The p-blocks of the Mackey algebra

Serge Bouc

Institut de Mathématiques de Jussieu Université Paris 7-Denis Diderot, 75251, Paris Cedex 05, France email: bouc@math.jussieu.fr

1 Introduction

Let p be a prime number, and O be a complete discrete valuation ring of characteristic 0 whith residue field of characteristic p. Let G be a finite group, and denote by μ_{O}(G) the Mackey algebra of G over O (see [8] for definition).

Formulae for the primitive idempotents in the center of μ_{O}(G) have been given by Yoshida (and slightly corrected by Oda [7]). However, those formulae are expressed in terms of ordinary irreducible characters of the centralizers of subgroups of G. The aim of this article is to state explicit formulae for the block idempotents of μ_{O}(G), in terms of the blocks of the group algebra OG.

The proof uses the natural ring homomorphism from the crossed Burnside ring B_{O}^{c}(G) to the center of the Mackey algebra, and a description of the prime spectrum and block idempotents of B_{O}^{c}(G).

The paper is organized as follows : section 2 is an exposition of definitions and basic results on the crossed Burnside ring. Section 3 describes the prime spectrum and p-blocks of this ring. Section 4 recalls the basic definitions on Mackey functors, and uses the action of the crossed Burnside ring to state explicit formulae for the block idempotents of the p-local Mackey algebra. Section 5 exposes some consequences on Mackey functors that follow from these formulae. In particular, one can show that a block b of G and the corresponding block b^{μ} of the Mackey algebra have the same defect groups.

2 The crossed Burnside ring

Most of the definitions and results of this section have already been discussed by Yoshida ([9]). However, they are still unpublished, or not currently available in published form. This is the reason for exposing this material here.

2.1 Definition

Let G be a finite group, and denote by G^{c} the set G, on which G acts by conjugation. The category G-set↓_{G^{c}} of crossed G-sets is the category of Gsets over G^{c} : a crossed G-set (X, α) is a a pair consisting of a finite G-set X (i.e. a finite set with a left G-action), together with a map of G-sets α from X to G^{c}, and a morphism of crossed G-sets from (X, α) to (Y, β) is a morphism of G-sets φ from X to Y such that β ∘ φ = α.

There is an obvious notion of disjoint union of crossed G-sets, and the crossed Burnside group B^{c}(G) is defined as the Grothendieck group of the category of crossed G-sets, for relation given by disjoint union decomposition : let I be the set of isomorphism classes of crossed G-sets, and denote by [X, α] the isomorphism class of the crossed G-set (X, α). Then

B^{c}(G) = ℤ^{I}/ <[X ⊔ Y, α ⊔ β] − [X, α] − [Y, β]> .

Let B denote the (ordinary) Burnside Mackey functor for G (see [3] or [1] Section 2.4 for definition). It follows from [1] Proposition 2.4.2 that B^{c}(G) is isomorphic (as a ℤ-module) to the evaluation B(G^{c}) of the Mackey functor B at the G-set G^{c}.

If (X, α) and (Y, β) are crossed G-set, then their product is the crossed G-set (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) .

This product on crossed G-sets clearly commutes with disjoint unions, hence it gives a product on B^{c}(G). This turns B^{c}(G) into a ring. The identity element of this ring is [•, u_{•}], where • is a G-set of cardinality one, and the map u_{•} sends the unique element of • to the identity element of the group G.

The ring B^{c}(G) is commutative : the map

(x, y) ∈ X × Y ↦ (α(x)y, x) ∈ Y × X

is an isomorphism from (X × Y, α.β) to (Y × X, β.α) in G-set↓_{G^{c}}, because for all (x, y) ∈ X × Y

β(α(x)y)α(x) = α(x)β(y)α(x)^{−1}α(x) = α(x)β(y) .

More generally, if R is a commutative ring, denote by B_{R}^{c}(G) the tensor product of B^{c}(G) with R over ℤ

B_{R}^{c}(G) = R ⊗_{ℤ} B^{c}(G) .

It is an R-algebra. Similarly, denote by B_{R}(G) the (ordinary) Burnside algebra of G over R.

Lemma 2.1.1: If X is a finite G-set, denote by u_{X} the map from X to G sending every element to 1 ∈ G. Then the correspondence X ↦ (X, u_{X}) extends linearly to a ring homomorphism from B_{R}(G) to B_{R}^{c}(G), which preserves identity elements.

Proof: This amounts to checking that if X and Y are finite G-sets, then the product (X × Y, u_{X}.u_{Y}) is isomorphic to (X × Y, u_{X×Y}), which is straightforward. Moreover the trivial G-set is mapped to (•, u_{•}).

2.2 Characterization of crossed G-sets

A crossed G-set (X, α) is called transitive if the G-set X is. In this case, let x be any element of X, denote by H the stabilizer of x in G, and set a = α(x). Then a lies in the centralizer C_{G}(H) of H in G. Define the map m_{a} from G/H to G^{c} by m_{a}(gH) = ^{g}a, where^{g}a = gag^{−1}. Then (G/H, m_{a}) is a crossed G-set, and the map gH ↦ gx from G/H to X is clearly an isomorphism of crossed G-sets from (G/H, m_{a}) to (X, α).

Conversely, if H is any subgroup of G, and if a ∈ C_{G}(H), then (G/H, m_{a}) is a transitive crossed G-set. If K is a subgroup of G, and b ∈ C_{G}(K), then the crossed G-sets (G/H, m_{a}) and (G/K, m_{b}) are isomorphic if and only if there exists an element g ∈ G such that ^{g}H = K and ^{g}a = b.

Notation 2.2.1: I will denote by s_{G} the set of subgroups of G. If X is a G-set and H ∈ s_{G}, I denote by X^{H} the set of fixed points of H on X. Let P_{G} denote the set of pairs (H, a) consisting of a subgroup H of G and and element a of C_{G}(H). The group G acts by conjugation on s_{G} and P_{G}, and I denote by [s_{G}] (resp. [P_{G}]) a set of representative of G-orbits on s_{G} (resp. on P_{G}). If (H, a) ∈ P_{G}, I denote by [H, a]_{G} the isomorphism class of the crossed G-set (G/H, m_{a}).

Now if (X, α) is any crossed G-set, choose a set S of representatives of the orbits of G on X. Then the map from ⨆_{s∈S} G/G_{s} to X sending gG_{s} to gs is clearly an isomorphism in G-set↓_{G^{c}} from ⨆_{s∈S}(G/G_{s}, m_{α(s)}) to X. Thus any crossed G-set is isomorphic to a disjoint union of transitive ones.

Lemma 2.2.2: Let (X, α) and (Y, β) be crossed G-sets. Then the following are equivalent:

1. The crossed G-sets (X, α) and (Y, β) are isomorphic.

2. For any crossed G-set (Z, γ)

|Hom_{G-set↓_{G^{c}}}((Z, γ), (X, α))| = |Hom_{G-set↓_{G^{c}}}((Z, γ), (Y, β))| .

3. For any (H, a) ∈ [P_{G}]
|α^{−1}(a)^{H}| = |β^{−1}(a)^{H}| .

Proof: Clearly 1) implies 2). Moreover 2) implies 3) since for any crossed G-set X

|Hom_{G-set↓_{G^{c}}}((G/H, m_{a}), (X, α))| = |α^{−1}(a)^{H}| .

To show that 3) implies 1), I can replace (X, α) and (Y, β) by isomorphic crossed G-sets, i.e. suppose that

(X, α) = ⨆_{(K,b)∈[P_{G}]} u_{K,b}(G/K, m_{b}) (Y, β) = ⨆_{(K,b)∈[P_{G}]} v_{K,b}(G/K, m_{b})

where u_{K,b} and v_{K,b} are natural integers, and the notation u_{K,b}(G/K, m_{b}) means a disjoint union of u_{K,b} copies of (G/K, m_{b}). Notice that if p = (H, a) and q = (K, b) are elements of P_{G}, then

|Hom_{G-set↓_{G^{c}}}((G/H, m_{a}), (G/K, m_{b}))| = |{g ∈ G/K | H^{g} ⊆ K, a^{g} = b}| .

Denote by M(p, q) this number. Condition 3) implies that

∑_{q∈[P_{G}]} u_{q}M(p, q) = ∑_{q∈[P_{G}]} v_{q}M(p, q)

for all p ∈ [P_{G}]. It follows that the sequence (u_{q} − v_{q})_{q∈[P_{G}]} is in the kernel of the square matrix M(p, q)_{p,q∈[P_{G}]}. Now with suitable ordering of [P_{G}], this matrix is upper triangular. The diagonal coefficient M(p, p) for p = (H, a) is equal to

M(p, p) = |{g ∈ N_{G}(H)/H | ^{g}a = a}| = |N_{G}(H) ∩ C_{G}(a) : H|

This is non-zero, and M is non singular. Thus u_{K,b} = v_{K,b} for any (K, b) in [P_{G}]. Hence (X, α) and (Y, β) are isomorphic.

Corollary 2.2.3: The elements [H, a]_{G}, for (H, a) ∈ [P_{G}], form a basis of B^{c}(G) over ℤ.

2.3 Brauer morphisms

Let K be a subgroup of G, and let C_{G}(K) be the centralizer of K in G. Denote by (X, α) ↦ (X^{K}, α^{K}) the fixed points functor from G-set↓_{G^{c}} to C_{G}(K)-set↓_{C_{G}(K)^{c}}, where X^{K} is viewed as a C_{G}(K)-set, and α^{K} is the map X^{K} → (G^{c})^{K} = C_{G}(K)^{c} induced by α.

This functor induces a map Br_{K}, called the Brauer morphism, from B_{R}^{c}(G) to B_{R}^{c}(C_{G}(K)), defined by linearity from Br_{K}([X, α]) = [X^{K}, α^{K}], which is clearly a ring homomorphism, which preserves identity elements.

If (X, α) is a crossed G-set, let s_{G}(X, α) denote the element of the center ZRG of the group algebra RG of G over R defined by

s_{G}(X, α) = ∑_{x∈X} α(x)

This clearly induces a morphism of R-algebras, still denoted by s_{G}, from B_{R}^{c}(G) to ZRG, which preserves identity elements.

Notation 2.3.1: If H is a subgroup of G, then I denote by z_{H} the ring homomorphism s_{C_{G}(H)} ∘ Br_{H} from B_{R}^{c}(G) to ZRC_{G}(H).

Thus if (X, α) is a crossed G-set, then

z_{H}([X, α]) = ∑_{x∈X^{H}} α(x) = ∑_{g∈C_{G}(H)} |α^{−1}(g)^{H}|g .

Lemma 2.3.2: If the characteristic of R is zero, then the ring homomorphism

Θ_{R} = ∏_{H∈[s_{G}]} z_{H} : B_{R}^{c}(G) → ∏_{H∈[s_{G}]} ZRC_{G}(H)

is injective.

Proof: Clearly Θ_{R} is a ring homomorphism. The injectivity assertion is just a reformulation of lemma 2.2.2 : if u = ∑_{(H,a)∈[P_{G}]} r_{H,a}[H, a]_{G} is a non-zero element the kernel of Θ_{R}, let K be a subgroup of G maximal such that there exists (K, b) ∈ [P_{G}] with r_{K,b} ≠ 0. Now

z_{K}(u) = ∑_{{a∈C_{G}(K) \atop a mod.N_{G}(K)}} r_{K,a} ∑_{g∈N_{G}(K)/K} ^{g}a = ∑_{a∈C_{G}(K)} r_{K,a}|N_{G}(K)∩C_{G}(a) : K|a = 0

Since R has characteristic zero, it follows that r_{K,a} = 0 for all a ∈ C_{G}(K). This contradiction proves the lemma.

The previous lemma can be considered from a slightly different point of view : let G^{c} denote the set G, on which G acts by conjugation. There is an isomorphism of G-sets

G^{c} ≅ ⨆_{g∈[G]} G/C_{G}(g)

where [G] is a set of representatives of conjugacy classes of G. Since B^{c}(G) is the value of the Burnside Mackey functor B at G^{c}, it follows that there is an isomorphism of ℤ-modules

B^{c}(G) ≅ ⨁_{g∈[G]} B(C_{G}(g))

sending the crossed G-set (X, α) to the sequence (α^{−1}(g))_{g∈[G]}. The inverse

isomorphism sends the element C_{G}(g)/L of B(C_{G}(g)) to [L, g]_{G}.

Now for any finite group H, it follows from Burnside’s theorem (see [3] Theorem 2.3.2) that there is an injective morphism

ϕ_{H} : B(H) → ∏_{K∈[s_{H}]} ℤ  (2.3.3)

defined by linearity by mapping the H-set X to the sequence (|X^{K}|)_{K∈[s_{H}]}. Hence there is an injective morphism of ℤ-modules

B^{c}(G) → ∏_{g∈[G]} ∏_{∈[s_{C_{G}(g)}]} ℤ ≅ ∏_{(H,g)∈[P_{G}]} ℤ

sending the crossed G-set (X, α) to the sequence (|α^{−1}(g)^{H}|)_{(H,g)∈[P_{G}]}. Now there is an isomorphism

∏_{(H,g)∈[P_{G}]} ℤ ≅ ∏_{H∈[s_{G}]} ZℤC_{G}(H) .

This gives the injective map

Θ_{ℤ} : B^{c}(G) → ∏_{H∈[s_{G}]} ZℤC_{G}(H)

and the map Θ_{R} of lemma 2.3.2 is obtained by tensoring this map with R, which is a flat ℤ-module if R has characteristic 0. As a consequence :

Proposition 2.3.4: Let K be a field of characteristic 0. Then the map

Ψ_{K} : ∏_{H⊆G} z_{H} : B_{K}^{c}(G) → ( ∏_{H⊆G} ZKC_{G}(H))^{G}

is an isomorphism of K-algebras. The inverse isomorphism maps the sequence (z_{H})_{H⊆G} to

\frac{1}{|G|} ∑_{{(L,g)∈P_{G} \atop L⊆H⊆C_{G}(g)}} |L|μ(L, H)z_{H}(g)[L, g]_{G}

where μ(L, H) denotes the Möbius function of the poset of subgroups of G, and z_{H}(g) denotes the coefficient of g in z_{H}.

Proof: The map Ψ_{K} is injective by lemma 2.3.2, and its image is contained in

( ∏_{H⊆G} ZKC_{G}(H))^{G} ≅ ∏_{H∈[s_{G}]} (KC_{G}(H))^{N_{G}(H)} .

Now this K-vector space has the same (finite) dimension as B_{K}^{c}(G), namely the cardinality of [P_{G}]. The first assertion follows.

To build the inverse map, note that for any finite group G, the Burnside algebra B_{K}(G) is a split semi-simple commutative K-algebra. The primitive idempotents of B_{K}(G) have been determined by Gluck ([5]). They are indexed by the (conjugacy classes of) subgroups of the group G. The idempotent e_{H}^{G} indexed by H is equal to (see [3] Theorem 3.3.2)

e_{H}^{G} = \frac{1}{|N_{G}(H)|} ∑_{L⊆H} |L|μ(L, H) G/L .  (2.3.5)

This idempotent is characterized by the fact that for any X ∈ B_{K}(G), one has that

X.e_{H}^{G} = |X^{H}|e_{H}^{G} .  (2.3.6)

It follows in particular that

X = ∑_{H∈[s_{G}]} X.e_{H}^{G} = ∑_{H∈[s_{G}]} |X^{H}|e_{H}^{G} = \frac{1}{|G|} ∑_{L⊆H⊆G} |L|μ(L, H)|X^{H}|G/L .

Now if (X, α) is a crossed G-set, the corresponding element (z_{H})_{H⊆G} is defined by

z_{H}(g) = |α^{−1}(g)^{H}|

thus

α^{−1}(g) = \frac{1}{|C_{G}(g)|} ∑_{L⊆H⊆C_{G}(g)} |L|μ(L, H)z_{H}(g) C_{G}(g)/L

in B(C_{G}(g)), and the second assertion follows.

Remark 2.3.7: Proposition 2.3.4 can be used to show that B_{K}^{c}(G) is a split semi-simple (commutative) algebra, and to state explicit formulae for its primitive idempotents. Those formulae are due to Yoshida.

It is possible to characterize the image of the above map Θ_{ℤ} :

Proposition 2.3.8: If (H, g) ∈ P_{G}, and if K is a subgroup of C_{G}(g), set

n_{g}(H, K) = |{x ∈ N_{G}(H) ∩ C_{G}(g)/H | <H, x>=_{C_{G}(g)} K}|

where <H, x> is the subgroup of G generated by H and x, and the notation < H, x >=_{C_{G}(g)} K means that < H, x > and K are conjugate by an element of C_{G}(g).

For H ∈ [s_{G}], let z_{H} = ∑_{g∈C_{G}(H)} z_{H}(g)g be an element of ZℤC_{G}(H). Then the sequence (z_{H})_{H∈[s_{G}]} belongs to the image of Θ_{ℤ} if and only if for any H in [s_{G}] the following two conditions hold :

1. The element z_{H} is invariant by N_{G}(H).

2. For any g ∈ C_{G}(H), the sum ∑_{K∈[s_{C_{G}(g)}]} n_{g}(H, K)z_{K}(g) is a multiple of |N_{G}(H) ∩ C_{G}(g) : H|.

Proof: This follows from a theorem of Dress ([4], or [3] Theorem 3.2.1), characterizing the image of the map ϕ_{H} defined in (2.3.3): giving an element z_{H} in (ℤC_{G}(H))^{N_{G}(H)}, for H ∈ [s_{G}] is equivalent to giving integers z_{H}(g), for (H, g) ∈ [P_{G}]. Now for a fixed g ∈ G, the sequence of integers (z_{H}(g))_{H∈[s_{C_{G}(g)}]} is in the image of ϕ_{C_{G}(g)} if and only if condition 2) holds.

3 The prime spectrum of B_{R}^{c}(G)

3.1 Prime spectrum

Lemma 3.1.1: The Krull dimension of the ring B_{R}^{c}(G) is equal to the Krull dimension of R.

Proof: Indeed, the ring B_{R}^{c}(G) is an extension of R, and it is a finitely generated (free) R-module. Hence it is integral over R. Thus dim B_{R}^{c}(G) = dim R.

Lemma 3.1.2: If R has characteristic zero, then the map Θ_{R} induces a surjection

Spec(Θ_{R}) : Spec( ∏_{H∈[s_{G}]} ZRC_{G}(H)) → Spec(B_{R}^{c}(G)) .

Proof: Here again, the ring C = ∏_{H∈[s_{G}]} ZRC_{G}(H) is an extension of B_{R}^{c}(G). Moreover, it is a finitely generated R-module, hence also a finitely generated B_{R}^{c}(G)-module. Thus C is integral over B_{R}^{c}(G), and Θ_{R} induces a surjective maps on the spectra.

Notation 3.1.3: If 𝔭 is a prime ideal of R, denote by k(𝔭) the field of fractions of R/𝔭. Denote by φ_{𝔭} the canonical morphism from ZRG to Zk(𝔭)G. If b is a block of k(𝔭)G, set

I_{𝔭,b} = {u ∈ ZRG | φ_{𝔭}(u)b ∈ J(Zk(𝔭)Gb)}

(where J(Zk(𝔭)Gb) denotes the Jacobson radical of the algebra Zk(𝔭)Gb). It is an ideal of ZRG.

Lemma 3.1.4: Let 𝔭 (resp. 𝔭′) denote a prime ideal of R, and b (resp. b′) denote a block of k(𝔭)G (resp. k(𝔭′)G). Then :

1. The ideal I_{𝔭,b} is a prime ideal of ZRG. Conversely, if I is a prime ideal of ZRG, there exist a unique prime ideal 𝔭 of R and a unique block b of k(𝔭)G such that I = I_{𝔭,b}.

2. If I_{𝔭,b} ⊆ I_{𝔭′,b′}, then 𝔭 ⊆ 𝔭′. If moreover 𝔭 = 𝔭′, then b = b′.

3. If 𝔭 ⊆ 𝔭′, and if the block b is given, then there exists a block b′ such that I_{𝔭,b} ⊆ I_{𝔭′,b′}.

4. If 𝔭 ⊆ 𝔭′, and if the block b′ is given, then there exists a block b such that I_{𝔭,b} ⊆ I_{𝔭′,b′}.

Proof: Clearly I_{𝔭,b} is the kernel of the canonical morphism

ZRG →^{φ_{𝔭}} Zk(𝔭)G →^{b} Zk(𝔭)Gb → Zk(𝔭)Gb/J(Zk(𝔭)Gb)

Since the ring on the right hand side is a field, it follows that I_{𝔭,b} is a prime ideal of ZRG.

The map u : r ∈ R ↦ r.1 ∈ ZRG is an injective ring homomorphism. Thus ZRG is an extension ring of R. If I is a prime ideal in ZRG, then 𝔭 = I ∩ R is a prime ideal of R. Moreover

Z𝔭G = 𝔭ZRG ⊆ I .

Let π denote the projection map ZRG → ZRG/I, and i : ZRG/I → K denote the embedding of ZRG/I in its field of fractions K. The kernel of the map ρ = i ∘ π ∘ u : R → K is equal to 𝔭. Hence ρ factors through the canonical map λ_{𝔭} : R → k(𝔭). Hence there is a map θ : Zk(𝔭)G → K and a commutative square

\begin{matrix}ZRG & \underrightarrow{π} & ZRG/I \\ φ_{𝔭} ↓ & & ↓ i \\ Zk(𝔭)G & \overrightarrow{θ} & K \end{matrix} .

The ideal I is the kernel of the map θ∘φ_{𝔭}, hence it is the inverse image under φ_{𝔭} of the prime ideal θ^{−1}(0) of Zk(𝔭)G. This ring has dimension zero. Hence there exists a primitive idempotent b of Zk(𝔭)G (i.e. a block of k(𝔭)G) such that θ^{−1}(0) = J(Zk(𝔭)Gb)+ ∑_{b′≠b} Zk(𝔭)Gb′ (where b′ denotes a block of kG).

Hence I = I_{𝔭,b}.

Finally if 𝔭′ is a prime ideal of R, and b′ is a block of k(𝔭′)G such that I_{𝔭,b} ⊆ I_{𝔭′,b′}, then

𝔭 = R ∩ I_{𝔭,b} ⊆ R ∩ I_{𝔭′,b′} = 𝔭′ .

Suppose moreover that 𝔭 = 𝔭′. The idempotent b can be written

b = ∑_{g∈G}(r_{g}/s_{g})g

where r_{g} ∈ R/𝔭, and s_{g} ∈ R/𝔭 − {0}, for g ∈ G. There is an element s ∈ R which maps to ∏_{g∈G} s_{g} in R/𝔭, and there is an element v ∈ ZRG such that

φ_{𝔭}(v) = λ_{𝔭}(s)b .

It follows that v − s.1 ∈ I_{𝔭,b} = I_{𝔭′,b′}. Thus φ_{𝔭}(v − s.1)b′ is nilpotent. But if b′ ≠ b, then

φ_{𝔭}(v − s.1)b′ = λ_{𝔭}(s)bb′ − λ_{𝔭}(s)b′ = −λ_{𝔭}(s)b′ .

This cannot be nilpotent since λ_{𝔭}(s) is non-zero in k(𝔭). Thus b = b′. This completes the proof of assertions 1) and 2).

Assertion 3) is nothing but the going up theorem, which holds between R and ZRG because ZRG is free as an R-module, hence flat over R (see [6] Theorem 9.5).

Similarly, assertion 4) is the going down theorem, which holds between R and ZRG because ZRG is a finitely generated R-module, hence integral over R (see [6] Theorem 9.4).

Notation 3.1.5: If H is a subgroup of G, if 𝔭 is a prime ideal of R, and if b is a block of k(𝔭)C_{G}(H), I set

I_{H,𝔭,b} = z_{H}^{−1}(I_{𝔭,b}) .

It is a prime ideal of B_{R}^{c}(G).

Corollary 3.1.6: If R has characteristic zero, and if I is a prime ideal of B_{R}^{c}(G), then there exists a subgroup H of G, a prime ideal 𝔭 of R, and a block b of k(𝔭)C_{G}(H), such that I = I_{H,𝔭,b}.

3.2 p-blocks

Notation 3.2.1: From now on, the letter O will denote a complete discrete valuation ring of characteristic 0, with maximal ideal π, with residue field k of characteristic p > 0, and field of fractions K, which will be supposed big enough (i.e. K contains the |G|^{th} roots of unity). I denote by x ↦ \overline{x} the reduction morphism from O to k, or from OG to kG.

If H is a group of order dividing |G|, the group algebra KG is split and semi-simple. If χ is an ordinary irreducible character of H, I denote by e_{χ} the corresponding block of KH, and by ω_{χ} the morphism from ZOH to O mapping u to χ(u)/χ(1).

I will say that χ is in the block e (or belongs to the block e) of kH if e acts as identity morphism on a reduction of a simple module affording the character χ, or equivalently, if there is a block E of ZOH such that e_{χ}.E = e_{χ} and \overline{E} = e.

If P is a p-subgroup of G, I denote by br_{P} the Brauer morphism from ZkG to ZkC_{G}(P), which is the k-linear map sending x ∈ G to itself if x ∈ C_{G}(P), and to 0 otherwise.

A Brauer pair (P, e) is a pair consisting of a p-subgroup P of G, and a block e of the algebra kC_{G}(P). If b is a block of kG, the Brauer pair (P, e) is a b-Brauer pair if e.br_{P} (b) = e.

For the remainder of this section, the ring R will be equal to O. The primitive idempotents of the Burnside algebra B_{O}(G) follow from a theorem of Dress ([4] or [3] Corollary 3.3.6). They are indexed by the (conjugacy classes of) p-perfect subgroups of G. The idempotent f_{H}^{G} indexed by the p-perfect subgroup H of G is equal to the sum of the idempotents e_{K}^{G} (see equation 2.3.5) for which O^{p}(K) is conjugate to H, each taken once (i.e. K is taken once up to conjugation in G). In particular

f_{1}^{G} = ∑_{P ∈\underline{s}_{p}(G)/G} e_{P}^{G}  (3.2.2)

where \underline{s}_{p}(G) is the set of p-subgroups of G. The corresponding block B_{O}(G)f_{1}^{G} of the Burnside algebra is the O-submodule generated by the Gsets G/P, for P ∈ \underline{s}_{p}(G).

Now the algebra homomorphism β : B_{O}(G) → B_{O}^{c}(G) of lemma 2.1.1 provides a decomposition of unity in B_{O}^{c}(G) as a sum of orthogonal idempotents β(f_{H}^{G}), for p-perfect subgroups H of G, up to conjugation. These idempotents are no longer primitive, and in this section I will show how the idempotent β(f_{1}^{G}) splits as a sum of primitive idempotents of B_{O}^{c}(G).

Notation 3.2.3: I will denote by A(G) the O-algebra B_{O}^{c}(G)β(f_{1}^{G}), and by \overline{A}(G) the k-algebra k ⊗_{O} A(G). The algebra A(G) will be called the p-local crossed Burnside algebra (over O).

Proposition 3.2.4: 1. The map

Ψ_{O} = ∏_{P ∈\underline{s}_{p}(G)} z_{P} : A(G) → ( ∏_{P ∈\underline{s}_{p}(G)} ZOC_{G}(P))^{G}

is an injective map of O-algebras, which preserves identity elements, and induces an isomorphism of K-algebras

Ψ_{K} : K ⊗_{O} A(G) ≅ ( ∏_{P ∈\underline{s}_{p}(G)} ZKC_{G}(P))^{G} .

2. The algebra A(G) is the O-submodule of B_{O}^{c}(G) with basis the set of elements [P, x]_{G} of [P_{G}] for which P is a p-group.

Proof: If X is a G-set and if H is a subgroup of G

(z_{H} ∘ β)(X) = z_{H}([X, u_{X}]) = ∑_{x∈X^{H}} u_{X}(x) = |X^{H}|.1 ∈ ZRC_{G}(H)

where u_{X} is defined in lemma 2.1.1. It follows from equations 2.3.6 and 3.2.2 that (z_{H} ∘ β)(f_{1}^{G}) = 0 if H is not a p-group, and that (z_{H} ∘ β)(f_{1}^{G}) = 1 ∈ ZRC_{G}(H) otherwise. Now the map Ψ_{O} is just the restriction to A(G) of the map Θ_{O} of lemma 2.3.2, hence it is injective. This proves the first part of assertion 1).

The previous argument actually shows that if P is a p-group and if x is an element of C_{G}(P), then

z_{H}([P, x]_{G}β(f_{1}^{G})) = \{ \begin{matrix}0 & if H is not a p-group \\ z_{H}([P, x]_{G}) & otherwise \end{matrix}

In both cases z_{H}([P, x]_{G}β(f_{1}^{G})) = z_{H}([P, x]_{G}), thus [P, x]_{G}β(f_{1}^{G}) = [P, x]_{G} by lemma 2.3.2. In other words [P, x]_{G} ∈ A(G) if P is a p-group. Conversely β(f_{1}^{G}) is a linear combination with coefficients in O of pairs [P, x]_{G} of P_{G} for which P is a p-group, and these linear combinations clearly form an ideal of B_{O}^{c}(G). Assertion 2) follows. The second part of assertion 1) also follows, since Ψ_{K} is an injective map of K-vector spaces of the same (finite) dimension.

Remark 3.2.5: Assertion 2) means that A(G) is generated over O by the images of the crossed G-sets (X, α) for which the stabilizer in G of any element of X is a p-group.

Notation 3.2.6: If P is a p-subgroup of G denote by \overline{z}_{P} the map \overline{A}(G) → ZkC_{G}(G) such that the square

A(G) \underrightarrow{z_{P}} ZOC_{G}(P)
↓  ↓
\overline{A}(G) \overrightarrow{\overline{z}_{P}} ZkC_{G}(P)

is commutative, where the vertical arrows are the reduction maps.

Lemma 3.2.7: For any p-subgroup P of G

\overline{z}_{P} = br_{P} ∘ \overline{z}_{1}

Proof: Indeed, if v is the image of the crossed G-set (X, α) in \overline{A}(G), then

\overline{z}_{1}(v) = ∑_{x∈X} α(x) = ∑_{g∈G} |α^{−1}(g)|g ∈ ZkG

whereas

\overline{z}_{P} (v) = ∑_{x∈X^{P}} α(x) = ∑_{g∈C_{G}(P)} |α^{−1}(g)^{P} |g ∈ ZkC_{G}(P)

and the lemma follows since |α^{−1}(g)| = |α^{−1}(g)^{P} | in k for any p-subgroup P of C_{G}(g) : this is because k has characteritic p, and because |α^{−1}(g)| and |α^{−1}(g)^{P} | are congruent modulo p, since P is a p-group.

The following theorem describes the prime spectrum of A(G), which is a ring of dimension 1 by lemma 3.1.1. If I is a prime ideal of A(G), and q ∈ {0, p}, I will say that I has co-characteristic q if the ring A(G)/I has characteristic q :

Theorem 3.2.8: Denote by b a block of kG, by P (resp P′) a p-subgroup of G, by χ (resp. χ′) an irreducible character of C_{G}(P) (resp. C_{G}(P′)), and by e a block of kC_{G}(P). Let I be a prime ideal of A(G).

1. The following are equivalent :

(a) The ideal I has co-characteristic 0.

(b) The ideal I is a minimal prime ideal.

(c) There exist a p-subgroup P of G and an irreducible character χ of C_{G}(P) such that I = I_{P,0,e_{χ}}.

2. The following are equivalent :

(a) The ideal I has co-characteristic p.

(b) The ideal I is a maximal prime ideal.

(c) There exist a p-subgroup P of G and a block e of kC_{G}(P) such that I = I_{P,π,e}.

3. The ideal I_{P,0,e_{χ}} is contained in I_{P′,0,e_{χ′}} if and only if the pairs (P, e_{χ}) and (P′, e_{χ′}) are conjugate in G. In this case moreover I_{P,0,e_{χ}} = I_{P′,0,e_{χ′}}.

4. The ideal I_{P,0,e_{χ}} is contained in I_{P,π,e} if and only if the character χ belongs to the block e. In this case, the inclusion is strict.

5. The ideal I_{1,π,b} is contained in I_{P,π,e} if and only if (P, e) is a b-Brauer pair, i.e. if e.br_{P} (b) = e. In this case moreover I_{1,π,b} = I_{P,π,e}.

6. The connected components of Spec(A(G)) are in one to one correspondence with the blocks of kG. The component C_{b} associated to the block b consists of the unique maximal prime ideal I_{1,π,b}, and of the ideals I_{P,0,e_{χ}}, where P is a p-subgroup of G, and χ is an irreducible character of C_{G}(P) belonging to a block e of C_{G}(P) such that (P, e) is a b-Brauer pair.

Proof: By corollary 3.1.6, any prime ideal I of A(G), which is also a prime ideal of B_{O}^{c}(G), is equal to some ideal I_{H,𝔭,b}, for a subgroup H of G, a prime ideal 𝔭 of O (hence 𝔭 = 0 or 𝔭 = π), and a block b of k(𝔭)C_{G}(H) (i.e. a block e_{χ} of KC_{G}(H) corresponding to an irreducible character χ if 𝔭 = 0, or a block b of kC_{G}(H) if 𝔭 = π). The ideal I_{H,𝔭,b} is an ideal of A(G) if and only if β(f_{1}^{G}) ∉ I_{H,𝔭,b}, or equivalently if H is a p-group.

Now the ideal I_{H,𝔭,b} has co-characteristic 0 if 𝔭 = 0, and p if 𝔭 = π. In particular, the ideals I_{H,0,e_{χ}} and I_{H,π,b} are distinct. By the last two assertions of lemma 3.1.4, any ideal I_{H,π,b} contains an ideal I_{H,0,e_{χ}}, and any ideal I_{H,0,e_{χ}} is contained in an ideal I_{H,π,b}. It follows that the minimal primes of A(G) are the ideals I_{P,0,e_{χ}}, for p-subgroups P of G, whereas the maximal primes are the ideals I_{P,π,b}. This proves assertions 1) and 2) of the theorem.

Now

I_{P,0,e_{χ}} = {u ∈ A(G) | ω_{χ} ∘ z_{P} (u) = 0}

It follows that I_{P,0,e_{χ}} = I_{P′,0,e_{χ′}} if the pairs (P, e_{χ}) and (P′, e_{χ′}) are conjugate. Conversely, if I_{P,0,e_{χ}} ⊆ I_{P′,0,e_{χ′}}, then I_{P,0,e_{χ}} = I_{P′,0,e_{χ′}} since both are minimal

prime ideals. Let f denote the idempotent of ( ∏_{Q∈\underline{s}_{p}(G)} ZKC_{G}(Q))^{G} with Q-component equal to the sum of idempotents e_{θ} for which (Q, e_{θ}) is Gconjugate to (P, e_{χ}). Let F = Ψ_{K}^{−1}(f) be the corresponding idempotent of B_{K}^{c}(G). Then there is a non-zero integer m such that mF lies in B_{O}(G). Moreover z_{Q}(mF) = 0 if Q is not conjugate to P in G, and z_{P} (mF) is m times the sum of the different conjugates of e_{χ} under N_{G}(P). In particular mF is not in I_{P,0,e_{χ}}, hence not in I_{P′,0,e_{χ′}}. It follows that ω_{χ′} ∘ z_{P′}(mF) ≠ 0. In particular P′ is conjugate to P, and I can suppose P = P′. Now there is a conjugate of e_{χ} under N_{G}(P) which is not in the kernel of ω_{χ′}. Hence e_{χ} and e_{χ′} are conjugate under N_{G}(P). This proves assertion 3).

The ideal I_{P,0,e_{χ}} is contained in the ideal I_{P,π,e} if and only if the ideal I_{0,e_{χ}} of ZOC_{G}(P) is contained in the ideal I_{π,e}. Now

I_{0,e_{χ}} = {u ∈ ZOC_{G}(P) | ω_{χ}(u) = 0} .

On the other hand, if θ is any character in the block e, then

I_{π,e} = \{u ∈ ZOC_{G}(P) | \overline{ω_{θ}(u)} = 0\} .

It follows that I_{0,e_{χ}} ⊆ I_{π,e} if χ belongs to e. Conversely, if I_{0,e_{χ}} ⊆ I_{π,e}, then let E be the block of OC_{G}(P) containing χ. Then ω_{χ}(1 − E) = 0, hence \overline{ω_{θ}(1 − E)} = 0 for any character θ in e. Thus ω_{θ}(E) = 1, and e = \overline{E}. Hence χ is in e, proving assertion 4).

Let b̂ be a block of OG lifting b (i.e. such that \overline{b̂} = b). Write

b̂ = ∑_{x∈G} r_{x}x

for coefficients r_{x} ∈ O. Consider

b̃ = ∑_{x∈[G]} r_{x}[C_{G}(x), x]_{G} .

If H is any subgroup of G, then

z_{H}(b̃) = ∑_{x∈[G]} r_{x} ∑_{w∈(G/C_{G}(x))^{H}} ^{w}x = ∑_{x∈C_{G}(H)} r_{x}x .

The second equality comes from the fact that w ∈ (G/C_{G}(x))^{H} if and only if ^{w}x ∈ C_{G}(H). Moreover r_{x} = r_{^{w}x} since b̂ is central in OG.

In particular for a p-subgroup P of G

\overline{z_{P} (b̃)} = br_{P} (b)

Suppose that I_{1,π,p} ⊆ I_{P,π,e}. For any character χ in b

\overline{ω_{χ}(z_{1}(1 − b̃))} = \overline{ω_{χ}(1 − b̂)} = 1 − \overline{ω_{χ}(b)} = 0

Thus 1 − b̃ ∈ I_{1,π,p}. Hence 1 − b̃ ∈ I_{P,π,e}, and e.\overline{z_{P} (1 − b̃)} must be nilpotent. But

e.\overline{z_{P} (1 − b̃)} = e − e.br_{P} (b)

This is nilpotent and idempotent, hence zero.

Conversely, suppose that e = e.br_{P} (b). Then

I_{1,π,b} = \{u ∈ B_{O}^{c}(G) | \overline{z_{1}(u)}b is nilpotent\}

whereas

I_{P,π,e} = \{u ∈ B_{O}^{c}(G) | \overline{z_{P} (u)}e is nilpotent\} .

Now if u ∈ I_{1,π,b}

\overline{z_{1}(u)}b = \overline{z_{1}(u)}br_{1}(b) = \overline{z_{1}(u)}\overline{z_{1}(b̃)} = \overline{z_{1}(ub̃)} .
\overline{z_{P} (u)}e = \overline{z_{P} (u)}br_{P} (b)e = \overline{z_{P} (u)}\overline{z_{P} (b̃)}e = \overline{z_{P} (ub̃)}e .

Now lemma 3.2.7 shows that I_{1,π,b} ⊆ I_{P,π,e}. Hence I_{1,π,b} = I_{P,π,e} since I_{1,π,b} is a maximal ideal in A(G). This completes the proof of assertion 5).

Assertion 5) also shows that all the maximal ideals of A(G) are of the form I_{1,π,b}, for a suitable bloc b of kG. Now the minimal primes are of the form I_{P,0,χ}, for a p-subgroup P of G, for an irreducible character χ of C_{G}(P). This ideal is contained in I_{P,π,e} for a block e of kC_{G}(P) if and only if e contains χ. Moreover I_{P,π,e} is equal to I_{1,π,b} for a block b of kG if and only if (P, e) is a b-Brauer pair. This shows that each minimal prime ideal is contained in a unique maximal ideal. Assertion 6) follows, and the proof of theorem 3.2.8 is complete.

Notation 3.2.9: If G is a finite group, if P is a p-subgroup of G, and b is an idempotent in ZkG, I denote by br_{P}^{O}(b) the unique idempotent in ZOC_{G}(P) lifting the idempotent br_{P} (b) of ZkC_{G}(P). If g ∈ C_{G}(P), I denote by br_{P}^{O}(b)(g) the element of O such that

br_{P}^{O}(b) = ∑_{g∈C_{G}(g)} br_{P}^{O}(b)(g) g

Theorem 3.2.10: If b is an idempotent of ZkG, let

b^{A} = \frac{1}{|G|} ∑_{{g∈G \atop Q⊆P ∈\underline{s}_{p}(C_{G}(g))}} |Q|μ(Q, P)br_{P}^{O}(b)(g) [Q, g]_{G}

Then b^{A} ∈ A(G), and as b runs through the blocks of kG, the elements b^{A} run through a complete set of primitive idempotents in A(G).

Proof: The primitive idempotents of the (commutative) ring A(G) are in one to one correspondence with the connected components of its spectrum, hence with the blocks of kG by theorem 3.2.8 : if b is a block of kG, then the idempotent b^{A} corresponding to the component C_{b} is characterised by the fact that b^{A} ∉ I for some I ∈ C_{b} (or equivalently for all I ∈ C_{b}). It follows that b^{A} ∉ I_{1,π,b}, but b^{A} ∈ I_{1,π,b′} for any block b′ of kG different from b.

Thus ω_{b}(\overline{z}_{1}(b^{A})) ≠ 0, but ω_{b′}(\overline{z}_{1}(b^{A})) = 0 for any block b′ ≠ b of kG. Since \overline{z}_{1}(b^{A}) is an idempotent of ZkG, it follows that \overline{z}_{1}(b^{A}) = b. Hence \overline{z}_{P} (b^{A}) = br_{P} (b) for any p-subgroup P of G, by lemma 3.2.7. Now z_{P} (b^{A}) is an idempotent of ZOC_{G}(P) lifting br_{P} (b), hence it is equal to br_{P}^{O}(b). Theorem 3.2.10 now follows from proposition 3.2.4 and from the inversion formula of proposition 2.3.4.

Corollary 3.2.11: Let b be an idempotent of ZkG. If Q is a p-subgroup of G, and g ∈ C_{G}(Q), then

∑_{Q⊆P ∈\underline{s}_{p}(G)} μ(Q, P)br_{P}^{O}(g) ∈ |N_{G}(Q) ∩ C_{G}(g) : Q|O

Proof: This follows from the following rewriting of the formula in theorem 3.2.10

b^{A} = ∑_{{(Q,g)∈[P_{G}] \atop Q⊆P ∈\underline{s}_{p}(C_{G}(g))}} \frac{μ(Q, P)br_{P}^{O}(b)(g)}{|N_{G}(Q) ∩ C_{G}(g) : Q|} [Q, g]_{G}

and from the fact that the elements [Q, g]_{G}, for (Q, g) ∈ [P_{G}] and Q ∈ \underline{s}_{p}(G), form a basis of A(G) over O, by proposition 3.2.4.

4 The blocks of the Mackey algebra

4.1 Mackey functors

Let R be a commutative ring, and G be a finite group. There are several definitions of the notion of Mackey functor for G over R (see [1] Chapter 1 for a summary of these definitions). One of the most conceptual is due to Dress:

Definition 4.1.1: Let G-set be the category of finite sets with a left Gaction. A Mackey functor for the group G, with values in R-Mod, is a bivariant functor from G-set to R-Mod, i.e. a pair of functors (M^{∗}, M_{∗}), with M^{∗} contravariant and M_{∗} covariant, which coincide on objects (i.e. M^{∗}(X) = M_{∗}(X) = M(X) for any G-set X). This bivariant functor is supposed to have the following two properties:

• If X and Y are G-sets, let i_{X} and i_{Y} be the respective injections from X and Y into X ∐ Y, then the maps M^{∗}(i_{X}) ⊕ M^{∗}(i_{Y}) and M_{∗}(i_{X}) ⊕ M_{∗}(i_{Y}) are mutual inverse R-module isomorphisms between M(X ∐ Y) and M(X) ⊕ M(Y).

• If

T \underrightarrow{γ} Y
δ ↓  ↓ α
Z \overrightarrow{β} X

is a cartesian (or pull-back) square of G-sets, then M^{∗}(β).M_{∗}(α) = M_{∗}(δ).M^{∗}(γ).

A morphism θ from the Mackey functor M to the Mackey functor N is a natural transformation of bivariant functors, consisting of a morphism θ_{X} :

M(X) → N(X) for any G-set X, such that for any morphism of G-sets f : X → Y, the squares

\begin{matrix}M(X) & \underrightarrow{θ_{X}} & N(X) \\ M_{∗}(f) ↓ & & ↓ N_{∗}(f) \\ M(Y) & \overrightarrow{θ_{Y}} & N(Y) \end{matrix}  \begin{matrix}M(X) & \underrightarrow{θ_{X}} & N(X) \\ M^{∗}(f) ↑ & & ↑ N^{∗}(f) \\ M(Y) & \overrightarrow{θ_{Y}} & N(Y) \end{matrix}

are commutative.

I will denote by Mack_{R}(G) the category of Mackey functors for G over R.

4.2 The Mackey algebra

The Mackey algebra μ_{R}(G) of R over G was defined by Thévenaz and Webb ([8]) : it is the R-algebra generated by the elements t_{K}^{H}, r_{K}^{H}, and c_{x,H}, where H and K are subgroups of G such that K ⊆ H, and x ∈ G, with the following relations:

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

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

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

t_{H}^{H} = r_{H}^{H} = c_{h,H} ∀ h, H such that h ∈ H
c_{x,H}t_{K}^{H} = t_{^{x}K}^{^{x}H}c_{x,K} ∀ x, K, H such that K ⊆ H
c_{x,K}r_{K}^{H} = r_{^{x}K}^{^{x}H}c_{x,H} ∀ x, K, H such that K ⊆ H

∑_{H} t_{H}^{H} = ∑_{H} r_{H}^{H} = 1

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

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

It can be shown from Proposition 3.4 of [8] that the map x ∈ G ↦ ∑_{H⊆G} c_{x,H} extends to an injective R-algebra homomorphism from RG to μ_{R}(G), and it is handy to identify G with its image in μ_{R}(G) via this map.

Thévenaz and Webb show that the category of μ_{R}(G)-modules is equivalent to the category of Mackey functors for G over R. This equivalence is build as follows : if K ⊆ H are subgroups of G, denote by p_{K}^{H} : G/K → G/H the map of G-sets sending xK to xH, for x ∈ G. If g ∈ G, denote by γ_{g} : G/H → G/^{g}H the map sending xH to xg.^{g}H, for x ∈ G. Then if M is a Mackey functor, the R-module ⊕_{L⊆G}M(L) can be endowed with a μ_{R}(G)- module structure : the generator t_{K}^{H} maps M(L) to 0 if L ≠ K, and M(K) to M(H) via the map M_{∗}(p_{K}^{H}). Similarly, the generator r_{K}^{H} maps M(L) to 0 if L ≠ H, and M(H) to M(K) via the map M^{∗}(p_{K}^{H}). Finally the generator c_{g,H} maps M(L) to 0 if L ≠ H, and M(H) to M(^{g}H) via the map M_{∗}(γ_{g}).

4.3 Action of crossed G-sets

Any crossed G-set gives an endofunctor of the category of Mackey functors for G over R, and this will lead to a ring homomorphism from B_{R}^{c}(G) to the center Zμ_{R}(G) of the Mackey algebra. This action of crossed G-sets on Mackey functors was already observed by Yoshida.

Let (X, α) be a crossed G-set. If M is any Mackey functor for G over R, and if Y is a finite G-set, let ζ(X, α)_{M,Y} denote the endomorphism of M(Y) defined by

ζ(X, α)_{M,Y} = M_{∗} (\begin{matrix}x,y \\ ↓ \\ α(x)y \end{matrix}) M^{∗} (\begin{matrix}x,y \\ ↓ \\ y \end{matrix})

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

and (\begin{matrix}x,y \\ ↓ \\ y \end{matrix}) is the projection map from X × Y to Y. This definition extends the one given by Thévenaz and Webb ([8] Section 9) for the action of B(G) on Mackey functors.

Proposition 4.3.1: Let (X, α) be a crossed G-set. The the maps ζ(X, α)_{M,Y} define a natural transformation ζ(X, α) of the identity functor of the category Mack_{R}(G). Moreover, if (Y, β) is another crossed G-set, then

ζ(X, α) + ζ(Y, β) = ζ(X ⊔ Y, α ⊔ β)
ζ(X, α) ∘ ζ(Y, β) = ζ(X × Y, α.β)

Proof: This amounts to a series of verifications : first the maps ζ(X, α)_{M,Y} define an endomorphism ζ(X, α)_{M} of the Mackey functor M. It means that for any morphism of G-sets f : Y → Z, one has that

ζ(X, α)_{M,Z} ∘ M_{∗}(f) = M_{∗}(f) ∘ ζ(X, α)_{M,Y}  (4.3.2)

ζ(X, α)_{M,Y} ∘ M^{∗}(f) = M^{∗}(f) ∘ ζ(X, α)_{M,Z}  (4.3.3)

Equation 4.3.2 reads

M_{∗} (\begin{matrix}x,z \\ ↓ \\ α(x)z \end{matrix}) M^{∗} (\begin{matrix}x,z \\ ↓ \\ z \end{matrix}) M_{∗}(f) = M_{∗}(f)M_{∗} (\begin{matrix}x,y \\ ↓ \\ α(x)y \end{matrix}) M^{∗} (\begin{matrix}x,y \\ ↓ \\ y \end{matrix})  (4.3.4)

The square

X × Y \underrightarrow{(\begin{matrix}x,y \\ ↓ \\ y \end{matrix})} Y
(\begin{matrix}x,y \\ ↓ \\ x,f(y) \end{matrix}) ↓  ↓ f
X × Z \overrightarrow{(\begin{matrix}x,z \\ ↓ \\ z \end{matrix})} Z

is a cartesian square of G-sets. Hence

M^{∗} (\begin{matrix}x,z \\ ↓ \\ z \end{matrix}) M_{∗}(f) = M_{∗} (\begin{matrix}x,y \\ ↓ \\ x,f(y) \end{matrix}) M^{∗} (\begin{matrix}x,y \\ ↓ \\ y \end{matrix})

so equation 4.3.4 is equivalent to

M_{∗} (\begin{matrix}x,y \\ ↓ \\ α(x)f(y) \end{matrix}) M^{∗} (\begin{matrix}x,y \\ ↓ \\ y \end{matrix}) = M_{∗} (\begin{matrix}x,y \\ ↓ \\ f(α(x)y) \end{matrix}) M^{∗} (\begin{matrix}x,y \\ ↓ \\ y \end{matrix})

which holds since f is a morphism of G-sets.

Similarly equation 4.3.3 reads

M_{∗} (\begin{matrix}x,y \\ ↓ \\ α(x)y \end{matrix}) M^{∗} (\begin{matrix}x,y \\ ↓ \\ y \end{matrix}) M^{∗}(f) = M^{∗}(f)M_{∗} (\begin{matrix}x,z \\ ↓ \\ α(x)z \end{matrix}) M^{∗} (\begin{matrix}x,z \\ ↓ \\ z \end{matrix})  (4.3.5)

Similarly the square

X × Y \underrightarrow{(\begin{matrix}x,y \\ ↓ \\ x,f(y) \end{matrix})} X × Z
(\begin{matrix}x,y \\ ↓ \\ α(x)y \end{matrix}) ↓  ↓ (\begin{matrix}x,z \\ ↓ \\ α(x)z \end{matrix})
Y  \overrightarrow{f}  Z

is cartesian, and both sides of equation 4.3.5 are equal to

M_{∗} (\begin{matrix}x,y \\ ↓ \\ α(x)y \end{matrix}) M^{∗} (\begin{matrix}x,y \\ ↓ \\ f(y) \end{matrix}) .

Hence ζ(X, α)_{M} is an endomorphism of the Mackey functor M, for any M. If ϕ : M → N is a morphism of Mackey functors, given by maps ϕ_{Y} : M(Y) → N(Y) for any G-set Y, then

ζ(X, α)_{N,Y} ∘ ϕ_{Y} = N_{∗} (\begin{matrix}x,y \\ ↓ \\ α(x)y \end{matrix}) ∘ N^{∗} (\begin{matrix}x,y \\ ↓ \\ y \end{matrix}) ∘ ϕ_{Y}
= N_{∗} (\begin{matrix}x,y \\ ↓ \\ α(x)y \end{matrix}) ∘ ϕ_{X×Y} ∘ M^{∗} (\begin{matrix}x,y \\ ↓ \\ y \end{matrix})

(since ϕ is a morphism of Mackey functors)

= ϕ_{Y} ∘ M_{∗} (\begin{matrix}x,y \\ ↓ \\ α(x)y \end{matrix}) ∘ M^{∗} (\begin{matrix}x,y \\ ↓ \\ y \end{matrix})
= ϕ_{Y} ∘ ζ(X, α)_{M,Y} .

It follows that the maps ζ(X, α)_{M} define an endomorphism of the identity functor of Mack_{R}(G).

Finally, if (Y, β) is another crossed G-set, then for any Mackey functor M and any G-set Z, the assertion

ζ(X, α)_{M,Z} ∘ ζ(Y, β)_{M,Z} = ζ(X ⊔ Y, α ⊔ β)_{M,Z}

follows easily from the first condition in definition 4.1.1. Concerning composition, one has that

ζ(X, α)_{M,Z} ∘ ζ(Y, β)_{M,Z} = M_{∗} (\begin{matrix}x,z \\ ↓ \\ α(x)z \end{matrix}) M^{∗} (\begin{matrix}x,z \\ ↓ \\ z \end{matrix}) M_{∗} (\begin{matrix}y,z \\ ↓ \\ β(y)z \end{matrix}) M^{∗} (\begin{matrix}y,z \\ ↓ \\ z \end{matrix}) .

Now the square

X × Y × Z \underrightarrow{(\begin{matrix}x,y,z \\ ↓ \\ y,z \end{matrix})} Y × Z
(\begin{matrix}x,y,z \\ ↓ \\ x,β(y)z \end{matrix}) ↓  ↓ (\begin{matrix}y,z \\ ↓ \\ β(y)z \end{matrix})
X × Z \overrightarrow{(\begin{matrix}x,z \\ ↓ \\ z \end{matrix})} Z

is cartesian. It follows that

ζ(X, α)_{M,Z} ∘ ζ(Y, β)_{M,Z} = M_{∗} (\begin{matrix}x,y,z \\ ↓ \\ α(x)β(y)z \end{matrix}) M^{∗} (\begin{matrix}x,y,z \\ ↓ \\ z \end{matrix}) = ζ(X × Y, α.β)_{M,Z}

as was to be shown.

4.4 The center of the Mackey algebra

In any category of modules over a ring A, the natural transformations from the identity functor to itself identify with the center of A : any element z of the center of A defines an endomorphism of any A-module M (by multiplication by z on M), and this gives a natural transformation of the identity functor of the category of A-modules.

Notation 4.4.1: The ring homomorphism from B_{R}^{c}(G) to Zμ_{R}(G) following from proposition 4.3.1 will be still denoted by ζ.

It is given as follows :

Proposition 4.4.2: Let (L, x) ∈ P_{G}. Then

ζ([L, x]_{G}) = ∑_{H⊆G} ∑_{w∈L\G/H} t_{L^{w}∩H}^{H} w^{−1}xw r_{L^{w}∩H}^{H} .

Proof: Let H be a subgroup of G, and M be a Mackey functor for G over R. The element [L, x]_{G} corresponds to the crossed G-set (G/L, m_{x}). The action of ζ(G/L, m_{x}) on M(G/H) is defined as

M_{∗} (\begin{matrix}uL,vH \\ ↓ \\ ^{u}x.vH \end{matrix}) M^{∗} (\begin{matrix}uL,vH \\ ↓ \\ vH \end{matrix}) : M(G/H) → M(G/H) .

This maps factors through M(G/L × G/H), which is isomorphic to

⨁_{w∈L\G/H} M(G/L^{w} ∩ H)

since the map

g(L^{w} ∩ H) ∈ ⨆_{w∈L\G/H} G/L^{w} ∩ H ↦ (gw^{−1}L, gH) ∈ G/L × G/H

is an isomorphism of G-sets.

Up to this isomorphism, the map

M^{∗} (\begin{matrix}uL,vH \\ ↓ \\ vH \end{matrix}) : M(G/H) → ⨁_{w∈L\G/H} M(G/L^{w} ∩ H)

is just the map ⨁_{w∈L\G/H} r_{L^{w}∩H}^{H}, and on the component M(G/L^{w} ∩ H), the

map M_{∗} (\begin{matrix}uL,vH \\ ↓ \\ ^{u}x.vH \end{matrix}) is equal to M_{∗}(ϕ_{w}), where ϕ_{w}(g.(L^{w} ∩ H)) = g.w^{−1}xwH.

Hence M_{∗}(ϕ_{w}) = t_{L^{w}∩H}^{H} w^{−1}xw, and the action of ζ([L, x]_{G}) on M(G/H) is equal to

∑_{w∈L\G/H} t_{L^{w}∩H}^{H} w^{−1}xw r_{L^{w}∩H}^{H} .

The proposition follows, taking for M the free Mackey functor μ_{R}(G).

Remark 4.4.3: Proposition 4.4.2 shows that the composition ζ ∘ β is equal to the morphism from the Burnside ring B_{R}(G) to the center of the Mackey algebra considered in Section 9 of [8].

4.5 The p-blocks of μ_{O}(G)

Notation 4.5.1: Let b be an idempotent of ZkG. I denote by b^{μ} the central idempotent of μ_{O}(G) defined by

b^{μ} = ζ(b^{A}) .

I denote by μ_{O}^{1}(G) the algebra

(ζ ∘ β)(f_{1}^{G})μ_{O}(G) .

This algebra will be called the p-local Mackey algebra.

This notation is consistent with the notation of [2]. It is motivated by Sections 9 and 10 of [8], in which Thévenaz and Webb show that the category Mack_{O}(G) splits as a direct sum of subcategories Mack_{O}(G, J), indexed by pperfect subgroups J of G (up to G-conjugation) : the category Mack_{O}(G, J) consists of μ_{O}(G)-modules M such that (ζ ∘ β)(f_{J}^{G})M = M. In particular, the category Mack_{O}(G, 1) is the category of μ_{O}^{1}(G)-modules. Moreover, the category Mack_{O}(G, J) is equivalent to Mack_{O} (N_{G}(J)/J, 1).

It follows that studying the p-blocks of μ_{O}(G) is equivalent to studying the p-blocks of the algebras μ_{O}^{1} (N_{G}(H)/H), for p-perfect subgroups H of G.

Theorem 4.5.2: Let b be an idempotent of ZkG. Then

b^{μ} = ∑_{H⊆G} \frac{1}{|H|} ∑_{{Q⊆P ∈\underline{s}_{p}(H) \atop x∈C_{G}(P)}} |Q|μ(Q, P)br_{P}^{O}(b)(x) t_{Q}^{H}xr_{Q}^{H} .

The idempotents b^{μ}, as b runs through the blocks of kG, are a complete set of primitive idempotents of the center of the p-local Mackey algebra μ_{O}^{1}(G).

Proof: By proposition 4.4.2 and theorem 3.2.10

b^{μ} = \frac{1}{|G|} ∑_{{g∈G \atop Q⊆P ∈\underline{s}_{p}(C_{G}(g))}} |Q|μ(Q, P)br_{P}^{O}(b)(g) ∑_{H⊆G} ∑_{w∈Q\G/H} t_{Q^{w}∩H}^{H}g^{w}r_{Q^{w}∩H}^{H} .

Changing the order of summations gives

b^{μ} = \frac{1}{|G|} ∑_{H⊆G} ∑_{g∈G ; {Q⊆P ∈\underline{s}_{p}(C_{G}(g)) \atop w∈Q\G/H}} |Q|μ(Q, P)br_{P}^{O}(b)(g)t_{Q^{w}∩H}^{H}g^{w}r_{Q^{w}∩H}^{H}

= \frac{1}{|G|} ∑_{H⊆G} ∑_{{g,w∈G \atop Q⊆P ∈\underline{s}_{p}(C_{G}(g))}} \frac{|Q^{w} ∩ H|}{|Q||H|}|Q|μ(Q, P)br_{P}^{O}(b)(g)t_{Q^{w}∩H}^{H}g^{w}r_{Q^{w}∩H}^{H} .

Summing on Q^{w}, P^{w} and g^{w} instead of Q, P, w leads to

b^{μ} = \frac{1}{|G|} ∑_{H⊆G} \frac{1}{|H|} ∑_{{g,w∈G \atop Q⊆P ∈\underline{s}_{p}(C_{G}(g))}} |Q ∩ H|μ(Q, P)br_{P}^{O}(b)(g)t_{Q∩H}^{H}gr_{Q∩H}^{H}

= ∑_{H⊆G} \frac{1}{|H|} ∑_{{g∈G \atop Q⊆P ∈\underline{s}_{p}(C_{G}(g))}} |Q ∩ H|μ(Q, P)br_{P}^{O}(b)(g)t_{Q∩H}^{H}gr_{Q∩H}^{H} .

Fix a subgroup H of G, an element g of G, and a p-subgroup P of C_{G}(g). By classical combinatorial formulae, if R is any subgroup of H ∩ P

∑_{{Q⊆P \atop Q∩H=R}} μ(Q, P) = \{ \begin{matrix}0 & if H ∩ P ≠ P \\ μ(R, P) & otherwise \end{matrix} .

It follows that

b^{μ} = ∑_{H⊆G} \frac{1}{|H|} ∑_{{R⊆P ∈\underline{s}_{p}(H) \atop g∈C_{G}(P)}} |R|μ(R, P)br_{P}^{O}(b)(g)t_{R}^{H}gr_{R}^{H}

which states the formula in theorem 4.5.2.

To complete the proof of this theorem, it remains to show that if b is a block of kG, i.e. a primitive idempotent of ZkG, then b^{μ} is a primitive idempotent of Zμ_{O}(G). Suppose that b^{μ} splits as the sum of two orthogonal idempotents c and d of Zμ_{O}(G). Then in particular

t_{1}^{1}b^{μ} = t_{1}^{1}b^{μ}t_{1}^{1} = t_{1}^{1}ct_{1}^{1} + t_{1}^{1}dt_{1}^{1}

is a decomposition of t_{1}^{1}b^{μ}t_{1}^{1} as a sum of orthogonal central idempotents in the algebra t_{1}^{1}μ_{O}(G)t_{1}^{1}. This algebra is isomorphic to OG (via x ∈ G ↦ t_{1}^{1}xt_{1}^{1}), and moreover

t_{1}^{1}b^{μ}t_{1}^{1} = t_{1}^{1}bt_{1}^{1} .

Indeed if R ⊆ H are subgroups of G, and if g ∈ G, then t_{1}^{1}.t_{R}^{H}gr_{R}^{H}.t_{1}^{1} = 0 in μ_{O}(G), unless H is the trivial subgroup of G.

If b is primitive, it follows that one of t_{1}^{1}ct_{1}^{1} or t_{1}^{1}dt_{1}^{1} is zero, say t_{1}^{1}ct_{1}^{1}. Now μ_{O}(G)c is a projective Mackey functor in Mack_{O}(G, 1) (i.e. a projective μ_{O}^{1}(G)-module), whose evaluation at the trivial subgroup is equal to

t_{1}^{1}μ_{O}(G)c = t_{1}^{1}ct_{1}^{1}μ_{O}(G) = 0 .

It follows that μ_{O}(G)c = 0, by Corollary 12.2 of [8]. Hence c = 0, and the proof is complete.

5 Consequences

5.1 Another formula

The following gives another expression for b^{μ}, which may be easier to remember :

Proposition 5.1.1: If H is a subgroup of G, denote by X ↦ [X] the Olinear morphism from B_{O}(G) to μ_{O}(G) mapping H/L to t_{L}^{H}r_{L}^{H}. If b is an idempotent of ZkG, then

b^{μ} = ∑_{H⊆G} \frac{1}{|H|} ∑_{P ∈\underline{s}_{p}(G)} |P |t_{P}^{H} ([e_{P}^{P}]br_{P}^{O}(b)) r_{P}^{H} .

Proof: Indeed in B_{K}(P)

e_{P}^{P} = \frac{1}{|P |} ∑_{Q⊆P} |Q|μ(Q, P) P/Q .

Thus

[e_{P}^{P}] = \frac{1}{|P |} ∑_{Q⊆P} |Q|μ(Q, P) t_{Q}^{P}r_{Q}^{P} .

Now by theorem 4.5.2

b^{μ} = ∑_{H⊆G} \frac{1}{|H|} ∑_{{Q⊆P ∈\underline{s}_{p}(H) \atop x∈C_{G}(P)}} |Q|μ(Q, P)br_{P}^{O}(b)(x) t_{Q}^{H}xr_{Q}^{H}

= ∑_{H⊆G} \frac{1}{|H|} ∑_{P ∈\underline{s}_{p}(H)} t_{P}^{H} ( ∑_{{Q⊆P \atop x∈C_{G}(P)}} |Q|μ(Q, P)t_{Q}^{P}br_{P}^{O}(b)(x)xr_{Q}^{P}) r_{P}^{H}

= ∑_{H⊆G} \frac{1}{|H|} ∑_{P ∈\underline{s}_{p}(H)} t_{P}^{H} ( ∑_{{Q⊆P \atop x∈C_{G}(P)}} |Q|μ(Q, P)t_{Q}^{P}r_{Q}^{P}br_{P}^{O}(b)(x)x) r_{P}^{H}

(since C_{G}(P) ⊆ C_{G}(Q) for Q ⊆ P)
= ∑_{H⊆G} \frac{1}{|H|} ∑_{P ∈\underline{s}_{p}(H)} t_{P}^{H} ((∑_{Q⊆P} |Q|μ(Q, P)t_{Q}^{P}r_{Q}^{P})( ∑_{x∈C_{G}(P)} br_{P}^{O}(b)(x)x)) r_{P}^{H}

as was to be shown.

5.2 Residues

Notation 5.2.1: Let M be a Mackey functor for G over R, and H be a subgroup of G. Then Brauer residue of M at H is defined as

\overline{M}(H) = M(H)/ ∑_{L⊂H} t_{L}^{H}M(L)

and the projection M(H) → \overline{M}(H) is denoted by br_{H}. By duality, the Brauer co-residue of M at H is defined as

\underline{M}(H) = ⋂_{L⊆H} Ker r_{L}^{H} ⊂ M(H)

Proposition 5.2.2: Let M be a Mackey functor for G over O, in Mack_{O}(G, 1), and b be a block of kG.

1. If H is a subgroup of G, then \overline{M}(H) = 0, unless H is a p-group. Similarly \underline{M}(H) = 0, unless H is a p-group.

2. If L is a p-subgroup of G, and if m ∈ M(L), then

br_{L}(b^{μ}.m) = br_{L}^{O}(b)br_{L}(m)

Similarly, if m ∈ \underline{M}(L), then

b^{μ}.m = br_{L}^{O}(b)m

3. The following are equivalent :

(a) The block b^{μ} acts as the identity of M.

(b) For any p-subgroup P of G, the idempotent br_{P}^{O}(b) acts as the identity of \overline{M}(P).

(c) For any p-subgroup P of G, the idempotent br_{P}^{O}(b) acts as the identity of \underline{M}(P).

Proof: The first assertion follows from the fact that (ζ ∘ β)(f_{1}^{G}) acts as the identity of M if M is in Mack_{O}(G, 1). Moreover, the action of this idempotent on M(H) is a linear combination with coefficients in O of elements t_{P}^{H}r_{P}^{H}, for p-subgroups P of H.

For assertion 2, observe that b^{μ} acts on M(L) via

b_{L}^{μ} = \frac{1}{|L|} ∑_{{Q⊆P ⊆L \atop x∈C_{G}(P)}} |Q|μ(Q, P)br_{P}^{O}(b)(x)t_{Q}^{L}xr_{Q}^{L} .

Thus if m ∈ M(L)

b_{L}^{μ}m = \frac{1}{|L|} ∑_{{Q⊆P ⊆L \atop x∈C_{G}(P)}} |Q|μ(Q, P)br_{P}^{O}(b)(x)t_{Q}^{L}xr_{Q}^{L}m .

In this summation, all the terms are traces from proper subgroups of L, except those for which Q = P = L. This gives :

br_{L}(b_{L}^{μ}m) = br_{L}( ∑_{x∈C_{G}(L)} br_{L}^{O}(b)(x)x.m) = br_{L}(br_{L}^{O}(b)m) = br_{L}^{O}(b)br_{L}(m) .

Similarly, if m ∈ \underline{M}(L), then all proper restrictions of m are zero, hence

b_{L}^{μ}.m = ∑_{x∈C_{G}(L)} br_{L}^{O}(b)(x)x.m = br_{L}^{O}(b)m

as was to be shown.

For assertion 3), suppose first that b^{μ} acts as the identity of M. Then by assertion 2), the idempotent br_{L}^{O}(b) acts as the identity of \overline{M}(L) and \underline{M}(L), for any p-subgroup L of G. Thus assertion (a) implies (b) and (c).

Conversely, suppose that (b) holds. Then in particular b^{μ} acts as the identity of \overline{M}(1) = M(1). By induction on the order of the subgroup H of G, this shows that b^{μ} acts as the identity of M(H) : let m ∈ M(H). If H is not a p-group, then by assertion 1) br_{H}(m) = 0, and m can be written

m = ∑_{L⊂H} t_{L}^{H}m_{L}

for proper subgroups L of H and elements m_{L} ∈ M(L). Now by induction

b^{μ}.m = b_{H}^{μ} ∑_{L⊂H} t_{L}^{H}m_{L} = ∑_{L⊂H} t_{L}^{H}(b_{L}^{μ}m_{L}) = ∑_{L⊂H} t_{L}^{H}m_{L} = m .

And if H is a p-subgroup of G, then by assertion (b)

m − b^{μ}.m ∈ Ker br_{H} .

Hence

m − b^{μ}.m = ∑_{L⊂H} t_{L}^{H}m_{L}

for proper subgroups L of H and elements m_{L} ∈ M(L). Now

b^{μ}.m − (b^{μ})^{2}.m = m − b^{μ}.m

by the same argument as above. Hence b^{μ}.m = m, and (b) implies (a).

Similarly if (c) holds, then in particular b^{μ} acts as the identity of \underline{M}(1) = M(1). By induction on the order of the subgroup H of G, this shows that b^{μ} acts as the identity of M(H) : let m ∈ M(H). If H is not a p-group, then m is a linear combination of traces from proper subgroups of H, hence b^{μ}.m = m by induction. And if H is a p-subgroup of G, then by induction for any proper subgroup L of H

r_{L}^{H}(b^{μ}.m) = r_{L}^{H}(b_{H}^{μ}.m) = b_{L}^{μ}.r_{L}^{H}m = r_{L}^{H}m

This shows that m − b^{μ}.m ∈ \underline{M}(L). Now (c) implies that

m − b^{μ}.m = b^{μ}(m − b^{μ}.m) = b^{μ}.m − b^{μ}.m = 0

thus m = b^{μ}.m, as was to be shown.

Corollary 5.2.3: Let M be a Mackey functor in Mack_{O}(G, 1). If M is indecomposable, and if there is a p-subgroup P such that \overline{M}(P) ≠ 0 and C_{G}(P) acts trivially on \overline{M}(P) (resp. \underline{M}(P) ≠ 0 and C_{G}(P) acts trivially on \underline{M}(P)), then M is in the principal block of Mack_{O}(G, 1).

Proof: Since M is indecomposable, there is a block b of kG such that b^{μ} acts as the identity of M. Hence br_{P}^{O}(b) acts as the identity of \overline{M}(P), for any psubgroup P of G. Moreover, since C_{G}(P) acts trivially on \overline{M}(P), the action of br_{P}^{O}(b) on \overline{M}(P) is multiplication by the sum ∑_{x∈C_{G}(P)} br_{P}^{O}(b)(x), which is zero if b is not the principal block of kG : indeed if e = ∑_{x∈C_{G}(P)} e_{x}x is any block of OC_{G}(P), then the sum ∑_{x∈C_{G}(P)} e_{x} is equal to 1 or 0, according to the fact that e is the principal block of OC_{G}(P) or not. And by Brauer main theorem, the principal block of OC_{G}(P) appears in the decomposition of br_{P}^{O}(b) if and only if b is the principal block of OG. The argument is the same with \underline{M}(P).

Remark 5.2.4: This corollary shows in particular that if M is an indecomposable Mackey functor in Mack_{O}(G, 1), and if C_{G}(P) acts trivially on M(P) for any p-subgroup P of G, then M is in the principal block of Mack_{O}(G, 1).

5.3 The defect of a block of μ_{O}(G)

In [1] Chapter 12, it is shown that there exists a natural Green functor for G whose evaluation at G is isomorphic to the center of the Mackey algebra. This functor is denoted by ζ_{B}, and its value at the G-set X is the set

ζ_{B}(X) = Hom_{F unct}(I, I_{X})

of natural transformations from the identity functor I on the category of Mackey functors to the endofunctor I_{X} on this category, mapping the Mackey functor M to its Dress construction M_{X}, defined on the G-set Y by

M_{X}(Y) = M(Y × X) .

This functor ζ_{B} has also been considered by Oda (the functor T of [7]) from a different point of view. The equivalence of those two points of view follows from Proposition 12.2.8 of [1] : the value of ζ_{B} at a subgroup H of G is the set of sequences (z_{L}), indexed by the subgroups L of H, such that

z_{L} ∈ t_{L}^{L}μ_{O}(G)t_{L}^{L}  t_{Q}^{L}z_{L} = z_{Q}t_{Q}^{L}  z_{Q}r_{Q}^{L} = r_{Q}^{L}z_{L}  xz_{L} = z_{^{x}L}x

for any subgroups Q ⊆ L of H and any x ∈ H. The isomorphism ζ_{B}(G) ≅ Zμ_{O}(G) is obtained by mapping the sequence (z_{L})_{L⊆G} to the element ∑_{L⊆G} z_{L}.

Also recall from Proposition 12.2.8 of [1] that if Q is a subgroup of G, and (z_{L})_{L⊆Q} is an element of ζ_{B}(Q), then for any subgroup H of G, the component of t_{Q}^{G}(z) at the subgroup H is equal to

t_{Q}^{G}(z)_{H} = ∑_{w∈Q\G/H} t_{Q^{w}∩H}^{H}w^{−1}z_{Q∩^{w}H}wr_{Q^{w}∩H}^{H} .

If g ∈ C_{G}(Q), one can define an element z(Q, g) of ζ_{B}(Q) by setting, for any subgroup L of Q

z(Q, g)_{L} = t_{L}^{L}gr_{L}^{L} .

Let b be a block of kG, with defect group D. With the previous notation, the expression of b^{μ} given in proposition 4.5.2 can be written as

b^{μ} = \frac{1}{|G|} ∑_{{Q⊆P ∈\underline{s}_{p}(G) \atop g∈C_{G}(P)}} |Q|μ(Q, P)br_{P}^{O}(b)(g)t_{Q}^{G}z(Q, g) .

The summation can be restricted to p-subgroups P of G for which br_{P}^{O}(b) ≠ 0. Such subgroups are all contained in D up to G-conjugation. It follows that

b^{μ} ∈ t_{D}^{G}ζ_{B}(D) .

Definition 5.3.1: A defect group of the block b^{μ} ∈ Zμ_{O}(G) = ζ_{B}(G) is a subgroup E of G, minimal subject to the condition

b^{μ} ∈ t_{E}^{G}ζ_{B}(E) .

The following argument is then classical : if D is any subgroup of G such that b^{μ} ∈ t_{D}^{G}ζ_{B}(D), then

b^{μ} = (b^{μ})^{2} ∈ (t_{E}^{G}ζ_{B}(E))(t_{D}^{G}ζ_{B}(D)) ⊆ ∑_{x∈E\G/D} t_{E∩^{x}D}^{G}ζ_{B}(E ∩^{x}D)

and by Rosenberg’s lemma and primitivity of b^{μ}, there exists x ∈ G such that E ⊆ ^{x}D. In particular E is unique up to G-conjugation.

Proposition 5.3.2: Let b be a block of kG, with defect group D. Then D is also a defect group of b^{μ}.

Proof: Let E be a defect of b^{μ}. The above argument shows that E is contained in D up to G-conjugation. The reverse inclusion has been proved by Oda ([7] Theorem 6). It also follows from the following fact : if b^{μ} = t_{E}^{G}(c), for c ∈ ζ_{B}(E), then

(b^{μ})_{1} = t_{1}^{1}T r_{E}^{G}(c_{1})r_{1}^{1} ∈ t_{1}^{1}T r_{E}^{G}(OG)^{E}r_{1}^{1}

where T r_{E}^{G} is the relative trace map. Since moreover (b^{μ})_{1} = t_{1}^{1}br_{1}^{O}(b)r_{1}^{1}, it follows that the block b^{O} = br_{1}^{O}(b) lifting b to OG belongs to T r_{E}^{G}(OG)^{E}. Hence D is contained in E up to G-conjugation.

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