The primitive idempotents of the p-permutation ring

Serge Bouc and Jacques Thévenaz

Abstract : Let G be a finite group, let p be a prime number, and let K be a field of characteristic 0 and k be a field of characteristic p, both large enough. In this note we state explicit formulae for the primitive idempotents of K ⊗_{ℤ} pp_{k}(G), where pp_{k}(G) is the ring of p-permutation kG-modules.

AMS Subject Classification : 19A22, 20C20.

Key words : p-permutation, idempotent, trivial source.

1. Introduction

Let G be a finite group, let p be a prime number, and let K be a field of characteristic 0 and k be a field of characteristic p, both large enough. In this note we state explicit formulae for the primitive idempotents of K ⊗_{ℤ}pp_{k}(G), where pp_{k}(G) is the ring of p-permutation kG-modules (also called the trivial source ring).

To obtain these formulae, we first use induction and restriction to reduce to the case where G is cyclic modulo p, i.e. G has a normal Sylow p-subgroup with cyclic quotient. Then we solve the easy and well known case where G is a cyclic p′-group. Finally we conclude by using the natural ring homomomorphism from the Burnside ring B(G) of G to pp_{k}(G), and the classical formulae for the primitive idempotents of K ⊗_{ℤ} B(G).

Our formulae are an essential tool in [2], where Cartan matrices of Mackey algebras are considered, and some invariants of these matrices (determinant, rank) are explicitly computed.

2. p-permutation modules

2.1. Notation.

• Throughout the paper, G will be a fixed finite group and p a fixed prime number. We consider a field k of characteristic p and we denote by kG the group algebra of G over k. We assume that k is large enough in the sense that it is a splitting field for every group algebra k(N_{G}(P)/P ), where P runs through the set of all p-subgroups of G.

• We let K be a field of characteristic 0 and we assume that K is large enough in the sense that it contains the values of all the Brauer characters of the groups N_{G}(P)/P, where P runs through the set of all p-subgroups of G.

We recall quickly how Brauer characters are defined. We let \overline{k} be an extension of k containing all the n-th roots of unity, where n is the p′-part of the exponent of G. We choose an isomorphism θ : μ_{n}(\overline{k}) → μ_{n}(ℂ) from the group of n-th roots of unity in \overline{k} and the corresponding group in ℂ. If V is an r-dimensional kH-module for the group H = N_{G}(P)/P and if s is an element of the set H_{p′} of all p′-elements of H, the matrix of the action of s on V has eigenvalues (λ_{1}, …, λ_{r}) in the group μ_{n}(\overline{k}). The Brauer character ϕ_{V} of V is the central function defined on H_{p′}, with values in the field ℚ[μ_{n}(ℂ)], sending s to ∑_{i=1}^{r} θ(λ_{i}). The actual values of Brauer characters may lie in a subfield of ℚ[μ_{n}(ℂ)] and we simply require that K contains all these values.

2.2. Remark : Let V be as above and let W be a t-dimensional kH-module. If s has eigenvalues (μ_{1}, …, μ_{t}) on W, its eigenvalues for the diagonal action of H on V ⊗_{k} W are (λ_{i}μ_{j})_{1≤i≤r, 1≤j≤t}. It follows that ϕ_{V ⊗_{k}W} (s) = ∑_{i=1}^{r} ∑_{j=1}^{t} θ(λ_{i}μ_{j}) = ϕ_{V} (s)ϕ_{W} (s).

• When H is a subgroup of G, and M is a kG-module, we denote by Res_{H}^{G}M the kH-module obtained by restricting the action of G to H. When L is a kH-module, we denote by Ind_{H}^{G}L the induced kG-module.

• When M is a kG-module, and P is a subgroup of G, the k-vector space of fixed points of P on M is denoted by M^{P}. When Q ≤ P are subgroups of G, the relative trace is the map tr_{Q}^{P} : M^{Q} → M^{P} defined by tr_{Q}^{P}(m) = ∑_{x∈[P/Q]} x · m.

• When M is a kG-module, the Brauer quotient of M at P is the k-vector space

M[P] = M^{P} / ∑_{Q<P} tr_{Q}^{P}M^{Q} .

This k-vector space has a natural structure of k\overline{N}_{G}(P)-module, where as usual \overline{N}_{G}(P) = N_{G}(P)/P. It is equal to zero if P is not a p-group.

• If P is a normal p-subgroup of G and M is a k(G/P)-module, denote by Inf_{G/P}^{G} M the kG-module obtained from M by inflation to G. Then there is an isomorphism

(Inf_{G/P}^{G} M)[P] ≅ M

of k(G/P)-modules.

• When G acts on a set X (on the left), and x, y ∈ X, we write x =_{G} y if x and y are in the same G-orbit. We denote by [G\X] a set of representatives of G-orbits on X, and by X^{G} the set of fixed points of G on X. For x ∈ X, we denote by G_{x} its stabilizer in G.

2.3. Review of p-permutation modules. We begin by recalling some definitions and basic results. We refer to [3], and to [1] Sections 3.11 and 5.5 for details :

2.4. Definition. A permutation kG-module is a kG-module admitting a Ginvariant k-basis. A p-permutation kG-module M is a kG-module such that Res_{S}^{G}M is a permutation kS-module, where S is a Sylow p-subgroup of G.

The p-permutation kG-modules are also called trivial source modules, because the indecomposable ones coincide with the indecomposable modules having a trivial source (see [3] 0.4). Moreover, the p-permutation modules also coincide with the direct summands of permutation modules (see [1], Lemma 3.11.2).

2.5. Proposition.

1. If H is a subgroup of G, and M is a p-permutation kG-module, then the restriction Res_{H}^{G}M of M to H is a p-permutation kH-module.

2. If H is a subgroup of G, and L is a p-permutation kH-module, then the induced module Ind_{H}^{G}L is a p-permutation kG-module.

3. If N is a normal subgroup of G, and L is a p-permutation k(G/N)- module, the inflated module Inf_{G/N}^{G}L is a p-permutation kG-module.

4. If P is a p-group, and M is a permutation kP -module with P -invariant basis X, then the image of the set X^{P} in M[P] is a k-basis of M[P].

5. If P is a p-subgroup of G, and M is a p-permutation kG-module, then the Brauer quotient M[P] is a p-permutation k\overline{N}_{G}(P)-module.

6. If M and N are p-permutation kG-modules, then their tensor product M ⊗_{k} N is again a p-permutation kG-module.

Proof : Assertions 1,2,3, and 6 are straightforward consequences of the same assertions for permutation modules. For Assertion 4, see [3] 1.1.(3). Assertion 5 follows easily from Assertion 4 (see also [3] 3.1).

This leads to the following definition :

2.6. Definition. The p-permutation ring pp_{k}(G) is the Grothendieck group of the category of p-permutation kG-modules, with relations corresponding to direct sum decompositions, i.e. [M] + [N] = [M ⊕ N]. The ring structure on pp_{k}(G) is induced by the tensor product of modules over k. The identity element of pp_{k}(G) is the class of the trivial kG-module k.

As the Krull-Schmidt theorem holds for kG-modules, the additive group pp_{k}(G) is a free (abelian) group on the set of isomorphism classes of indecomposable p-permutation kG-modules. These modules have the following properties :

2.7. Theorem. [ [3] Theorem 3.2]

1. The vertices of an indecomposable p-permutation kG-module M are the maximal p-subgroups P of G such that M[P] ≠ {0}.

2. An indecomposable p-permutation kG-module has vertex P if and only if M[P] is a non-zero projective k\overline{N}_{G}(P)-module.

3. The correspondence M ↦ M[P] induces a bijection between the isomorphism classes of indecomposable p-permutation kG-modules with vertex P and the isomorphism classes of indecomposable projective k\overline{N}_{G}(P)- modules.

2.8. Notation. Let P_{G,p} denote the set of pairs (P, E), where P is a p-subgroup of G, and E is an indecomposable projective k\overline{N}_{G}(P)-module. The group G acts on P_{G,p} by conjugation, and we denote by [P_{G,p}] a set of representatives of G-orbits on P_{G,p}.

For (P, E) ∈ P_{G,p}, let M_{P,E} denote the (unique up to isomorphism) indecomposable p-permutation kG-module such that M_{P,E}[P] ≅ E.

2.9. Corollary. The classes of the modules M_{P,E}, for (P, E) ∈ [P_{G,p}] form a ℤ-basis of pp_{k}(G).

2.10. Notation. The operations Res_{H}^{G}, Ind_{H}^{G}, Inf_{G/N}^{G} extend linearly to maps between the corresponding p-permutations rings, denoted with the same symbol.

The maps Res_{H}^{G} and Inf_{G/N}^{G} are ring homomorphisms, whereas Ind_{H}^{G} is not in general. Similarly :

2.11. Proposition. Let P be a p-subgroup of G. Then the correspondence M ↦ M[P] induces a ring homomorphism Br_{P}^{G} : pp_{k}(G) → pp_{k}(\overline{N}_{G}(P)).

Proof : Let M and N be p-permutation kG-modules. The canonical bilinear map M × N → M ⊗_{k} N is G-equivariant, hence it induces a bilinear map β_{P} : M[P] × N[P] → (M ⊗_{k} N)[P] (see [3] 1.2), which is \overline{N}_{G}(P)-equivariant. Now if X is a P -invariant k-basis of M, and Y a P -invariant k-basis of N, then X ×Y is a P -invariant basis of M ⊗_{k} N. The images of the sets X^{P}, Y^{P}, and (X × Y)^{P} are bases of M[P], N[P], and (M ⊗_{k} N)[P], respectively, and the restriction of β_{P} to theses bases is the canonical bijection X^{P} × Y^{P} → (X × Y)^{P}. It follows that β_{P} induces an isomorphism M[P] ⊗_{k} N[P] → (M ⊗_{k} N)[P] of k\overline{N}_{G}(P)-modules. Proposition 2.11 follows.

2.12. Notation. Let Q_{G,p} denote the set of pairs (P, s), where P is a p-subgroup of G, and s is a p′-element of \overline{N}_{G}(P). The group G acts on Q_{G,p}, and we denote by [Q_{G,p}] a set of representatives of G-orbits on Q_{G,p}.

If (P, s) ∈ Q_{G,p}, we denote by N_{G}(P, s) the stabilizer of (P, s) in G, and by <P s> the subgroup of N_{G}(P) generated by P s (i.e. the inverse image in N_{G}(P) of the cyclic group <s> of \overline{N}_{G}(P)).

2.13. Remarks :

• When H is a subgroup of G, there is a natural inclusion of Q_{H,p} into Q_{G,p}, as \overline{N}_{H}(P) ≤ \overline{N}_{G}(P) for any p-subgroup P of H. We will consider Q_{H,p} as a subset of Q_{G,p}.

• When (P, s) ∈ Q_{G,p}, the group N_{G}(P, s) is the set of elements g in N_{G}(P) whose image in \overline{N}_{G}(P) centralizes s. In other words, there is a short exact sequence of groups

(2.14)  1 → P → N_{G}(P, s) → C_{\overline{N}_{G}(P)}(s) → 1 .

In particular N_{G}(P, s) is a subgroup of N_{G}(<P s>).

2.15. Notation. Let (P, s) ∈ Q_{G,p}. Let τ_{P,s}^{G} denote the additive map from pp_{k}(G) to K sending the class of a p-permutation kG-module M to the value at s of the Brauer character of the \overline{N}_{G}(P)-module M[P].

2.16. Remarks :

• It is clear that τ_{P,s}^{G}(M) only depends on the restriction of M to the group <P s>. In other words

τ_{P,s}^{G} = τ_{P,s}^{<P s>} ∘ Res_{<P s>}^{G} .

Furthermore, it is clear from the definition that

(2.17)  τ_{P,s}^{G} = τ_{1,s}^{<P s>/P} ∘ Br_{P}^{<P s>} ∘ Res_{<P s>}^{G} .

• It is easy to see that τ_{P,s}^{G} only depends on the G-orbit of (P, s), that is, τ_{P^{g},s^{g}}^{G} = τ_{P,s}^{G} for every g ∈ G.

The following proposition is Corollary 5.5.5 in [1], but our construction of the species is slightly different (but equivalent, of course). For this reason, we sketch an independent proof :

2.18. Proposition.

1. The map τ_{P,s}^{G} is a ring homomorphism pp_{k}(G) → K and extends to a K-algebra homomorphism (a species) τ_{P,s}^{G} : K ⊗_{ℤ} pp_{k}(G) → K.

2. The set {τ_{P,s}^{G} | (P, s) ∈ [Q_{G,p}]} is the set of all distinct species from K ⊗_{ℤ} pp_{k}(G) to K. These species induce a K-algebra isomorphism

T = ∏_{(P,s)∈[Q_{G,p}]} τ_{P,s}^{G} : K ⊗_{ℤ} pp_{k}(G) → ∏_{(P,s)∈[Q_{G,p}]} K .

Proof : By 2.17, to prove Assertion 1, it suffices to prove that τ_{1,s}^{<P s>/P} is a ring homomorphism, since both Res_{<P s>}^{G} and Br_{P}^{<P s>} are ring homomorphisms. In other words, we can assume that P = 1. Now the value of τ_{1,s}^{G} on the class of a kG-module M is the value ϕ_{M}(s) of the Brauer character of M at s, so Assertion 1 follows from Remark 2.2.

For Assertion 2, it suffices to prove that T is an isomorphism. Since [P_{G,p}| and [Q_{G,p}] have the same cardinality, the matrix M of T is a square matrix. Let (P, E) ∈ P_{G,p}, and (Q, s) ∈ Q_{G,p}. Then τ_{Q,s}(M_{P,E}) is equal to zero if Q is not contained in P up to G-conjugation, because in this case M_{P,E}(Q) = {0} by Theorem 2.7. It follows that M is block triangular. As moreover M_{P,E}[P] ≅ E, we have that τ_{P,s}(M_{P,E}) = ϕ_{E}(s). This means that the diagonal block of M corresponding to P is the matrix of Brauer characters of projective k\overline{N}_{G}(P)-modules, and these are linearly independent by Lemma 5.3.1 of [1]. It follows that all the diagonal blocks of M are non singular, so M is invertible, and T is an isomorphism.

2.19. Corollary. The algebra K ⊗_{ℤ} pp_{k}(G) is a split semisimple commutative K-algebra. Its primitive idempotents F_{P,s}^{G} are indexed by [Q_{G,p}], and the idempotent F_{P,s}^{G} is characterized by

∀(R, u) ∈ Q_{G,p}, τ_{R,u}^{G}(F_{P,s}^{G}) = \{ \begin{matrix}1 & if (R, u) =_{G} (P, s) \\ 0 & otherwise. \end{matrix}

3. Restriction and induction

3.1. Proposition. Let H ≤ G, and (P, s) ∈ Q_{G,p}. Then

Res_{H}^{G}F_{P,s}^{G} = ∑_{(Q,t)} F_{Q,t}^{H} ,

where (Q, t) runs through a set of representatives of H-conjugacy classes of G-conjugates of (P, s) contained in H.

Proof : Indeed, as Res_{H}^{G} is an algebra homomorphism, the element Res_{H}^{G}F_{P,s}^{G} is an idempotent of K ⊗_{ℤ} pp_{k}(H), hence it is equal to a sum of some distinct primitive idempotents F_{Q,t}^{H}. The idempotent F_{Q,t}^{H} appears in this decomposition if and only if τ_{Q,t}^{H}(Res_{H}^{G}F_{P,s}^{G}) = 1. By Remark 2.16

τ_{Q,t}^{H}(Res_{H}^{G}F_{P,s}^{G}) = τ_{Q,t}^{<Qt>}(Res_{<Qt>}^{H}Res_{H}^{G}F_{P,s}^{G})
= τ_{Q,t}^{<Qt>}(Res_{<Qt>}^{G}F_{P,s}^{G})
= τ_{Q,t}^{G}(F_{P,s}^{G}) .

Now τ_{Q,t}^{G}(F_{P,s}^{G}) is equal to 1 if and only if (Q, t) and (P, s) are G-conjugate. This completes the proof.

3.2. Proposition. Let H ≤ G, and (Q, t) ∈ Q_{H,p}. Then

Ind_{H}^{G}F_{Q,t}^{H} = |N_{G}(Q, t) : N_{H}(Q, t)|F_{Q,t}^{G} .

Proof : Since K ⊗_{ℤ} pp_{k}(G) is a split semisimple commutative K-algebra, any element X in K ⊗_{ℤ} pp_{k}(G) can be written

(3.3)  X = ∑_{(P,s)∈[Q_{G,p}]} τ_{P,s}^{G}(X)F_{P,s}^{G} ,

and moreover for any (P, s) ∈ Q_{G,p}

τ_{P,s}^{G}(X)F_{P,s}^{G} = X · F_{P,s}^{G} .

Setting X = Ind_{H}^{G}F_{Q,t}^{H} in this equation gives

τ_{P,s}^{G}(Ind_{H}^{G}F_{Q,t}^{H})F_{P,s}^{G} = (Ind_{H}^{G}F_{Q,t}^{H}) · F_{P,s}^{G}
= Ind_{H}^{G}(F_{Q,t}^{H} · Res_{H}^{G}F_{P,s}^{G}) .

By Proposition 3.1, the element Res_{H}^{G}F_{P,s}^{G} is equal to the sum of the distinct idempotents F_{P^{y},s^{y}}^{H} associated to elements y of G such that <P s>^{y} ≤ H. The product F_{Q,t}^{H} · F_{P^{y},s^{y}}^{H} is equal to zero, unless (Q, t) is H-conjugate to (P^{y}, s^{y}), which implies that (Q, t) and (P, s) are G-conjugate. It follows that the only non zero term in the right hand side of Equation 3.3 is the term corresponding to (Q, t). Hence

Ind_{H}^{G}F_{Q,t}^{H} = τ_{Q,t}^{G}(Ind_{H}^{G}F_{Q,t}^{H})F_{Q,t}^{G} .

Now by Remark 2.16 and the Mackey formula

τ_{Q,t}^{G}(Ind_{H}^{G}F_{Q,t}^{H}) = τ_{Q,t}^{<Qt>}(Res_{<Qt>}^{G}Ind_{H}^{G}F_{Q,t}^{H})
= τ_{Q,t}^{<Qt>}( ∑_{x∈<Qt>\G/H} Ind_{<Qt>∩^{x}H}^{<Qt>}^{x}Res_{<Qt>^{x}∩H}^{H}F_{Q,t}^{H}) .

By Proposition 3.1, the element Res_{<Qt>^{x}∩H}^{H}F_{Q,t}^{H} is equal to the sum of the distinct idempotents F_{Q^{y},t^{y}}^{<Qt>^{x}∩H} corresponding to elements y ∈ H such that <Qt>^{y} ≤ <Qt>^{x} ∩H. This implies <Qt>^{y} = <Qt>^{x}, i.e. y ∈ N_{G}(<Qt>)x, thus x ∈ N_{G}(<Qt>) · H. This gives

τ_{Q,t}^{G}(Ind_{H}^{G}F_{Q,t}^{H}) = τ_{Q,t}^{<Qt>}( ∑_{{x∈N_{G}(<Qt>)H/H \atop y∈N_{H}(Q,t)\N_{G}(<Qt>)x}} ^{x}F _{Q^{y},t^{y}}^{<Qt>})

= ∑_{{x∈N_{G}(<Qt>)/N_{H}(<Qt>) \atop y∈N_{H}(Q,t)\N_{G}(<Qt>)x}} τ_{Q,t}^{<Qt>}(F_{Q^{yx^{−1}},t^{yx^{−1}}}^{<Qt>})

= ∑_{z∈N_{H}(Q,t)\N_{G}(<Qt>)} τ_{Q,t}^{<Qt>}(F_{Q^{z},t^{z}}^{<Qt>}) ,

where z = yx^{−1}. Finally τ_{Q,t}^{<Qt>}(F_{Q^{z},t^{z}}^{<Qt>}) is equal to 1 if (Q^{z}, t^{z}) is conjugate to (Q, t) in <Qt>, and to zero otherwise.

If u ∈ <Qt> is such that (Q^{z}, t^{z})^{u} = (Q, t), then zu ∈ N_{G}(Q, t). But since [<Qt>, t] ≤ Q, we have <Qt> ≤ N_{G}(Q, t), so u ∈ N_{G}(Q, t), hence z ∈ N_{G}(Q, t), and (Q^{z}, t^{z}) = (Q, t). It follows that

τ_{Q,t}^{G}(Ind_{H}^{G}F_{Q,t}^{H}) = |N_{G}(Q, t) : N_{H}(Q, t)| ,

which completes the proof of the proposition.

3.4. Corollary. Let (P, s) ∈ Q_{G,p}. Then

F_{P,s}^{G} = \frac{|s|}{|C_{\overline{N}_{G}(P)}(s)|}Ind_{<P s>}^{G}F_{P,s}^{<P s>} .

Proof : Apply Proposition 3.2 with (Q, t) = (P, s) and H = <P s>. Then N_{H}(Q, t) = <P s>, thus by Exact sequence 2.14

|N_{G}(Q, t) : N_{H}(Q, t)| = \frac{|P ||C_{\overline{N}_{G}(P)}(s)|}{|P ||s|} = \frac{|C_{\overline{N}_{G}(P)}(s)|}{|s|} ,

and the corollary follows.

4. Idempotents

It follows from Corollary 3.4 that, in order to find formulae for the primitive idempotents F_{P,s}^{G} of K⊗_{ℤ}pp_{k}(G), it suffices to find the formula expressing the idempotent F_{P,s}^{<P s>}. In other words, we can assume that G = <P s>, i.e. that G has a normal Sylow p-subgroup P with cyclic quotient generated by s.

4.1. A morphism from the Burnside ring. When G is an arbitrary finite group, there is an obvious ring homomorphism L_{G} from the Burnside ring B(G) to pp_{k}(G), induced by the linearization operation, sending a finite G-set X to the permutation module kX, which is obviously a p-permutation module. This morphism also commutes with restriction and induction : if H ≤ G, then

(4.2)  L_{H} ∘ Res_{H}^{G} = Res_{H}^{G} ∘ L_{G} ,  L_{G} ∘ Ind_{H}^{G} = Ind_{H}^{G} ∘ L_{H} .

Indeed, for any G-set X, the kH-modules kRes_{H}^{G}X and Res_{H}^{G}(kX) are isomorphic, and for any H-set Y, the kG-modules kInd_{H}^{G}Y and Ind_{H}^{G}(kY) are isomorphic.

Similarly, when P is a p-subgroup of G, the ring homomorphism Φ_{P} : B(G) → B(\overline{N}_{G}(P)) induced by the operation X ↦ X^{P} on G-sets, is compatible with the Brauer morphism Br_{P}^{G} : pp_{k}(G) → pp_{k}(\overline{N}_{G}(P)) :

(4.3)  L_{\overline{N}_{G}(P)} ∘ Φ_{P} = Br_{P}^{G} ∘ L_{G} .

This is because for any G-set X, the k\overline{N}_{G}(P)-modules k(X^{P}) and (kX)[P] are isomorphic.

The ring homomorphism L_{G} extends linearly to a K-algebra homomorphism K ⊗_{ℤ} B(G) → K ⊗_{ℤ} pp_{k}(G), still denoted by L_{G}. The algebra K ⊗_{ℤ} B(G) is also a split semisimple commutative K-algebra. Its species are the K-algebra maps

K ⊗_{ℤ} B(G) → K, X ↦ |X^{H}| ,

where H runs through the set of all subgroups of G up to conjugation. Here we denote by |X^{H}| the number of H-fixed points of a G-set X and this notation is then extended K-linearly to any X ∈ K ⊗_{ℤ} B(G). The primitive idempotents e_{H}^{G} of K ⊗_{ℤ} B(G) are indexed by subgroups H of G, up to conjugation. They are given by the following formulae, found by Gluck ([4]) and later independently by Yoshida ([5]) :

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

where μ denotes the Möbius function of the poset of subgroups of G. The idempotent e_{H}^{G} is characterized by the fact that for any X ∈ K ⊗_{ℤ} B(G)

X · e_{H}^{G} = |X^{H}|e_{H}^{G} .

4.5. Remark : Since |X^{H}| only depends on Res_{H}^{G}X, it follows in particular that X is a scalar multiple of the “top” idempotent e_{G}^{G} if and only if Res_{H}^{G}X = 0 for any proper subgroup H of G. In particular, if N is a normal subgroup of G, then

(4.6)  (e_{G}^{G})^{N} = e_{G/N}^{G/N} .

This is because for any proper subgroup H/N of G/N

Res_{H/N}^{G/N}(e_{G}^{G})^{N} = (Res_{H}^{G}e_{G}^{G})^{N} = 0 .

So (e_{G}^{G})^{N} is a scalar multiple of e_{G/N}^{G/N}. As it is also an idempotent, it is equal

to 0 or e_{G/N}^{G/N}. Finally

|((e_{G}^{G})^{N})^{G/N}| = |(e_{G}^{G})^{G}| = 1 ,

so (e_{G}^{G})^{N} is non zero.

4.7. The case of a cyclic p′-group. Suppose that G is a cyclic p′-group, of order n, generated by an element s. In this case, there are exactly n group homomorphisms from G to the multiplicative group k^{×} of k. For each of these group homomorphisms φ, let k_{φ} denote the kG-module k on which the generator s acts by multiplication by φ(s). As G is a p′-group, this module is simple and projective. The (classes of the) modules k_{φ}, for φ ∈ Ĝ = Hom(G, k^{×}), form a basis of pp_{k}(G).

Since moreover for φ, ψ ∈ Ĝ, the modules k_{φ}⊗_{k}k_{ψ} and k_{φψ} are isomorphic, the algebra K ⊗_{ℤ} pp_{k}(G) is isomorphic to the group algebra of the group Ĝ. This leads to the following classical formula :

4.8. Lemma. Let G be a cyclic p′-group. Then for any t ∈ G,

F_{1,t}^{G} = \frac{1}{n} ∑_{φ∈Ĝ} φ̃(t^{−1})k_{φ} ,

where φ̃ is the Brauer character of k_{φ}.

Proof : Indeed for s, t ∈ G

τ_{1,t}^{G}(\frac{1}{n} ∑_{φ∈Ĝ} φ̃(s^{−1})k_{φ}) = \frac{1}{n} ∑_{φ∈Ĝ} φ̃(s^{−1})φ̃(t) = δ_{s,t} ,

where δ_{s,t} is the Kronecker symbol.

4.9. The case G = <P s>. Suppose now more generally that G = <P s>, where P is a normal Sylow p-subgroup of G and s is a p′-element. In this case, by Proposition 3.1, the restriction of F_{P,s}^{G} to any proper subgroup of G is equal to zero. Moreover, since N_{G}(P, t) = G for any t ∈ G/P, the conjugacy class of the pair (P, t) reduces to {(P, t)}.

4.10. Lemma. Suppose G = <P s>, and set E_{G}^{G} = L_{G}(e_{G}^{G}). Then

E_{G}^{G} = ∑_{<t>=<s>} F_{P,t}^{G} .

Proof : By 4.2 and by Remark 4.5, the restriction of E_{G}^{G} to any proper subgroup of G is equal to zero. Let (Q, t) ∈ Q_{G,p}, such that the group L = <Qt> is a proper subgroup of G. By Proposition 3.2, there is a rational number r such that

F_{Q,t}^{G} = r Ind_{L}^{G}F_{Q,t}^{L} .

It follows that

E_{G}^{G} · F_{Q,t}^{G} = r Ind_{L}^{G}((Res_{L}^{G}E_{G}^{G}) · F_{Q,t}^{L}) = 0 .

Now E_{G}^{G} is an idempotent of K ⊗_{ℤ} pp_{k}(G), hence is it a sum of some of the primitive idempotents F_{Q,t}^{G} associated to pairs (Q, t) for which <Qt> = G. This condition is equivalent to Q = P and <t> = <s>.

It remains to show that all these idempotents F_{P,t}^{G} appear in the decomposition of E_{G}^{G}, i.e. equivalently that τ_{P,t}^{G}(E_{G}^{G}) = 1 for any generator t of <s>. Now by 4.6 and Remark 2.16

τ_{P,t}^{G}(E_{G}^{G}) = τ_{1,t}^{G/P} (Br_{P}^{G}(E_{G}^{G})) = τ_{1,t}^{<s>}(E_{<s>}^{<s>}) .

Now the value at t of the Brauer character of a permutation module kX is equal to the number of fixed points of t on X. By K-linearity, this gives

τ_{1,t}^{<s>}(E_{<s>}^{<s>}) = |(e_{<s>}^{<s>})^{t}| ,

and this is equal to 1 if t generates <s>, and to 0 otherwise, as was to be shown.

4.11. Proposition. Let (P, s) ∈ Q_{G,p}, and suppose that G = <P s>. Then

F_{P,s}^{G} = E_{G}^{G} · Inf_{G/P}^{G} F_{1,s}^{G/P} .

Proof : Set E_{s} = E_{G}^{G}·Inf_{G/P}^{G} F_{1,s}^{G/P}. Then E_{s} is an idempotent of K⊗_{ℤ}pp_{k}(G), as it is the product of two (commuting) idempotents. Let (Q, t) ∈ Q_{G,p}. If <Qt> ≠ G, then τ_{Q,t}^{G}(E_{G}^{G}) = 0 by Lemma 4.10, thus τ_{Q,t}^{G}(E_{s}) = 0. And if <Qt> = G, then Q = P and <t> = <s>. In this case

τ_{Q,t}^{G}(E_{s}) = τ_{P,t}^{G}(E_{G}^{G}) · τ_{P,t}^{G}(Inf_{G/P}^{G} F_{1,s}^{G/P}) .

By Lemma 4.10, the first factor in the right hand side is equal to 1. The second factor is equal to

τ_{P,t}^{G}(Inf_{G/P}^{G} F_{1,s}^{G/P}) = τ_{1,t}^{G/P} Br_{P}^{G}(Inf_{G/P}^{G} F_{1,s}^{G/P})
= τ_{1,t}^{G/P} (F_{1,s}^{G/P}) = δ_{t,s} ,

where δ_{t,s} is the Kronecker symbol. Hence τ_{P,t}^{G}(E_{s}) = δ_{t,s}, and this completes the proof.

4.12. Theorem. Let G be a finite group, and let (P, s) ∈ Q_{G,p}. Then the primitive idempotent F_{P,s}^{G} of the p-permutation algebra K ⊗_{ℤ} pp_{k}(G) is given by the following formula :

F_{P,s}^{G} = \frac{1}{|P ||s||C_{\overline{N}_{G}(P)}(s)|} ∑_{φ∈ ˆ<s> ; {L≤<P s> \atop P L=<P s>}} φ̃(s^{−1})|L|μ(L, <P s>) Ind_{L}^{G}k_{L,φ}^{<P s>} ,

where k_{L,φ}^{<P s>} = Res_{L}^{<P s>}Inf_{<s>}^{<P s>}k_{φ}.

Proof : By Corollary 3.4, and Proposition 4.11

F_{P,s}^{G} = \frac{|s|}{|C_{\overline{N}_{G}(P)}(s)|}Ind_{<P s>}^{G}(E_{<P s>}^{<P s>} · Inf_{<s>}^{<P s>}F_{1,s}^{<s>}) .

By Equation 4.4, this gives

F_{P,s}^{G} = \frac{|s|}{|C_{\overline{N}_{G}(P)}(s)|}Ind_{<P s>}^{G} \frac{1}{|P ||s|} ∑_{L≤<P s>}|L|μ(L, <P s>)Ind_{L}^{<P s>}k·Inf_{<s>}^{<P s>}F_{1,s}^{<s>} .

Moreover for each L ≤ <P s>

Ind_{L}^{<P s>}k · Inf_{<s>}^{<P s>}F_{1,s}^{<s>} ≅ Ind_{L}^{<P s>}(Res_{L}^{<P s>}Inf_{<s>}^{<P s>}F_{1,s}^{<s>})
≅ Ind_{L}^{<P s>}Inf_{L/L∩P}^{L} Iso_{LP/P}^{L/L∩P} Res_{LP/P}^{<s>} F_{1,s}^{<s>} .

Here we have used the fact that if L and P are subgroups of a group H, with P \underline{◁} H, then there is an isomorphism of functors

Res_{L}^{H} ∘ Inf_{H/P}^{H} ≅ Inf_{L/L∩P}^{L} ∘ Iso_{LP/P}^{L/L∩P} ∘ Res_{LP/P}^{H/P} ,

which follows from the isomorphism of (L, H/P)-bisets

H ×_{H} (H/P) ≅ _{L}(H/P)_{H/P} ≅ (L/L ∩ P) ×_{L/L∩P} (LP/P) ×_{LP/P} (H/P) .

Now Proposition 3.1 implies that Res_{LP/P}^{<s>} F_{1,s}^{<s>} = 0 if LP/P ≠ <s>, i.e. equivalently if P L ≠ <P s>. It follows that

F_{P,s}^{G} = \frac{1}{|P ||C_{\overline{N}_{G}(P)}(s)|} ∑_{{L≤<P s> \atop P L=<P s>}}|L|μ(L, <P s>)Ind_{L}^{G}(Res_{L}^{<P s>}Inf_{<s>}^{<P s>}F_{1,s}^{<s>}) .

By Lemma 4.8, this gives

F_{P,s}^{G} = \frac{1}{|P ||s||C_{\overline{N}_{G}(P)}(s)|} ∑_{φ∈{ \atop <s>} ; {L≤<P s> \atop P L=<P s>}}φ̃(s^{−1})|L|μ(L, <P s>)Ind_{L}^{G}k_{L,φ}^{<P s>} ,

where k_{L,φ}^{<P s>} = Res_{L}^{<P s>}Inf_{<s>}^{<P s>}k_{φ}, as was to be shown.

References

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

[2] S. Bouc. On the Cartan matrix of Mackey algebras. Preprint, 2009.

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

[4] D. Gluck. Idempotent formula for the Burnside ring with applications to the p-subgroup simplicial complex. Illinois J. Math., 25:63–67, 1981.

[5] T. Yoshida. Idempotents in Burnside rings and Dress induction theorem. J. Algebra, 80:90–105, 1983.

Serge Bouc, CNRS-LAMFA, Université de Picardie - Jules Verne, 33, rue St Leu, F-80039 Amiens Cedex 1, France.

serge.bouc@u-picardie.fr

Jacques Thévenaz, Institut de Géométrie, Algèbre et Topologie, EPFL, Bâtiment BCH, CH-1015 Lausanne, Switzerland.

Jacques.Thevenaz@epfl.ch