On the Cartan matrix of Mackey algebras

Serge Bouc

Abstract : Let k be a field of characteristic p > 0, and G be a finite group. The first result of this paper is an explicit formula for the determinant of the Cartan matrix of the Mackey algebra μ_{k}(G) of G over k. The second one is a formula for the rank of the Cartan matrix of the cohomological Mackey algebra coμ_{k}(G) of G over k, and a characterization of the groups G for which this matrix is non singular. The third result is a generalization of this rank formula and characterization to blocks of coμ_{k}(G) : in particular, if b is a block of kG, the Cartan matrix of the corresponding block coμ_{k}(b) of coμ_{k}(G) is non singular if and only if b is nilpotent with cyclic defect groups.

AMS Subject Classification : 18G05, 20C20, 20J06.

Keywords : Cartan matrix, cohomological, Mackey functor, block.

1 Introduction

1.1. The theory of Mackey functors for a finite group G over a commutative ring R is by many aspects very similar to the theory of RG-modules. It has been shown by J. Thévenaz and P. Webb ([13]), among many fundamental other results, that the category of Mackey functors for G over R is equivalent to the category of modules over the Mackey algebra μ_{R}(G). This algebra shares many properties with the group algebra RG: it is free as an R-module, and its R-rank does not depend on R ; if K is a field of characteristic 0 or coprime to the order of G, the algebra μ_{K}(G) is semisimple ; when (K, O, k) is a p-modular system, there is a decomposition theory from Mackey functors for G over K to Mackey functors for G over k ; the Cartan matrix of μ_{k}(G) is symmetric and non singular. This list of common properties between the Mackey algebra and the group algebra is far from exhaustive… 1.2. However some well known results for group algebras are no longer true for Mackey algebras. It was observed on small examples in particular by M. Nicollerat in her thesis ([12] Chapitre 5) that the determinant of the Cartan matrix of μ_{k}(G), when k is a field of characteristic p, is generally not a power of p, even when G itself is a p-group. Instead, some rather strange prime factors appear in this determinant. One of the motivations of this paper is to give an explanation for these strange factors, by stating an explicit formula for the determinant of the Cartan matrix of μ_{k}(G) (Theorem 2.28). 1.3. The other motivation is the similar problem for the cohomological Mackey algebra coμ_{k}(G), which was defined by Thévenaz and Webb as a specific quotient of μ_{k}(G), with the property that the modules over coμ_{k}(G) are exactly the cohomological Mackey functors : the first major difference is that in general, the Cartan matrix of this algebra is singular. This raises the question of characterizing those finite groups G for which the Cartan matrix of coμ_{k}(G) is non singular, and the answer is the second main result of this paper (Theorem 3.4) : these groups are exactly the p-nilpotent groups with cyclic Sylow p-subgroups. The other possibly interesting result is that for an arbitrary finite group G, the rank of the Cartan matrix of coμ_{k}(G) is equal to the number of conjugacy classes of pairs (R, s), where R is a cyclic p-subgroup of G, and s is a p′-element of the centralizer of R in G.

The third result of this paper is a natural generalization of Theorem 3.4 to blocks, suggested by the one to one correspondence b ↦ coμ_{k}(b) between blocks of kG and blocks of coμ_{k}(G) : Theorem 4.2 gives a formula for the rank of the Cartan matrix of coμ_{k}(b), in terms of b-Brauer pairs, and shows that this matrix is non singular if and only if the block b is nilpotent with cyclic defect groups. 1.4. The paper is organized according to these results : Section 2 is devoted to the case of μ_{k}(G), starting by recalling some standard notation, definitions and properties, and Section 3 deals with coμ_{k}(G). Finally, Section 4 is devoted to the case of blocks of cohomological Mackey functors.

Acknowledgment : I wish to thank Jacques Thévenaz for very fruitful discussions and friendly collaboration on all these questions also, at the EPFL in June 2009.

2 The Mackey algebra

2.1. Throughout the paper when G is a finite group and p is a prime number, the set of p′-elements of G is denoted by G_{p′}, and the symbol [G_{p′}] denotes a set of representatives of G-conjugacy classes in G_{p′}. The set of p-subgroups of G is denoted by S_{p}(G), and [S_{p}(G)] denotes similarly a set of representatives of G-conjugacy classes in S_{p}(G). 2.2. From now on G will be a fixed finite group, and p a fixed prime number. Let (K, O, k) be a p-modular system : thus O is a discrete valuation ring with residue field k of characteristic p, and field of fractions K of characteristic 0. Assume that K and k are splitting fields for all the groups N_{G}(Q)/Q, for Q ∈ S_{p}(G) (e.g. assume that K contains the e-th roots of unity, where e is the exponent of G). 2.3. When R is a commutative ring with identity element, the Mackey algebra μ_{R}(G) of G over R has been defined by Thévenaz and Webb ([13] Section 3). It is an associative algebra with the property that the category μ_{R}(G)-Mod of left μ_{R}(G)-modules is equivalent to the category Mack_{R}(G) of Mackey functors for G over R. 2.4. Thévenaz and Webb have also shown ([13] Theorem 10.1) that there is an equivalence of abelian categories

Mack_{k}(G) ≅ ∏_{H} Mack_{k}(N_{G}(H)/H, 1) ,

where H runs through a set of representatives of conjugacy classes of pperfect subgroups of G, and Mack_{k}(N_{G}(H)/H, 1) denotes the subcategory of Mack_{k}(N_{G}(H)/H) consisting of Mackey functors which are projective relative to p-subgroups. The category Mack_{k}(G, 1) is equivalent to μ_{k}(G, 1)-Mod, where μ_{k}(G, 1) is a direct summand of μ_{k}(G) of the form μ_{k}(G)f, for a specific central idempotent f of μ_{k}(G). 2.5. ([13] Theorem 12.7 and Corollary 12.8) The correspondence M ↦ M(1), that is evaluation at the trivial subgroup of G, induces a one to one correspondence between the set of isomorphism classes of indecomposable projective μ_{k}(G, 1)-modules and the set of isomorphism classes of indecomposable p-permutation kG-modules (also called trivial source kG-modules).

2.6. Let pp_{k}(G) denote the Green ring of p-permutation kG-modules : as a group, it is the Grothendieck group of the category of finitely generated ppermutation kG-modules, for relations given by direct sum decompositions. The product on pp_{k}(G) is induced by the tensor product of kG-modules over k.

If W is a finitely generated p-permutation kG-module, its dual W^{∗} = Hom_{k}(W, k) is also a p-permutation kG-module, and this duality extends to a ring automorphism W ↦ W^{∗} of pp_{k}(G).

When Q is a p-subgroup of G, the Brauer quotient W [Q] is a p-permutation k\overline{N}_{G}(Q)-module, where \overline{N}_{G}(Q) = N_{G}(Q)/Q (see [7]). This construction commutes with duality and tensor product of p-permutation modules : if V and W are (finitely generated) p-permutation kG-modules, there are isomorphisms of kN_{G}(Q)/Q-modules

(2.7)  W [Q]^{∗} ≅ W^{∗}[Q], V [Q] ⊗ W [Q] ≅ (V ⊗ W)[Q] .

In particular, the Brauer quotient induces a ring homomorphism from pp_{k}(G) to pp_{k}(\overline{N}_{G}(Q)), still denoted by W ↦ W [Q] (see e.g. [5] Proposition 2.11).

There is a ℤ-valued bilinear form ⟨ , ⟩ on pp_{k}(G) defined for p-permutation kG-modules V and W by

⟨V, W ⟩_{G} = dim_{k} Hom_{kG}(V, W) .

It is worth noticing that

(2.8)  ⟨V, W ⟩_{G} = dim_{k} Hom_{kG}(k, V^{∗} ⊗_{k} W) .

The bilinear form ⟨ , ⟩_{G} extends to a K-valued bilinear form on K⊗_{ℤ}pp_{k}(G), still denoted by ⟨ , ⟩_{G}. 2.9. Let ⟨⟨ , ⟩⟩_{G} denote the bilinear form on pp_{k}(G) defined for p-permutation kG-modules V and W by

⟨⟨V, W ⟩⟩_{G} = ∑_{Q∈[S_{p}(G)]}⟨V [Q], W [Q]⟩_{\overline{N}_{G}(Q)}
= ∑_{Q∈[S_{p}(G)]} dim_{k} Hom_{k\overline{N}_{G}(Q)}(V [Q], W [Q]) .

It follows from 2.7 that

(2.10)  ⟨⟨V, W ⟩⟩_{G} = ⟨⟨k, V^{∗} ⊗_{k} W ⟩⟩_{G} .

It was shown in [3] Proposition 5.11 that if L and M are projective Mackey functors in Mack_{k}(G, 1), then

dim_{k} Hom_{Mack_{k}(G,1)}(L, M) = ⟨⟨L(1), M(1)⟩⟩_{G} .

When L and M are indecomposable, this is equal to the coefficient c_{L,M} of the Cartan matrix of the algebra μ_{k}(G, 1). 2.11. Let Q_{G,p} denote the set of pairs (R, s) consisting of a p-subgroup R of G and a p′-element s of \overline{N}_{G}(R). The group G acts by conjugation on Q_{G,p}. Let [Q_{G,p}] denote a set of representatives of G-orbits on Q_{G,p}, and let N_{G}(R, s) denote the stabilizer of (R, s) ∈ Q_{G,p} in G : it is the set of elements g ∈ N_{G}(R) such that the image \overline{g} of g in \overline{N}_{G}(R) centralizes s. In other words, there is an exact sequence of groups

(2.12)  1 → R → N_{G}(R, s) → C_{\overline{N}_{G}(R)}(s) → 1 .

It was shown in [5] that the primitive idempotents of the (commutative) ring K ⊗_{ℤ} pp_{k}(G) are indexed by the orbits of G on Q_{G,p} : the idempotent F_{R,s}^{G} associated to the orbit of the pair (R, s) is equal to (2.13)

F_{R,s}^{G} = \frac{1}{|R||s||C_{\overline{N}_{G}(R)}(s)|} ∑_{φ,L} φ̃(s^{−1})|L|μ(L, <sR>) Ind_{L}^{G}Res_{L}^{<sR>}k_{φ} ,

where • the group <sR> is the inverse image in N_{G}(R) of the subgroup <s>

of \overline{N}_{G}(R) under the map x ↦ xR.

• the morphism φ runs through group homomorphisms <s> → k^{×}, and φ̃ lifts φ to K. The module k_{φ} is the vector space k on which <sR> acts by <sR> → <s> →^{φ} k^{×} (thus φ̃ is the Brauer character of the module k_{φ}).

• the group L runs through the set of subgroups of <sR> such that LR = <sR>.

The idempotents F_{R,s}^{G}, for (R, s) ∈ [Q_{G,p}], form a K-basis of K ⊗_{ℤ} pp_{k}(G). Any element W of K ⊗_{ℤ} pp_{k}(G) can be expressed in this basis as

(2.14)  W = ∑_{(R,s)∈[Q_{G,p}]} F_{R,s}^{G} ⊗_{k} W = ∑_{(R,s)∈[Q_{G,p}]} t_{R,s}^{G}(W) F_{R,s}^{G} ,

where t_{R,s}^{G} is the extension to K ⊗_{ℤ} pp_{k}(G) of the species (i.e. the ring homomorphism, see [1] Lemma 2.2.1, page 26) pp_{k}(G) → K associated to the pair (R, s). 2.15. Recall that the value of t_{R,s}^{G} on a p-permutation kG-module W is equal to the value at s of the Brauer character of the Brauer quotient W [R] (see [5] Notation 2.15 and Proposition 2.18). This will be also be called the Brauer trace of s on W [R], and denoted by BrTr(s | W [R]). 2.16. The ring automorphism W ↦ W^{∗} extends by K-linearity to an automorphism of the ring K ⊗_{ℤ} pp_{k}(G). This automorphism preserves the set of primitive idempotents. The bilinear form ⟨⟨ , ⟩⟩_{G} also extends to a K-valued bilinear form on K ⊗_{ℤ} pp_{k}(G). Since the dual of k_{φ} is isomorphic to k_{φ^{−1}}, and since { \atop (φ^{−1})}(s^{−1}) = φ̃(s), it follows from 2.13 that (F_{R,s}^{G})^{∗} = F_{R,s^{−1}}^{G}. Now Equation 2.10 shows that

⟨⟨F_{R′,s′}^{G}, F_{R,s}^{G}⟩⟩_{G} = \{ \begin{matrix}⟨⟨k, F_{R,s}^{G}⟩⟩_{G} & if (R′, s′) =_{G} (R, s^{−1}) \\ 0 & otherwise \end{matrix}

(where =_{G} denotes G-conjugacy). Thus, if V and W are any elements of K ⊗_{ℤ} pp_{k}(G), it follows from 2.14 that

⟨⟨V, W ⟩⟩_{G} = ∑_{(R,s)∈[Q_{G,p}]} t_{R,s^{−1}}^{G}(V)t_{R,s}^{G}(W)⟨⟨k, F_{R,s}^{G}⟩⟩_{G} .

2.17. The indecomposable p-permutation kG-modules are indexed by conjugacy classes of pairs (P, E), where P is a p-subgroup of G, and E is an indecomposable projective k\overline{N}_{G}(P)-module ([7] Theorem 3.2). Let [P_{G,p}] denote a set of representatives of such conjugacy classes in G.

The indecomposable module M_{P,E} indexed by the pair (P, E) has vertex P, and the projective module E is isomorphic to the Brauer quotient M_{P,E}[P]. In particular, if Q is a subgroup of G, then M_{P,E}[Q] is non zero only if Q ≤_{G} P (and in fact, if and only if Q ≤_{G} P). 2.18. The modules M_{P,E}, for (P, E) ∈ [P_{G,p}], form a basis of pp_{k}(G), hence also a basis of K ⊗_{ℤ} pp_{k}(G). The Cartan matrix C(μ_{k}(G, 1)) of the algebra μ_{k}(G, 1) can be viewed as the square matrix indexed by [P_{G,p}], where the coefficient indexed by the elements (P, E) and (Q, F) of [P_{G,p}] is equal to

c_{(P,E),(Q,F)} = ⟨⟨M_{P,E}, M_{Q,F} ⟩⟩_{G}
= ∑_{(R,s)∈[Q_{G,p}]} t_{R,s^{−1}}^{G}(M_{P,E})t_{R,s}^{G}(M_{Q,F})⟨⟨k, F_{R,s}^{G}⟩⟩_{G} .

2.19. Let T and T′ denote the matrices indexed by the product [P_{G,p}]×[Q_{G,p}], defined by

T_{(P,E),(R,s)} = t_{R,s}^{G}(M_{P,E}) ,
T′_{(P,E),(R,s)} = t_{R,s^{−1}}^{G}(M_{P,E}) ,

respectively. Note that T and T′ are square matrices. Let moreover S be the (square) diagonal matrix indexed by [Q_{G,p}], where the diagonal term indexed by (R, s) is equal to ⟨⟨F_{R,s}^{G}, k⟩⟩_{G}. It follows that

C(μ_{k}(G, 1)) = T′ · S ·^{t}T .

Thus

(2.20)  det C(μ_{k}(G, 1)) = det T′ det S det^{t}T ,

and moreover, since S is diagonal

(2.21)  det S = ∏_{(R,s)∈[Q_{G,p}]}⟨⟨k, F_{R,s}^{G}⟩⟩_{G} = ∏_{R∈[S_{p}(G)]} ∏_{s∈[\overline{N}_{G}(R)_{p′}]}⟨⟨k, F_{R,s}^{G}⟩⟩_{G}

2.22. Now since M_{P,E}[R] = {0} unless R ≤_{G} P, the matrices T and T′ are actually block triangular, with diagonal blocks Δ(P) indexed by the conjugacy classes of p-subgroups P of G. Thus

det T = ∏_{P ∈[S_{p}(G)]} det Δ(P) .

The block matrix Δ(P) is a (square) matrix with rows indexed by the isomorphism classes of indecomposable projective k\overline{N}_{G}(P)-modules E, and columns indexed by the conjugacy classes of p′-elements s of \overline{N}_{G}(P). The coefficient Δ(P)_{E,s} is equal to

Δ(P)_{E,s} = t_{P,s}^{G}(M_{P,E})
= Brauer trace(s | M_{P,E}[P])
= Φ_{E}(s) ,

since M_{P,E}[P] ≅ E, where Φ_{E} is the Brauer character of the projective k\overline{N}_{G}(P)-module E.

Similarly,

det T′ = ∏_{P ∈[S_{p}(G)]} det Δ′(P) ,

where the coefficients of the diagonal block Δ′(P) of T′ are given by

Δ′(P)_{E,s} = Φ_{E}(s^{−1}) .

2.23. Let Σ(P) denote the (square) diagonal matrix indexed by the conjugacy classes of p′-elements of \overline{N}_{G}(P), with diagonal coefficient Σ(P)_{s,s} equal to the inverse of the order of the centralizer C_{\overline{N}_{G}(P)}(s) of s in \overline{N}_{G}(P), and let U(P) = Δ(P) · Σ(P) · ^{t}Δ′(P). The coefficient of U(P) indexed by the indecomposable projective k\overline{N}_{G}(P)-modules E and F is equal to

U(P)_{E,F} = ∑_{s∈[\overline{N}_{G}(P)_{p′}]} Φ_{E}(s)\frac{1}{|C_{\overline{N}_{G}(P)}(s)|}Φ_{F} (s^{−1})

= \frac{1}{|\overline{N}_{G}(P)|} ∑_{s∈\overline{N}_{G}(P)_{p′}} Φ_{E}(s)Φ_{F} (s^{−1})
= ⟨E, F ⟩_{\overline{N}_{G}(P)} .

In other words, the matrix U(P) is equal to the Cartan matrix of the group algebra k\overline{N}_{G}(P). It is then well known (see [6] III.16 or [10]) that

det U(P) = ∏_{s∈[\overline{N}_{G}(P)_{p′}]} |C_{\overline{N}_{G}(P)}(s)|_{p} ,

and it follows that

det Δ(P) det^{t}Δ′(P) = \frac{∏_{s∈[\overline{N}_{G}(P)_{p′}]} |C_{\overline{N}_{G}(P)}(s)|_{p}}{det Σ(P)}
= ∏_{s∈[\overline{N}_{G}(P)_{p′}]} |C_{\overline{N}_{G}(P)}(s)|_{p} ∏_{s∈[\overline{N}_{G}(P)_{p′}]} |C_{\overline{N}_{G}(P)}(s)| .

Together with 2.20 and 2.21, this gives (2.24)

det C(μ_{k}(G, 1)) = ∏_{R∈[S_{p}(G)]} ∏_{s∈[\overline{N}_{G}(R)_{p′}]} (|C_{\overline{N}_{G}(R)}(s)|_{p}|C_{\overline{N}_{G}(R)}(s)|⟨⟨k, F_{R,s}^{G}⟩⟩_{G}) .

2.25. Lemma : Let Q ∈ S_{p}(G) and (R, s) ∈ Q_{G,p}. Then

F_{R,s}^{G}[Q] = ∑_{{x∈N_{G}(Q)\G/N_{G}(R,s) \atop Q ⊴^{x}<Rs>}} F_{^{x}R/Q,^{x}s}^{\overline{N}_{G}(Q)} ,

(where ^{x}s denotes the image in \overline{N}_{G}(Q,^{x}R)/^{x}R of the x-conjugate of s).

Proof : The Brauer map W ↦ W [Q] is a ring homomorphism from pp_{k}(G) to pp_{k}(\overline{N}_{G}(Q)). Thus F_{R,s}^{G}[Q] is an idempotent of pp_{k}(\overline{N}_{G}(Q)), hence a sum

of distinct primitive idempotents F_{X,u}^{\overline{N}_{G}(Q)} associated to some pairs (X, u) ∈ Q_{\overline{N}_{G}(Q),p}. In other words X is a p-subgroup of \overline{N}_{G}(Q), of the form Y/Q, for some p-subgroup Y of G such that Q ⊴ Y, and u is a p′-element of \overline{N}_{\overline{N}_{G}(Q)}(X) ≅ N_{G}(Q, Y)/Y.

The idempotent F_{X,u}^{\overline{N}_{G}(Q)} appears in F_{R,s}^{G}[Q] if and only if the species t_{X,u}^{\overline{N}_{G}(Q)}

takes the value 1 when evaluated at F_{R,s}^{G}[Q]. But t_{X,u}^{\overline{N}_{G}(Q)}(F_{R,s}^{G}[Q]) is equal to the Brauer trace (see 2.15) of u on F_{R,s}^{G}[Q][Y/Q] = Res_{N_{G}(Q,Y)/Y}^{N_{G}(Y)/Y} F_{R,s}^{G}[Y]. In other words

t_{X,u}^{\overline{N}_{G}(Q)}(F_{R,s}^{G}[Q]) = t_{Y,u}^{G}(F_{R,s}^{G}) ,

and this is equal to 1 if and only if the pair (Y, u) is G-conjugate to the pair (R, s). The lemma follows.

2.26. Lemma : Let (R, s) ∈ Q_{G,p}. Then

⟨k, F_{R,s}^{G}⟩_{G} = \{ \begin{matrix}\frac{ϕ(|R|)}{|N_{G}(R, s)|} & if <sR> is cyclic \\ 0 & otherwise \end{matrix} ,

where ϕ is the Euler totient function.

Proof : Recall from 2.13 that

F_{R,s}^{G} = \frac{1}{|R||s||C_{\overline{N}_{G}(R)}(s)|} ∑_{φ,L} φ̃(s^{−1})|L|μ(L, <sR>) Ind_{L}^{G}Res_{L}^{<sR>}k_{φ} ,

where φ runs through homomorphisms <s> → k^{×} (i.e. equivalently homomorphisms <sR> → k^{×}), and L through subgroups of <sR> such that LR = <sR>. Now

⟨k, Ind_{L}^{G}Res_{L}^{<sR>}k_{φ}⟩_{G} = ⟨k, k_{φ}⟩_{L}

is equal to zero unless the restriction of φ to L is trivial, i.e. if L ≤ Ker φ. Since R ≤ Ker φ, this implies that <sR> ≤ Ker φ, i.e. that φ is trivial. Thus

⟨k, F_{R,s}^{G}⟩_{G} = \frac{1}{|R||s||C_{\overline{N}_{G}(R)}(s)|} ∑_{LR=<sR>} |L|μ(L, <sR>) .

Now a subgroup L of H = <sR> is such that LR = H if and only if it contains some conjugate of s in H, i.e. if it is of the form Q<^{x}s>, for some x ∈ R and some subgroup Q of R normalized by ^{x}s. In this case, there are |Q : C_{Q}(^{x}s)| conjugates of s contained in L. Moreover, since LR = H and L ∩ R = O_{p}(H) = Q, the map X ↦ X ∩ R, from the poset ]L, H[ of proper subgroups of H strictly containing L, to the poset ]Q, R[^{L}=]Q, R[^{^{x}s} of L-invariant proper subgroups of R strictly containing Q, is an isomorphism : the inverse isomorphism is the map Y ↦ Y · L. Thus μ(L, <sR>) is equal to the value μ((Q, R)^{^{x}s}) of the Möbius function of the poset of subgroups normalized by ^{x}s, and

⟨k, F_{R,s}^{G}⟩_{G} = \frac{1}{|R||s||C_{\overline{N}_{G}(R)}(s)|} ∑_{x∈R/C_{R}(s)} ∑_{{Q≤R \atop Q^{^{x}s}=Q}}|Q||s|μ((Q, R)^{^{x}s})/|Q : C_{Q}(^{x}s)|

= \frac{1}{|R||C_{\overline{N}_{G}(R)}(s)|} ∑_{x∈R/C_{R}(s)} ∑_{{Q≤R \atop Q^{^{x}s}=Q}} |C_{Q}(^{x}s)|μ((Q, R)^{^{x}s}) .

Now Q is normalized by ^{x}s if and only if Q^{x} is normalized by s. Moreover μ((Q, R)^{^{x}s}) = μ((Q^{x}, R)^{s}), and |C_{Q}(^{x}s)| = |C_{Q^{x}}(s)|. Thus

⟨k, F_{R,s}^{G}⟩_{G} = \frac{1}{|R||C_{\overline{N}_{G}(R)}(s)|} ∑_{x∈R/C_{R}(s)} ∑_{{Q≤R \atop Q^{s}=Q}} |C_{Q}(s)|μ((Q, R)^{s})

= \frac{1}{|C_{R}(s)||C_{\overline{N}_{G}(R)}(s)|} ∑_{{Q≤R \atop Q^{s}=Q}} |C_{Q}(s)|μ((Q, R)^{s}) .

Now

∑_{{Q≤R \atop Q^{s}=Q}} |C_{Q}(s)|μ((Q, R)^{s}) = ∑_{y∈C_{R}(s)} ∑_{{y∈Q≤R \atop Q^{s}=Q}} μ((Q, R)^{s})

= ∑_{y∈C_{R}(s)} δ_{<y>,R}
= |{y ∈ C_{R}(s) | <y> = R}| .

This is non zero if and only if R is cyclic and centralized by s, i.e. if the group <sR> is cyclic. In this case C_{R}(s) = R, and

⟨k, F_{R,s}^{G}⟩_{G} = \frac{1}{|R||C_{\overline{N}_{G}(R)}(s)|}ϕ(|R|) = \frac{ϕ(|R|)}{|N_{G}(R, s)|} ,

since |R||C_{\overline{N}_{G}(R)}(s)| = |N_{G}(R, s)|, in view of Exact Sequence 2.12. This completes the proof of the lemma.

2.27. Lemma : Let (R, s) ∈ Q_{G,p}. Then

⟨⟨k, F_{R,s}^{G}⟩⟩_{G} = \frac{1}{|C_{\overline{N}_{G}(R)}(s)|} ∑_{x∈R/[<sR>,R]} \frac{1}{|x|} .

Proof : By Definition 2.9

⟨⟨k, F_{R,s}^{G}⟩⟩_{G} = ∑_{Q∈[S_{p}(G)]}⟨k, F_{R,s}^{G}[Q]⟩_{\overline{N}_{G}(Q)} .

Hence, by Lemma 2.25

⟨⟨k, F_{R,s}^{G}⟩⟩_{G} = ∑_{Q∈S_{p}(G)} \frac{|N_{G}(Q)|}{|G|} ∑_{{x∈N_{G}(Q)\G/N_{G}(R,s) \atop Q ⊴^{x}<sR>}}⟨k, F_{^{x}R/Q,^{x}s}^{\overline{N}_{G}(Q)}⟩_{\overline{N}_{G}(Q)}

= ∑_{Q∈S_{p}(G) ; {x∈G \atop Q^{x} ⊴ <sR>}} \frac{|N_{G}(Q)|}{|G|} \frac{|N_{G}(Q^{x}) ∩ N_{G}(R, s)|}{|N_{G}(Q)||N_{G}(R, s)|}⟨k, F_{R/Q^{x},s}^{\overline{N}_{G}(Q^{x})}⟩_{\overline{N}_{G}(Q^{x})}

= ∑_{{Q≤R \atop Q ⊴ <sR>}} \frac{|N_{G}(Q) ∩ N_{G}(R, s)|}{|N_{G}(R, s)|}⟨k, F_{R/Q,s}^{\overline{N}_{G}(Q)}⟩_{\overline{N}_{G}(Q)}

Now ⟨k, F_{R/Q,s}^{\overline{N}_{G}(Q)}⟩_{\overline{N}_{G}(Q)} = 0 by Lemma 2.26, unless R/Q is cyclic and centralized by s, i.e. if R/Q is cyclic and Q contains the commutator subgroup [<sR>, R], in which case it is equal to \frac{ϕ(|R/Q|)}{|M|}, where M = N_{\overline{N}_{G}(Q)}(R/Q, s). Now the two exact sequences

1 → R → N_{G}(Q) ∩ N_{G}(R, s) → C_{N_{G}(Q,R)/R}(s) → 1

1 → R/Q → N_{\overline{N}_{G}(Q)}(R/Q, s) → C_{N_{G}(Q,R)/R}(s) → 1

show that

\frac{|N_{G}(Q) ∩ N_{G}(R, s)|}{|N_{\overline{N}_{G}(Q)}(R/Q, s)|} = |Q| .

It follows that

⟨⟨k, F_{R,s}^{G}⟩⟩_{G} = ∑_{{[<sR>,R]≤Q≤R \atop R/Q cyclic}} \frac{|Q|ϕ(|R/Q|)}{|N_{G}(R, s)|}

= \frac{1}{|C_{\overline{N}_{G}(R)}(s)|} ∑_{{[<sR>,R]≤Q≤R \atop R/Q cyclic}} \frac{ϕ(|R/Q|)}{|R/Q|} .

Now the summation is equal to the summation over all cyclic quotients X = R/Q of the abelian group A = R/[<sR>, R] of \frac{ϕ(|X|)}{|X|}, i.e. by a classical duality argument, to the summation of this quantity over the cyclic subgroups of A, i.e. finally to ∑_{x∈A} \frac{1}{|x|}. Thus

⟨⟨k, F_{R,s}^{G}⟩⟩_{G} = \frac{1}{|C_{\overline{N}_{G}(R)}(s)|} ∑_{x∈R/[<sR>,R]} \frac{1}{|x|} ,

as was to be shown.

2.28. Theorem : Let G be a finite group, let p be a prime number, and k be a field of characteristic p, big enough to be a splitting field for all the groups N_{G}(Q)/Q, for Q ∈ S_{p}(G). Then the determinant of the Cartan matrix of the algebra μ_{k}(G, 1) is equal to

det C(μ_{k}(G, 1)) = ∏_{R∈[S_{p}(G)]} ∏_{s∈[\overline{N}_{G}(R)_{p′}]} (|C_{\overline{N}_{G}(R)}(s)|_{p} ∑_{x∈R/[<sR>,R]} \frac{1}{|x|}) .

Proof : The Cartan matrix of μ_{k}(G, 1) is independent of the field k, as long as it is big enough. So one can assume e.g. that k is algebraically closed, and choose a corresponding p-modular system (K, O, k), where K is also big enough. Then the formula for det C(μ_{k}(G, 1)) follows from Equation 2.24 and Lemma 2.27.

2.29. Examples : Recall that k is a (big enough) field of characteristic p.

• Let G be a cyclic group of order p^{n}. Then

det C(μ_{k}(G, 1)) = p^{\binom{n}{2}} ∏_{i=1}^{n} (p + i(p − 1)) .

• Let G be an elementary abelian group of order p^{2}. Then

det C(μ_{k}(G, 1)) = p(2p − 1)^{p+1}(p^{2} + p − 1) .

3 The cohomological Mackey algebra

3.1. The cohomological Mackey algebra coμ_{R}(G) of a finite group G over a commutative ring R has also been introduced by Thévenaz and Webb (see [13] Section 16). It is an associative algebra with the property that the category of left coμ_{R}(G)-modules is equivalent to the category coMack_{R}(G) of cohomological Mackey functors. The algebra coμ_{R}(G) is defined as a quotient of the Mackey algebra μ_{R}(G).

In the case where R is a field k of characteristic p, the cohomological Mackey functors for G over k are projective relative to p-subgroups. In other words, the algebra coμ_{k}(G) is a quotient of μ_{k}(G, 1). 3.2. Thévenaz and Webb have shown that the projective cohomological Mackey functors for G over R are exactly the fixed points functors F P_{W}, where W is a direct summand of some permutation RG-module. Moreover, for any RG-modules V and W,

Hom_{coMack_{R}(G)}(F P_{W} , F P_{V}) ≅ Hom_{RG}(W, V) .

In particular, the indecomposable projective coμ_{k}(G)-modules are the functors F P_{W}, where W is an indecomposable p-permutation kG-module. Thus, the Cartan matrix of the algebra coμ_{k}(G) is the square matrix indexed by the set [P_{G,p}], defined by

c_{(P,E),(Q,F)} = dim_{k} Hom_{coMack_{k}(G)}(F P_{M_{P,E}}, F P_{M_{Q,F}}) = ⟨M_{P,E}, M_{Q,F} ⟩_{G} .

3.3. Thus by the same argument already used in Section 2, the Cartan matrix of the algebra coμ_{k}(G) is the matrix of the bilinear form ⟨ , ⟩ on pp_{k}(G). This extends to a bilinear form on K ⊗_{ℤ} pp_{k}(G), and by 2.8, for (R, s) and (R′, s′) in Q_{G,p}

⟨F_{R′,s′}^{G}, F_{R,s}^{G}⟩ = \{ \begin{matrix}⟨k, F_{R,s}^{G}⟩_{G} & if (R′, s′) =_{G} (R, s^{−1}) \\ 0 & otherwise \end{matrix}

By Lemma 2.26, this is non zero if and only if the group <sR> is cyclic.

The argument of paragraph 2.19 shows that the Cartan matrix of coμ_{k}(G) can be expressed as

C(coμ_{k}(G)) = T′ · S′ ·^{t}T ,

where S′ is the diagonal matrix indexed by [Q_{G,p}] with (R, s)-diagonal entry equal to ⟨F_{R,s}^{G}, k⟩_{G}. Since T and T′ are invertible, it follows that the rank of C(coμ_{k}(G)) is equal to the number of elements (R, s) of [Q_{G,p}] such that <sR> is cyclic.

3.4. Theorem : Let G be a finite group, let p be a prime number, and k be a field of characteristic p, big enough to be a splitting field for all the groups N_{G}(Q)/Q, for Q ∈ S_{p}(G). Then :

1. The rank of the Cartan matrix of the algebra coμ_{k}(G) is equal to the number of G-conjugacy classes of pairs (R, s), where R ∈ S_{p}(G) and s ∈ \overline{N}_{G}(R)_{p′}, such that <sR> is cyclic. In other words

rk C(coμ_{k}(G)) = ∑_{R∈[C_{p}(G)]} |N_{G}(R)\C_{G}(R)_{p′}|
= ∑_{s∈[G_{p′}]} c_{p}(C_{G}(s)) ,

where [C_{p}(G)] denotes a set of representatives of conjugacy classes of cyclic p-subgroups of a group G, and c_{p}(G) is the number of such conjugacy classes. Moreover, in the first summation, |N_{G}(R)\C_{G}(R)_{p′}| denotes the number of N_{G}(R)-conjugacy classes of p-regular elements

of C_{G}(R).

2. The Cartan matrix of the algebra coμ_{k}(G) is non singular if and only if the group G is p-nilpotent with cyclic Sylow p-subgroups. In this case, if G = N ⋊P, where N is a p′-group and P is a cyclic group of order p^{n}

det C(coμ_{k}(G)) = ∏_{R≤P} (\frac{ϕ(|R|)}{|R|})^{l_{p}(\overline{N}_{G}(R))} det C(k\overline{N}_{G}(R))

= ∏_{R≤P} ∏ ; x∈[P \[C_{N} (R)]] (\frac{ϕ(|R|)}{|R|}|C_{C_{N} (R)⋊P/R}(x)|_{p}) ,

where l_{p}(\overline{N}_{G}(R)) is the number of p-regular classes of \overline{N}_{G}(R), where C(k\overline{N}_{G}(R)) is the Cartan matrix of the group algebra k\overline{N}_{G}(R), and [P \[C_{N}(R)]] is a set of representatives of C_{N}(R)⋊P -conjugacy classes

in C_{N}(R).

Proof : The first sentence of Assertion 1 follows from the above arguments, and the second one follows by counting the number of pairs (R, s) of [Q_{G,p}] such that <sR> is cyclic, in two different ways. For Assertion 2, observe that the Cartan matrix C(coμ_{k}(G)) is non singular if and only if for any (R, s) ∈ Q_{G,p}, the group <sR> is cyclic. This amounts to saying that the Sylow p-subgroups of G are cyclic, and that whenever s ∈ G_{p′} normalizes a p-subgroup R, it centralizes it. In other words, the group N_{G}(R)/C_{G}(R) is a p-group, for any R ∈ S_{p}(G). This is equivalent to saying that G is p-nilpotent, by the theorem of Frobenius ([11] Theorem 4.5).

In the case G = N ⋊ P, where P is cyclic of order p^{n}, then as in 2.24

det C(coμ_{k}(G)) = det T′ det S′ det^{t}T
= ∏_{R∈[S_{p}(G)]} ∏_{s∈[\overline{N}_{G}(R)_{p′}]} (|C_{\overline{N}_{G}(R)}(s)|_{p}|C_{\overline{N}_{G}(R)}(s)|⟨k, F_{R,s}^{G}⟩_{G})

= ∏_{R∈[S_{p}(G)]}(det C(k\overline{N}_{G}(R)) ∏_{s∈[\overline{N}_{G}(R)_{p′}]}(|C_{\overline{N}_{G}(R)}(s)|⟨k, F_{R,s}^{G}⟩_{G})) .

The set [S_{p}(G)] can be chosen to be the set of subgroups of P. Moreover by Lemma 2.26, for R ≤ P and s ∈ \overline{N}_{G}(R)_{p′}

⟨k, F_{R,s}^{G}⟩_{G} = \frac{ϕ(|R|)}{|N_{G}(R, s)|} = \frac{ϕ(|R|)}{|R||C_{\overline{N}_{G}(R)}(s)|} .

It follows that

det C(coμ_{k}(G)) = ∏_{R≤P} (\frac{ϕ(|R|)}{|R|})^{l_{p}(\overline{N}_{G}(R))} det C(k\overline{N}_{G}(R)) .

Finally the p′-elements of the group \overline{N}_{G}(R) ≅ C_{N}(R) ⋊ (P/R) are the elements of C_{N}(R). The last formula of the theorem follows.

3.5. Remark : Thus when it is non zero, the determinant of the Cartan matrix of coμ_{k}(G) is equal to (p − 1)^{n}p^{m}, for suitable non negative integers n and m.

4 Blocks of cohomological Mackey functors

4.1. It was shown by Thévenaz and Webb (see [13] Theorem 17.1 and its proof, see also [4])) that the blocks of the algebra μ_{k}(G, 1) and the blocks of the algebra coμ_{k}(G) are in one to one correspondence with the blocks of the group algebra kG. If b is a block of kG, denote by coμ_{k}(b) the corresponding block of coμ_{k}(G). When R is a p-subgroup of G, let Br_{R} : (kG)^{R} → kC_{G}(R) denote the Brauer morphism. If moreover R ⊴ G, denote by u ↦ \overline{u} the projection map kG → k(G/R). When A is a k-algebra, denote by Irr_{k}(A) the set of isomorphism classes of simple A-modules.

This section is devoted to the proof of the following block version of Theorem 3.4 for such blocks of cohomological Mackey functors :

4.2. Theorem : Let G be a finite group, let p be a prime number, and k be an algebraically closed field of characteristic p. Let moreover b be a block

of kG. Then :

1. The rank of the Cartan matrix of the algebra coμ_{k}(b) is equal to

rk C(coμ_{k}(b)) = ∑_{R∈[C_{p}(G)]} |N_{G}(R)\Irr_{k}(kC_{G}(R)Br_{R}(b))|
= ∑_{(R,c)∈[C_{p}(b)]} |N_{G}(R, c)\Irr_{k}(kC_{G}(R)c)| ,

where [C_{p}(b)] is a set of representatives of G-conjugacy classes of b-Brauer pairs (R, c) for which R is cyclic.

2. The Cartan matrix of the block coμ_{k}(b) is non singular if and only if b is a nilpotent block with cyclic defect groups.

4.3. The Cartan matrix C of coμ_{k}(b) is non singular if and only if the rows of the decomposition matrix D are linearly independent (since C = D ·^{t}D) : indeed, if a vector u is such that Cu = 0, then ^{t}uCu = ^{t}(^{t}Du) · (^{t}Du) = 0, thus ^{t}Du = 0. Conversely, if ^{t}Du = 0, then obviously Cu = 0. 4.4. The decomposition matrix D of coμ_{k}(b) has rows indexed by the (isomorphism classes of) indecomposable p-permutation kG-modules in the block b, and columns indexed by the (isomorphism classes of) simple KG-modules in the block b. The coefficient D_{W,χ} corresponding to the indecomposable p-permutation kGb-module W and the simple KG-module χ is equal to the multiplicity of χ in K ⊗_{O} W̃, where W̃ is an OG-module lifting W to O (i.e. such that k ⊗_{O} W̃ ≅ W). Such an OG-module is unique up to isomorphism, since W is a p-permutation module. The character of the module K ⊗_{O} W̃ will be called the (ordinary) character of W.

It follows that the rank of the Cartan matrix of coμ_{k}(b) is equal to the dimension of the subspace of ℚ ⊗_{ℤ} R_{K}(b) generated by characters of p-permutation modules in the block b.

In particular, the Cartan matrix of coμ_{k}(b) is non singular if and only if the ordinary characters of the indecomposable p-permutation modules in b are linearly independent. 4.5. Recall (see 2.15, and [8] Proposition 3.3) that the value of the ordinary character of a p-permutation module W on an element s of G is equal to the Brauer trace BrTr(s_{p′} | W [<s_{p}>]), where s_{p} and s_{p′} are the p-part and p′-part of s, respectively.

4.6. Notation : Let Z_{p}(G) denote the set of pairs (P, E) consisting of a cyclic p-subgroup P of G and an indecomposable projective k\overline{C}_{G}(P)- module E, where \overline{C}_{G}(P) = C_{G}(P)/P, and let [Z_{p}(G)] be a set of representatives of G-orbits on Z_{p}(G).

Let Z_{p}(b) denote the subset of Z_{p}(G) consisting of pairs (P, E) such that

Br_{P} (b)Inf_{\overline{C}_{G}(P)}^{C_{G}(P)}E = Inf_{\overline{C}_{G}(P)}^{C_{G}(P)}E .

Set moreover [Z_{p}(b)] = Z_{p}(b) ∩ [Z_{p}(G)].

4.7. Lemma : The characters of the modules Ind_{C_{G}(P)}^{G}Inf_{\overline{C}_{G}(P)}^{C_{G}(P)}E, for (P, E) ∈ [Z_{p}(G)], form a basis of the subspace of ℚ ⊗_{ℤ} R_{K}(G) generated by the characters of the p-permutation modules.

Proof : Since projective modules are p-permutation modules, and since induction and inflation preserves this class of modules, the module L_{P,E} = Ind_{C_{G}(P)}^{G}Inf_{\overline{C}_{G}(P)}^{C_{G}(P)}E, for (P, E) ∈ Z_{p}(G), is a p-permutation module. Up to isomorphism, this module depends only on the G-orbit of (P, E). Moreover the number of G-orbits on the set Z_{p}(G) is equal to

|G\Z_{p}(G)| = ∑_{P ∈[C_{p}(G)]} |N_{G}(P)\Irr_{k}(kC_{G}(P))| .

Indeed, the number of isomorphism classes of indecomposable projective k\overline{C}_{G}(P)-modules is equal to the number of isomorphism classes of simple k\overline{C}_{G}(P)-modules, i.e. to the number of isomorphism classes of simple kC_{G}(P)-modules, since P is a normal p-subgroup of C_{G}(P).

By Theorem 3.4, it follows that the cardinality of the set [Z_{p}(G)] is precisely equal to the rank of the Cartan matrix of the algebra coμ_{k}(G), i.e. to the dimension of the subspace of ℚ ⊗_{ℤ}R_{K}(G) generated by the characters of the p-permutation modules. Thus, to prove Lemma 4.7, it is enough to prove that the characters of the modules L_{P,E}, for (P, E) ∈ [Z_{p}(G)], are linearly independent.

The character χ_{P,E} of the module L_{P,E} is equal to Ind_{C_{G}(P)}^{G}Inf_{\overline{C}_{G}(P)}^{C_{G}(P)}Φ_{E}, where Φ_{E} is the character of the module E. Suppose that some non trivial linear combination of these characters is equal to 0, i.e. that there are integers n_{P,E} ∈ ℤ, for (P, E) ∈ [Z_{p}(G)], not all equal to 0, such that

(4.8)  ∑_{(P,E)∈[Z_{p}(G)]} n_{P,E}χ_{P,E} = 0 .

Let Q be maximal such that there exists (Q, F) ∈ [Z_{p}(G)] with n_{Q,F} ≠ 0, and let s be a generator of the cyclic group Q. By 4.5, for any t ∈ C_{G}(s)_{p′}, the value of the linear combination 4.8 at the element st is equal to

0 = ∑_{(P,E)∈[Z_{p}(G)]} n_{P,E}BrTr(t | L_{P,E}[Q]) .

But L_{P,E}[Q] = 0, unless some conjugate of Q is contained in P. By maximality of Q, it follows that

0 = ∑_{E} n_{Q,E}BrTr(t | L_{Q,E}[Q]) ,

where E runs through a set [P] of indecomposable projective k\overline{C}_{G}(Q)-modules, up to isomorphism and conjugation by N_{G}(Q). Since moreover

L_{Q,E}[Q] ≅ Ind_{C_{G}(Q)}^{N_{G}(Q)}Inf_{\overline{C}_{G}(Q)}^{C_{G}(Q)}E ,

it follows that for any t ∈ C_{G}(Q)_{p′}

∑_{E∈[P]} n_{Q,E}(Ind_{C_{G}(Q)}^{N_{G}(Q)}Inf_{\overline{C}_{G}(Q)}^{C_{G}(Q)}Φ_{E})(t) = 0 .

This is also equal to

∑_{E∈[P]} n_{Q,E}(Res_{C_{G}(Q)}^{N_{G}(Q)}Ind_{C_{G}(Q)}^{N_{G}(Q)}Inf_{\overline{C}_{G}(Q)}^{C_{G}(Q)}Φ_{E})(t) = ∑_{{E∈[P] \atop x∈N_{G}(Q)/C_{G}(Q)}} n_{Q,E}Φ_{E}(\overline{^{x}t})

= ∑_{{E∈[P] \atop x∈N_{G}(Q)/C_{G}(Q)}} n_{Q,E}Φ_{E^{x}}(\overline{t}) ,

where \overline{t} is the image of t in \overline{C}_{G}(Q), and E^{x} is the image of E by conjugation by x ∈ N_{G}(Q).

Since the map t ↦ \overline{t} is a surjection from C_{G}(Q)_{p′} to \overline{C}_{G}(Q)_{p′}, it follows that the modular character ∑_{{E∈[P] \atop x∈N_{G}(Q)/C_{G}(Q)}} n_{Q,E}Φ_{E^{x}} of \overline{C}_{G}(Q) is equal to zero.

But the set of modules E^{x}, for E ∈ [P] and x ∈ N_{G}(Q)/C_{G}(Q), is exactly the set of projective indecomposable k\overline{C}_{G}(Q)-modules, up to isomorphism. Thus

∑_{F ∈Q} n_{Q,F} m_{F} Φ_{F} = 0 ,

where Q is a set of representatives of isomorphism classes of indecomposable projective k\overline{C}_{G}(Q)-modules, where n_{Q,F} is defined as n_{Q,E} if (Q, E) ∈ [P] and if there exists x ∈ N_{G}(Q) such that E^{x} ≅ F, and where m_{F} is the number of elements x ∈ N_{G}(Q)/C_{G}(Q) such that F^{x} ≅ F.

Now the characters Φ_{F}, for F ∈ Q, are linearly independent, and it follows that n_{Q,F} = 0 for any F. This contradicts the definition of Q, and completes the proof of Lemma 4.7.

4.9. Notation : Let P ∈ S_{p}(G), and E be any projective k\overline{N}_{G}(P)-module. Then E splits as a direct sum ⊕_{i∈I} E_{i} of indecomposable k\overline{N}_{G}(P)-modules E_{i}.

In this situation, set M_{P,E} = ⊕_{i∈I} M_{P,E_{i}}.

4.10. Lemma : The characters of the modules M_{P,Ind_{\overline{C}_{G}(P)}^{\overline{N}_{G}(P)}E}, for (P, E) ∈

[Z_{p}(b)], form a basis of the subspace of ℚ⊗_{ℤ}R_{K}(b) generated by the characters of the p-permutation modules in the block b.

Proof : Let (P, E) ∈ Z_{p}(b). The module M = Ind_{\overline{C}_{G}(P)}^{\overline{N}_{G}(P)}E splits as a direct sum ⊕_{i∈I} E_{i} of indecomposable k\overline{N}_{G}(P)-modules E_{i}. Since Br_{P} (b) is N_{G}(P)-

invariant and belongs to kC_{G}(P), it follows that \overline{Br_{P} (b)} acts as the identity on M, hence on every direct summand E_{i}. Recall that the indecomposable module M_{P,E_{i}} belongs to the block b if and only if \overline{Br_{P} (b)}E_{i} = E_{i} (see [2] Corollary 6.3.2). Hence all the indecomposable kG-modules M_{P,E_{i}} are in the block b, and their direct sum M_{P,Ind_{\overline{C}_{G}(P)}^{\overline{N}_{G}(P)}E} is also in b.

To prove Lemma 4.10, it suffices to observe that the sets Z_{p}(b), when b runs through the blocks of kG, form a partition of Z_{p}(G), and to prove that the characters of the modules M_{P,Ind_{\overline{C}_{G}(P)}^{\overline{N}_{G}(P)}E}, for (P, E) ∈ [Z_{p}(G)], form

a basis of the subspace of ℚ ⊗_{ℤ} R_{K}(G) generated by the characters of the p-permutation modules.

For this, observe that

Ind_{C_{G}(P)}^{G}Inf_{\overline{C}_{G}(P)}^{C_{G}(P)}E ≅ Ind_{N_{G}(P)}^{G}Inf_{\overline{N}_{G}(P)}^{N_{G}(P)}Ind_{\overline{C}_{G}(P)}^{\overline{N}_{G}(P)}E

is a direct sum of M_{P,Ind_{\overline{C}_{G}(P)}^{\overline{N}_{G}(P)}E} and of indecomposable modules with vertex

strictly contained in P up to conjugation. This yields a triangular transition matrix, with non zero diagonal coefficients, and such a matrix changes a basis to another basis.

4.11. The first equality in Assertion 1 of Theorem 4.2 follows trivially from Lemma 4.10. The second one follows from the fact that, for any b-Brauer pair (R, c)

|N_{G}(R)\Irr_{k}(kC_{G}(R)Tr_{N_{G}(R,c)}^{N_{G}(R)}c)| = |N_{G}(R, c)\Irr_{k}(kC_{G}(R)c)| .

This is because the algebra kC_{G}(R)Tr_{N_{G}(R,c)}^{N_{G}(R)}c is isomorphic to the direct sum of the block algebras kC_{G}(R)^{x}c, for x ∈ [N_{G}(R)/N_{G}(R, c)], which are transitively permuted by N_{G}(R). Moreover the stabilizer in N_{G}(R) of kC_{G}(R)c is equal to N_{G}(R, c). 4.12. For any p-subgroup R of G, induction from C_{G}(R) to N_{G}(R) induces an inequality

(4.13)  |N_{G}(R)\Irr_{k}(kC_{G}(R)Br_{R}(b))| ≤ |Irr_{k}(kN_{G}(R)Br_{R}(b))| .

Indeed, the left hand side is equal to the rank of the group

P_{k}(kC_{G}(R)Br_{R}(b))_{N_{G}(R)}

of N_{G}(R)-coinvariants on the group of projective kC_{G}(R)Br_{R}(b)-modules. It is easy to see that Ind_{C_{G}(R)}^{N_{G}(R)} induces an injective map from this group to the group P_{k}(kN_{G}(R)Br_{R}(b)) of projective kN_{G}(R)Br_{R}(b)-modules, and the rank of this group is equal to the right hand side of 4.13.

The Cartan matrix of coμ_{k}(b) is non singular if and only if

∑_{R∈[C_{p}(G)]} |N_{G}(R)\Irr_{k}(kC_{G}(R)Br_{R}(b))| = ∑_{R∈[S_{p}(G)]} |Irr_{k}(kN_{G}(R)Br_{R}(b))| .

Indeed, the left hand side is the rank of the Cartan matrix, and the right hand side the size of this matrix, i.e. the number of indecomposable p-permutation modules in the block b, up to isomorphism.

By inequality 4.13, this is in turn equivalent to the following equality, for any R ∈ S_{p}(G) :

|Irr_{k}(kN_{G}(R)Br_{R}(b))| = \{ \begin{matrix}|N_{G}(R)\Irr_{k}(kC_{G}(R)Br_{R}(b))| & if R is cyclic \\ 0 & otherwise . \end{matrix}

4.14. Hence if the Cartan matrix of coμ_{k}(b) is non singular, and b has defect D, then in particular Br_{D}(b) ≠ 0, and |Irr_{k}(kN_{G}(D)Br_{D}(b))| ≠ 0, so D is cyclic. Let (D, c) be a maximal b-Brauer pair. Then Br_{D}(b) = Tr_{N_{G}(D,c)}^{N_{G}(D)}c, and the algebra kN_{G}(D)Br_{D}(b) is isomorphic to a matrix algebra over kN_{G}(D, c)c. In particular

(4.15)  |Irr_{k}(kN_{G}(D)Br_{D}(b))| = |Irr_{k}(kN_{G}(D, c)c)| .

Moreover Br_{D}(b) splits as a sum of blocks of C_{G}(D), which are the distinct N_{G}(D)-conjugates of c. Hence, the orbits of N_{G}(D) on Irr_{k}(kC_{G}(D)Br_{D}(b)) are in one to one correspondence with the orbits of N_{G}(D, c) on Irr_{k}(kC_{G}(D)c). It follows that

|N_{G}(D, c)\Irr_{k}(kC_{G}(D)c)| = |Irr_{k}(kN_{G}(D, c)c)| .

Now the group N_{G}(D, c)/C_{G}(D) is the inertial quotient of b. It is a cyclic group of order e dividing p − 1. The block c is a nilpotent block of C_{G}(D), so in particular there is a unique simple kC_{G}(D)c-module S, which is invariant by N_{G}(D, c). This simple module can be extended to a simple kN_{G}(D, c)c module in e different ways, by the following argument (see [2], proof of Proposition 6.5.4) : let g be a generator of the group N_{G}(Q)/C_{G}(Q), and θ : S → S^{g} an isomorphism of kC_{G}(Q)-modules. Then the map g^{e}(s) ↦ θ^{e}(s) is a kC_{G}(Q) automorphism of S, because g^{e} ∈ C_{G}(Q). As k is algebraically closed, there is a scalar μ ∈ k such that g^{e}(s) = μθ^{e}(s), for any s ∈ S. Moreover there are e distinct elements λ ∈ k such that λ^{e} = μ. For each such λ, one can let g act on S by λθ, and this gives e mutually non isomorphic extensions of S to a simple kN_{G}(Q, c)-module, which are all in the block c since c ∈ kC_{G}(Q).

These modules are not isomorphic to each other (since their restrictions to <g> are not). Hence e = 1 by 4.15, since |Irr_{k}(kC_{G}(D)c)| = 1, and this is equivalent to saying that b is nilpotent, since b has cyclic defect. 4.16. Conversely, if b is a nilpotent block with a cyclic defect group D, then for each R ∈ S_{p}(G), either Br_{R}(b) = 0 if R is not contained in D up to G conjugation, or Br_{R}(b) is a sum ∑_{i∈I} b_{i} of blocks b_{i} of N_{G}(R), if R is contained

in D up to conjugation, and in that case R is cyclic since D is. Each of the blocks b_{i} is equal to Tr_{N_{G}(R,c_{i})}^{N_{G}(R)}c_{i}, where (R, c_{i}) is some b-Brauer pair. In particular

|Irr_{k}(kN_{G}(R)Br_{R}(b))| = ∑_{i∈I} |Irr_{k}(kN_{G}(R)Br_{R}(b_{i}))|
= ∑_{i∈I} |Irr_{k}(kN_{G}(R, c_{i})c_{i})| .

Since the block b is nilpotent, all the groups N_{G}(R, c_{i})/C_{G}(R) are p-groups, so |Irr_{k}(kN_{G}(R, c_{i})c_{i})| = |N_{G}(R, c_{i})\Irr_{k}(kC_{G}(R)c_{i})|, and this is equal to 1 since c_{i} is a nilpotent block of C_{G}(R) by Theorem 1.2 of [9]. It follows that

|Irr_{k}(kN_{G}(R)Br_{R}(b))| = |I| = |N_{G}(R)\Irr_{k}(kC_{G}(R)Br_{R}(b))| .

This implies that the Cartan matrix of coμ_{k}(b) is non singular, and completes the proof of Theorem 4.2.

References

[1] D. Benson. Modular representation theory: New trends and methods, volume 1081 of Lecture Notes in Mathematics. Springer, 1984.

[2] D. Benson. Representations and cohomology I, volume 30 of Cambridge studies in advanced mathematics. Cambridge University Press, 1991.

[3] S. Bouc. Résolutions de foncteurs de Mackey. In Group representations: Cohomology, Group Actions and Topology, volume 63, pages 31–83. AMS Proceedings of Symposia in Pure Mathematics, 1998.

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

[5] S. Bouc and J. Thévenaz. The primitive idempotents of the ppermutation ring. Journal of Algebra, 323:2905-2915, 2010.

[6] R. Brauer and C. Nesbitt. On the modular characters of groups. Ann. of Maths, 42(2):556–590, 1941.

[7] M. Broué. On Scott modules and p-permutation modules: an approach through the Brauer morphism. Proc.AMS, 93(3):401–408, March 1985.

[8] M. Broué. Rickard equivalences and block theory, volume 211 of London. Math. Soc. Lecture Note, pages 58–79. Cambridge Univ. Press, 1995.

[9] M. Broué and L. Puig. A Frobenius theorem for blocks. Invent. Math., 56:117–128, 1980.

[10] W. Feit. The representation theory of finite groups. North Holland Publishing Company, 1982.

[11] W. Gorenstein. Finite groups. Chelsea, 1968.

[12] M. Nicollerat. Foncteurs de Mackey projectifs. PhD thesis, Ecole Polytechnique Fédérale de Lausanne, 2008.

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

Serge Bouc LAMFA - CNRS UMR 6140 Université de Picardie Jules Verne, 33, rue St Leu 80039 - Amiens - FRANCE.

email :  serge.bouc@u-picardie.fr
web :  http://www.mathinfo.u-picardie.fr/bouc/