof of the lemma. □

We will denote by ℂ[X] the ℂ-permutation module associated to a set X. The endofunctor of the category of Green functors of Theorem (3.4) applied to the morphism θ from Ω to R_{ℂ} leads to the following lemma.

(3.6) Lemma. Let θ : Ω → R_{ℂ} be the natural morphism from the Burnside Green functor to the Grothendieck ring Green functor. Then the morphism θ_{G^{c}} : Ω_{G^{c}} → R_{ℂ}_{G^{c}} given by the Bouc’s construction is a morphism of Green functors.

(3.7) Lemma. There is a morphism

θ_{G^{c}} : XΩ(∗, G^{c}) → R_{ℂ}(D_{G}(∗))

of Green functors.

Proof: Lemma (3.6) shows that there is a morphism θ_{G^{c}} of Green functors from Ω_{G^{c}} to R_{ℂG^{c}}, whose evaluation at the transitive G-set G/H is the map

(θ_{G^{c}})_{G/H} = θ_{G/H×G^{c}} : Ω_{G^{c}}(G/H) = Ω(G/H × G^{c}) → R_{ℂ}(G/H × G^{c}) = R_{ℂG^{c}}(G/H)

defined for the transitive G-set [K, s] : G/K → G/H × G^{c} over G/H × G^{c} by

( ∐_{aH∈G/H}[K, s]^{−1}(aH, α))_{α∈G^{c}} ↦ ( ⨁_{aH∈G/H} ℂ[[K, s]^{−1}(aH, α)])_{α∈G^{c}} .

The maps f_{H} of the proof of Lemma (3.5) and θ_{G/H×G^{c}} induce maps

[figure: XΩ(H, G^{c}); f_{H}; Ω_{G^{c}}(G/H); θ_{G/H×G^{c}}; R_{ℂG}(G/H)]

for a subgroup H of G. The compositions θ_{G/H×G^{c}} ∘f_{H} form a morphism of Green functors from XΩ(∗, G^{c}) to R_{ℂG}(∗). The isomorphism (see Remark 2.3 of [Bo03a])

R_{ℂG^{c}}(G/H) = (  ⨁_{(aH,α)∈G/H×G^{c}} R_{ℂ}(G_{aH,α}))^{G}
→ ⨁_{(H,α)∈[G\(G/H×G^{c})]} R_{ℂ}(G_{H,α}) = ⨁_{(H,α)∈[G\(G/H×G^{c})]} R_{ℂ}(C_{H}(α))

defined by

(U_{aH,α})_{(aH,α)∈G/H×G^{c}} ↦ (U_{H,α})_{(H,α)∈[G\(G/H×G^{c})]}

and Theorem 2.4 of [Wi96] induce an isomorphism of Green functors R_{ℂG^{c}} ≅ R(D_{G}(∗)), and complete the proof of the lemma. □

Let (H/L)_{g} be an element of the basis of XΩ(H, G^{c}). Then the previous lemma shows that θ_{G^{c}}((H/L)_{g}) is an H-vector bundle on G^{c}. We denote by [H/L]_{g} this H-vector bundle. Lemma (3.5) shows the following lemma.

(3.8) Lemma. The H-vector bundle [H/L]_{g} is the ℂC_{H}(^{x}g)-module ℂ[[^{x}L,^{x}g]^{−1}(^{x}g)] in the ^{x}g-component, for x ∈ [H/C_{H}(g)], and 0 in all other components.

We recall the maps Incl_{J,h} : R_{ℂ}(J) → R_{ℂ}(D_{G}(J)), where J is a subgroup of G and h ∈ C_{G}(J), introduced in Section 2 of [Wi96]: Given a ℂJ-module V, Incl_{J,h}(V) is the D_{G}(J)-module which is V in the h-component and 0 elsewhere.

(3.9) Lemma. Let θ_{G^{c}} be the ring homomorphism θ_{(G/G)×G^{c}}from the crossed Burnside ring XΩ(G, G^{c}) to the Grothendieck ring R_{ℂ}(D_{G}(G)) given by the previous lemma. Then the D(G)-module corresponding to the G-vector bundle θ_{G^{c}}((G/L)_{g}) is the induced module

D(G) ⊗_{D_{G}(L)} Incl_{L,g}(ℂ[L/L]).

Proof: By the previous lemma, the ring homomorphism θ_{G^{c}} assigns to a transitive

crossed G-set (G/L)_{g} the G-vector bundle [G/L]_{g} which is ℂ[[^{x}L,^{x}g]^{−1}(^{x}g)] ≅ ℂ[C_{G}(^{x}g)/^{x}L] as ℂC_{G}(^{x}g)-modules in the ^{x}g-component for x ∈ [G/C_{G}(g)] and 0 in all other components. Section 2 of [Wi96] shows that the ℂD_{G}(G)-module corresponding to the G-vector bundle [G/L]_{g} is the induced module

D(G) ⊗_{D_{G}(C_{G}(g))} Incl_{C_{G}(g),g}(ℂ[C_{G}(g)/L]).

Since L ≤ C_{G}(g) and g ∈ C_{G}(C_{G}(g)), we have

Incl_{C_{G}(g),g} ∘ Ind_{L}^{C_{G}(g)} = Dind_{L}^{C_{G}(g)} ∘ Incl_{L,g}

by Lemma 5.4 of [Wi96]. Finally, we see that

D_{G}(G) ⊗_{D_{G}(J)} Incl_{J,g}(ℂ[J/L]) ≅ D(G) ⊗_{D_{G}(J)} Incl_{J,g} ∘ Ind_{L}^{J}(ℂ[L/L])
≅ D(G) ⊗_{D_{G}(J)} Dind_{L}^{J} ∘ Incl_{L,g}(ℂ[L/L])
≅ D(G) ⊗_{D_{G}(J)} D_{G}(J) ⊗_{D_{G}(L)} Incl_{L,g}(ℂ[L/L])
≅ D(G) ⊗_{D_{G}(L)} Incl_{L,g}(ℂ[L/L]),

where J = C_{G}(g). □

(3.10) Sub-Green functors. There is a sub-Green functor XΩ(∗, G^{c})_{1} which assigns to each subgroup H of G the subring XΩ(H, G^{c})_{1} of XΩ(H, G^{c}) generated by the elements (H/L)_{1_{G}}. There is also a sub-Green functor R_{ℂ}(D_{G}(∗)_{1}) which assigns to each subgroup H of G the subring R_{ℂ}(D_{G}(H)_{1}) of R_{ℂ}(D_{G}(H)) generated by Incl_{H,1_{G}}(V)′s, where Incl_{H,1_{G}} is a functor embedding the category of ℂH-modules as a full subcategory of the category of D_{G}(H)-modules (see, [Wi96] Section 1) and V is a ℂH-module. It is easy to see that XΩ(H, G^{c})_{1} is isomorphic to the Burnside ring Ω(H) and R_{ℂ}(D_{G}(H)_{1}) is isomorphic to the ordinary character ring R_{ℂ}(H). The homomorphism θ_{G^{c}} ↓_{XΩ(H,G^{c})_{1}} is the natural ring homomorphism from Ω(H) to R_{ℂ}(H).

(3.11) Characters. Witherspoon pointed out the character of a ℂD(G)-module in [Wi96], that appeared in [Lu87]. For g ∈ G and an irreducible character ρ of C_{G}(g), a character χ_{g,ρ} of a ℂD(G)-module U = {U_{h}}_{h∈G^{c}} is given by the formula

χ_{g,ρ}(U) = \frac{1}{degρ} ∑_{h∈C_{G}(g)} Tr(g, U_{h})ρ(h).

The characters of the crossed Burnside ring have been considered by Oda and Yoshida ([OY01], Section 5). For a subgroup H of G and an irreducible character θ of C_{G}(H), the linear map ω_{H,θ} of XΩ(G, G^{c}) to ℂ is the composite of Burnside homomorphism φ_{H} and a central character ω̃_{H,θ} : given a crossed G-set X over G^{c}, H ≤ G, and an irreducible character ρ of the group algebra ℂC_{G}(H), ω_{H,ρ}(X) = ω̃_{H,ρ} ∘ φ_{H}(X).

For each h ∈ G^{c} the h-component of the crossed G-set (X, α) is defined by

X[h] = {x ∈ X|α(x) = h}.

(3.12) Lemma. Let g be an element of G, ρ an irreducible character of ℂC_{G}(g), and θ_{G^{c}} the homomorphism from XΩ(G, G^{c}) to R_{ℂ}(D(G)). Then χ_{g,ρ}θ_{G^{c}} = ω_{⟨g⟩,ρ}, where ⟨g⟩ is the cyclic subgroup generated by g.

Proof: Let (G/L)_{s} be a transitive crossed G-set over G^{c}. Then 3.11 shows that

ω_{⟨g⟩,ρ}((G/L)_{s}) = ∑_{xL∈(G/L)^{⟨g⟩}} \frac{ρ(^{x}s)}{ρ(1_{G})}

= ∑_{h∈C_{G}(⟨g⟩)} |G/L[h]^{⟨g⟩}| · \frac{ρ(h)}{ρ(1_{G})}

= \frac{1}{deg ρ} ∑_{h∈C_{G}(g)} ♯\{xL ∈ (G/L)^{⟨g⟩}|^{x}s = h\}ρ(h)

= \frac{1}{degρ} ∑_{h∈C_{G}(g)} Tr(g, ([G/L]_{s})_{h})ρ(h)
= χ_{g,ρ}([G/L]_{s})
= χ_{g,ρ}(θ_{G^{c}}((G/L)_{s})).
□

4 Induction theorems

The proof of the following theorem is similar to the proof of Theorem 3.5.2 in [Bo00].

(4.1) Theorem. Let G be a finite group. Then

ℂR_{ℂ}(D(G)) = ∑_{H∈C(G)} Dind_{H}^{G}ℂR_{ℂ}(D_{G}(H)),

where C(G) is the family of cyclic subgroups of G. In other words, any complex character of D(G) is a linear combination with rational coefficients of characters induced from cyclic groups of G.

Proof: It suffices to show that the left hand side is contained in the right hand side. There is a natural homomorphism θ_{G^{c}} from the crossed Burnside ring XΩ(G, G^{c}) to the Grothendieck ring R_{ℂ}(D(G)) by Lemma 3.8. This extends to a homomorphism of ℂalgebra from ℂXΩ(G, G^{c}) to ℂR_{ℂ}(D(G)).

Let H be a subgroup of G and s an element of G. Let e_{H,ψ} be the primitive idempotent corresponding to a ℂ-irreducible character ψ of C_{G}(H) (see Theorem 2.3). Since a character ω_{⟨s⟩,ρ} of ℂXΩ(G, G^{c}) is the composite of a central character ω̃_{⟨s⟩,ρ} and a Burnside homomorphism φ_{⟨s⟩} (see 3.11), ω_{⟨s⟩,ρ}(e_{H,ψ}) = 0 if H is not conjugate in G to the cyclic subgroup ⟨s⟩ generated by s. Lemma 3.12 implies that χ_{s,ρ}(θ_{G^{c}}(e_{H,ψ})) = 0, and it follows that θ_{G^{c}}(e_{H,ψ}) in ℂR_{ℂ}(D(G)) is zero unless H is cyclic. If H is cyclic, by Lemma 3.9, θ_{G^{c}}(e_{H,ψ}) is a linear combination with coefficients of induced ℂD(G)-modules D(G) ⊗_{D_{G}(L)} Incl_{L,s}(ℂ[L/L]), for a subgroup L of the cyclic group H. Let

(G/G)_{1_{G}} = ∑_{H∈[G\S(G)]} ∑_{ψ∈[N_{G}(H)\Irr(C_{G}(H))]} e_{H,ψ}

be the decomposition of the unit element of ℂXΩ(G, G^{c}). Then Lemma 3.8 shows that θ_{G^{c}}((G/G)_{1_{G}}) is the unit of ℂR_{ℂ}(D(G)) and θ_{G^{c}}((G/G)_{1_{G}}) is a linear combination with ℂ-coefficients of such induced modules D(G) ⊗_{D_{G}(L)} Incl_{L,g}(ℂ[L/L]), which are induced from cyclic subgroups L of G. Then if U is any D(G)-module

U ⊗ Incl_{L,s}(ℂ[L/L]) ↑^{D_{G}(G)}≅ (U ↓_{D_{G}(L)} ⊗Incl_{L,s}(ℂ[L/L])) ↑^{D_{G}(G)}

by Lemma 5.2 of [Wi96]. Therefore, the theorem follows by the unitarity of θ_{G^{c}}. □

The previous theorem and (3.10) show the following corollary.

(4.2) Corollary (Artin). Let G be a finite group. Then

ℚR_{ℂ}(G) = ∑_{H∈C(G)} Ind_{H}^{G}ℚR_{ℂ}(H).

References

[Bo97] S. Bouc, Green functors and G-sets, Lecture Notes in Mathematics, vol. 1671, Springer, 1997.

[Bo00] S. Bouc, Burnside rings, in Handbook of Algebra, Vol. 2, Elsevier Science B.V. (2000), 439–804.

[Bo03a] S. Bouc, Hochschild constructions for Green functors, Comm. Algebra, 31 (2003), 419–453.

[Bo03b] S. Bouc, The p-blocks of the Mackey algebra, Algebras and Representation Theory, 6 (2003), 515–543.

[Dr73] A.W.M. Dress, Contributions to the theory of induced representations, Lecture Notes in Math., 342, Springer–Verlag, Berlin, 1973, 183-240.

[Dr86] V. G. Drinfel’d, Quantum groups, in Proceedings International Congress of Mathematicians, Berkeley, pp. 798–820, American Mathematical Society, Providence, Rhode Island, 1986.

[Lu87] G. Lusztig, Leading coefficients of character values of Hecke algebras, Proc. Symp. Pure. Math. 47 (1987), 235–262.

[OY01] F. Oda and T. Yoshida, Crossed Burnside rings I. The Fundamental Theorem, J. Algebra 236 (2001), 29–79.

[OY04] F. Oda and T. Yoshida, Crossed Burnside rings II. The Dress construction of a Green functor, J. Algebra 282 (2004), 58–82.

[Ro95] M. Rosso, Groupes quantiques et algébres de battage quantiques, C. R. Acad. Sc. Paris 320 (1995), 145–148.

[Th88] J. Thévenaz, Some remarks on G-functors and the Brauer morphism, J. Reine Angew. Math. 384 (1988), 24–56.

[TW95] J. Thévenaz and P. Webb, The structure of Mackey functors, Trans. Amer. Math. soc. 347 (6) (1995), 1865–1961.

[We00] P. Webb, A guide to Mackey functor, In Handbook of Algebra Vol. 2, Elsevier Science B.V. (2000), 805–836.

[Wi04] S.J.Witherspoon, Products in Hochschild cohomology and Grothendieck rings of group crossed products, Adv. Math. 185 (2004), 136–158.

[Wi96] S.J.Witherspoon, The representation ring of the quantum double of a finite group, J. Algebra 179 (1996), 305–329.