The diagonal p-permutation functor kR_{k}

Serge Bouc

Abstract Let k be an algebraically closed field of positive characteristic p. We describe the full lattice of subfunctors of the diagonal p-permutation functor kR_{k} obtained by k-linear extension from the functor R_{k} of linear representations over k. This leads to the description of the “composition factors” S_{P} of kR_{k}, which are parametrized by finite p-groups (up to isomorphism), and of the evaluations of these particular simple diagonal p-permutation functors over k.

MSC2020: 16S50, 18B99, 20C20, 20J15.

Keywords: diagonal p-permutation functor, simple functor.

1 Introduction

Let k be an algebraically closed field of positive characteristic p, and R be a commutative and unital ring. We consider^{1} the assignment G ↦ RR_{k}(G) sending a finite group G to RR_{k}(G) = R ⊗_{ℤ} R_{k}(G), where R_{k}(G) is the Grothendieck group of finite dimensional kG-modules. When G and H are finite groups, if M is a (kH, kG)-bimodule which is projective as a right kG-module, tensoring with M over kG preserves exact sequences, hence induces a well defined group homomorphism R_{k}(M) : R_{k}(G) → R_{k}(H), and an R-linear map RR_{k}(M) : RR_{k}(G) → RR_{k}(H), both simply denoted V ↦ M ⊗_{kG} V. In particular, one checks easily that this endows RR_{k} with a structure of diagonal p-permutation functor over R (see [2], [3], [4]). Our main goal is Theorem 1.3, describing the full lattice of subfunctors of kR_{k} (ordered by inclusion of subfunctors).

As a byproduct of this description, we get a “composition series” of kR_{k}, with simple subquotients S_{P} indexed by finite p-groups P up to isomorphism. From this, the evaluations of these particular simple functors can be computed, giving a new proof of the last corollary of [4].

To be more precise, we first need some notation and some definitions:

Notation 1.1: Let P be the set of isomorphism classes of finite p-groups. For a finite p-group P, we denote by [P] its isomorphism class. For a subset C of P, we say (abusively) that P belongs to C, and we write P ∈ C, if [P] ∈ C.

For a finite group G, we denote by k_{G} the trivial kG-module k, and by ⟨k_{G}⟩ the subfunctor of kR_{k} generated by k_{G} ∈ kR_{k}(G). We observe that ⟨k_{G}⟩ only depends on the isomorphism class of G.

1mainly in the case R = k, but a few results deal with the case of an arbitrary R.

For a finite group G, we denote by E_{R}(G) the essential algebra of G over R, defined by

E_{R}(G) = RT^{Δ}(G, G)/ ∑_{|H|<|G|} RT^{Δ}(G, H) ∘ RT^{Δ}(H, G),

where T^{Δ}(H, G) denotes the Grothendieck group of diagonal p-permutation (kH, kG)-bimodules (see [2], [3], [4] for details).

Definition 1.2: We say that a subset C of P is closed if for any finite p-groups P and Q

Q ≤ P and [P] ∈ C ⟹ [Q] ∈ C.

In other words, if a p-group is isomorphic to a subgroup of a p-group in C, then it belongs to C.

Our main theorem is the following:

Theorem 1.3: Let L denote the lattice of subfunctors of kR_{k}, ordered by inclusion of subfunctors. Let F denote the lattice of closed subsets of P, ordered by inclusion of subsets. Then the maps

Ψ : F ∈ L ↦ Ψ(F) = {[P] ∈ P | k_{P} ∈ F (P)} ∈ F, and
Θ : C ∈ F ↦ Θ(C) = ∑_{[P]∈C}⟨k_{P} ⟩ ∈ L,

are well defined isomorphisms of lattices, inverse to each other.

2 Some evaluations of diagonal p-permutation functors.

We start with a result of independent interest on diagonal p-permutation functors. It gives a way to compute the evaluation of a diagonal p-permutation functor at a direct product of a p′-group and a p-group. This requires the following notation:

Notation 2.1: Let L be a p′-group and Q be a p-group, and set H = L×Q. For a (finite dimensional) kL-module V, let \overrightarrow{V} = V ⊗_{k} kQ, viewed as a (kH, kQ)-bimodule for the action

∀(l, x) ∈ H = L × Q, ∀y, z ∈ Q, ∀v ∈ V, (l, x) · (v ⊗ z) · y = lv ⊗ xzy.

Similarly, let V^{∗} denote the k-dual of V, viewed as a right kL-module, and \overleftarrow{V} = V^{∗} ⊗_{k} kQ, viewed as a (kQ, kH)-bimodule for the action

∀(l, x) ∈ H = L × Q, ∀y, z ∈ Q, ∀α ∈ V^{∗}, y · (α ⊗ z) · (l, x) = αl ⊗ yzx.

Let also Irr_{k}(L) denote a set of representatives of isomorphism classes of irreducible kL-modules. With this notation:

Proposition 2.2:

1. The bimodules \overrightarrow{V} and \overleftarrow{V} are diagonal p-permutation bimodules.

2. If V and W are irreducible kL-modules, then \overleftarrow{W} ⊗_{kH} \overrightarrow{V} is isomorphic to the identity (kQ, kQ)-bimodule kQ, if W and V are isomorphic, and it is zero otherwise.

3. The direct sum ⊕_{V} (\overrightarrow{V} ⊗_{kQ} \overleftarrow{V}), for V ∈ Irr_{k}(L), is isomorphic to the

identity (kH, kH)-bimodule kH.

Proof: 1. Indeed, the group H×Q has a normal Sylow subgroup S = Q×Q, and the restrictions to S of \overrightarrow{V} and \overleftarrow{V} are both isomorphic to a direct sum of dim_{k} V copies of the identity (kQ, kQ)-bimodule kQ.

2. One checks easily that

\overleftarrow{W} ⊗_{kH} \overrightarrow{V} = (W^{∗} ⊗_{k} kQ) ⊗_{k(L×Q)} (V ⊗_{k} kQ) ≅ (W^{∗} ⊗_{kL} V) ⊗_{k} (kQ ⊗_{kQ} kQ)

as (kQ, kQ)-bimodules. Assertion 2 follows, since W^{∗} ⊗_{kL} V = 0 unless V and W are isomorphic, and since V^{∗} ⊗_{kL} V ≅ k as k is algebraically closed. Moreover kQ ⊗_{kQ} kQ ≅ kQ.

3. Similarly

\overrightarrow{V} ⊗_{kQ} \overleftarrow{V} = (V ⊗_{k} kQ) ⊗_{kQ} (V^{∗} ⊗_{k} kQ) ≅ (V ⊗_{k} V^{∗}) ⊗_{k} (kQ ⊗_{kQ} kQ)
≅ End_{k}(V) ⊗_{k} kQ

as (kH, kH)-bimodules. Now the direct sum ⊕_{V ∈Irr_{k}(L)} End_{k}(V) is the decom-

position of the semisimple group algebra kL as a direct product of its Wedderburn components. It follows that ⊕_{V} (\overrightarrow{V} ⊗_{kQ} \overleftarrow{V}) ≅ kL ⊗ kQ ≅ k(L × Q)

is isomorphic to the identity (kH, kH)-bimodule.

In the following corollary, we view \overrightarrow{V} and \overleftarrow{V} as elements of T^{Δ}(H, Q) and T^{Δ}(Q, H), respectively.

Corollary 2.3: Let R be a commutative (unital) ring, and F be a diagonal p-permutation functor over R. Let L be a p′-group, and Q be a p-group. Then:

1. F (L × Q) ≅ ⨁_{V ∈Irr_{k}(L)} F (Q). More precisely, the maps

φ ∈ F (L × Q) ↦ ⨁_{V ∈Irr_{k}(L)} F (\overleftarrow{V})(φ) ∈ ⨁_{V ∈Irr_{k}(L)} F (Q)

and

ψ = ⨁_{V ∈Irr_{k}(L)} ψ_{V} ∈ ⨁_{V ∈Irr_{k}(L)} F (Q) ↦ ∑_{V ∈Irr_{k}(L)} F (\overrightarrow{V})(ψ_{V}) ∈ F (L × Q)

are isomorphisms of R-modules, inverse to each other.

2. If L is non-trivial, then the essential algebra E_{R}(L × Q) is zero.

Proof: 1. Indeed, setting H = L × Q as above, for irreducible kL-modules V and W, we have that

F (\overleftarrow{W}) ∘ F (\overrightarrow{V}) = F (\overleftarrow{W} ⊗_{kH} \overrightarrow{V}), and F (\overrightarrow{V}) ∘ F (\overleftarrow{V}) = F (\overrightarrow{V} ⊗_{kQ} \overleftarrow{V}),

so the result follows from Assertions 2 and 3 of Proposition 2.2.

2. Assertion 3 of Proposition 2.2 tells us that the identity bimodule kH is equal to a sum of morphisms which factor through Q, and |Q| < |H| if L is non-trivial. Assertion 2 of Corollary 2.3 follows.

Remark 2.4:

1. In short, Corollary 2.3 says that

F (L × Q) ≅ R_{k}(L) ⊗_{ℤ} F (Q).

2. The (non-)vanishing of E_{R}(H) for an arbitrary finite group H is studied in detail in Section 3 of [4], under additional conditions on R. The proof given here is much simpler and explicit for the case of a direct product H = L × Q of a p′-group L and a p-group Q, and doesn’t require any additional assumption on R.

We now consider a version of Brauer’s induction theorem relative to the prime p:

Theorem 2.5: Let G be a finite group. Then the cokernel of the induction map

⊕_{H} Ind_{H}^{G} : ⨁_{H} R_{k}(H) → R_{k}(G)

where H runs through Brauer p-elementary subgroups of G, i.e. subgroups of the form H = P × C, where P is a p-group and C is a cyclic p′-group, is finite and of order prime to p.

Proof: Let (K, O, k) be a p-modular system, such that K is big enough for G. We have a commutative diagram

\xymatrix{⨁_{H} R_{K}(H) \ar[r]^{^{⊕_{H} Ind^{G}^{H}}} \ar[d]_{⊕_{H} d_{H}} & R_{K}(G) \ar[d]^{d_{G}} \\ ⨁_{H} R_{k}(H) \ar[r]_{_{⊕_{H} Ind_{G}_{H}}} & R_{k}(G)}

where the vertical arrows d_{H} and d_{G} are the respective decomposition maps. Let C_{K} (resp. C_{k}) denote the cokernel of the top (resp. bottom) horizontal map. By Theorem 12.28 of [9], the group C_{K} is finite, and of order prime to p. By Theorem 16.33 of [9], the vertical arrows are surjective. So d_{G} induces a surjective group homomorphism C_{K} ↠ C_{k}, hence C_{k} is finite and of order prime to p.

Corollary 2.6: Let G be a finite group. Then

kR_{k}(G) = ∑_{H} Ind_{H}^{G} kR_{k}(H),

where H runs through the Brauer p-elementary subgroups of G.

Corollary 2.7: Let F be a subfunctor of kR_{k}, and G be a finite group. Then F (G) is an ideal of the algebra kR_{k}(G).

Proof: By Corollary 2.6, there is a set S of Brauer elementary subgroups of G and elements w_{H} ∈ kR_{k}(H), for H ∈ S, such that

k_{G} = ∑_{H∈S} Ind_{H}^{G}w_{H}.

Now let u ∈ F (G). Then in kR_{k}(G), we have

u = k_{G} · u = ∑_{H∈S} Ind_{H}^{G}(w_{H} · Res_{H}^{G}u).

Since F is a subfunctor of kR_{k}, we know that Res_{H}^{G}F (G) ⊆ F (H) and Ind_{H}^{G}F (H) ⊆ F (G). So it is enough to prove that F (H) is an ideal of kR_{k}(H), for each H in S.

Now each H ∈ S is in particular of the form H = L × Q, where L is a (cyclic) p′-group, and Q is a p-group. In particular R_{k}(H) = Inf_{L}^{L×Q}R_{k}(L). Let V and W be kL-modules, and let Ṽ denote the (kL, kL)-bimodule Ind_{Δ(L)}^{L×L}V. Then Ṽ ⊗_{k} kQ is a (kH, kH)-diagonal p-permutation bimodule.

Moreover, one can check easily that Ṽ ⊗_{kL} W ≅ V ⊗_{k} W as kL-modules, and it follows that there is an isomorphism of kH-modules

(Ṽ ⊗_{k} kQ) ⊗_{k(L×Q)} Inf_{L}^{L×Q}W ≅ Inf_{L}^{L×Q}(V ⊗_{k} W).

Written differently, this reads

kR_{k}(Ṽ ⊗_{k} kQ)(Inf_{L}^{L×Q}W ) = Inf_{L}^{L×Q}(V ⊗_{k} W).

In the algebra kR_{k}(H) = Inf_{L}^{L×Q}kR_{k}(L), the right hand side is nothing but the product V ·W, and the left hand side is given by the action of the diagonal p-permutation bimodule Ṽ ⊗_{k} kQ on Inf_{L}^{L×Q}W. It follows more generally that if φ ∈ F (H), then the product V · φ is obtained from φ by applying Ṽ ⊗_{k} kQ ∈ T^{Δ}(H, H), so V · φ ∈ F (H). Thus kR_{k}(H) · F (H) ⊆ F (H), as was to be shown.

Notation 2.8: For finite groups G and H, we write H ↪ G if H is isomorphic to a subgroup of G.

Lemma 2.9: Let P and Q be finite p-groups. Then

⟨k_{P} ⟩(Q) = \{ \begin{matrix}k & if Q ↪ P \\ 0 & otherwise. \end{matrix}

In particular ⟨k_{Q}⟩ ≤ ⟨k_{P} ⟩ if and only if Q ↪ P.

Proof: By definition of ⟨k_{P} ⟩, we have ⟨k_{P} ⟩(Q) = kT^{Δ}(Q, P)(k_{P}). Moreover since P and Q are p-groups, the group T^{Δ}(Q, P) is equal to the Burnside group B^{Δ}(Q, P) of left-right free (Q, P)-bisets, so kT^{Δ}(Q, P) = kB^{Δ}(Q, P). Now if X is a diagonal subgroup of Q × P, the set of (right-)orbits of P on the (Q, P)-biset (Q × P)/X is a Q-set isomorphic to Q/p_{1}(X), where p_{1}(X) is the first projection of X, so we have

k((Q × P)/X) ⊗_{kP} k ≅ k(Q/p_{1}(X)).

As an element of R_{k}(Q), this is equal to |Q : p_{1}(X)|k_{Q}, so it is equal to 0 in kR_{k}(Q) unless p_{1}(X) = Q, and then it is equal to k_{Q}. But saying that there exists a diagonal subgroup X ≤ Q × P such that p_{1}(X) = Q amounts to saying that Q is isomorphic to a subgroup of P.

Since ⟨k_{P} ⟩ is a subfunctor of kR_{k}, and since kR_{k}(Q) is one dimensional, generated by k_{Q}, it follows that ⟨k_{P} ⟩(Q) is non-zero if and only if Q is isomorphic to a subgroup of P, and this occurs if and only if k_{Q} ∈ ⟨k_{P} ⟩(Q), that is ⟨k_{Q}⟩ ≤ ⟨k_{P} ⟩. This completes the proof.

3 The subfunctors ⟨k_{P} ⟩

The following result is a major step in the proof of Theorem 1.3:

Theorem 3.1: Let F be a subfunctor of kR_{k}. Then for any finite group G

F (G) = ∑_{\begin{matrix}P ∈P, P ≤G \\ k_{P} ∈F (P) \end{matrix}}⟨k_{P} ⟩(G).

Proof: Let G be a finite group. Saying that k_{P} ∈ F (P) amounts to saying that ⟨k_{P} ⟩ ≤ F, so ∑_{\begin{matrix}P ∈P,P ≤G \\ k_{P} ∈F (P) \end{matrix}}⟨k_{P} ⟩(G) ≤ F (G). We have to prove the reverse

inclusion.

By Theorem 2.6, there is a set S of Brauer p-elementary subgroups of G, and elements w_{H} ∈ kR_{k}(H), for H ∈ S, such that

k_{G} = ∑_{H∈S} Ind_{H}^{G}w_{H}

in kR_{k}(G). Let u ∈ F (G). Then

u = k_{G} ⊗_{k} u = ∑_{H∈S} Ind_{H}^{G}(w_{H} ⊗_{k} Res_{H}^{G}u).  (3.2)

Now Res_{H}^{G}u ∈ F (H), for each H ∈ S, since F is a subfunctor of kR_{k}. Then w_{H} ⊗_{k} Res_{H}^{G}u ∈ F (H) also, by Lemma 2.7, since each H ∈ S is a product L_{H} × Q_{H}, where L_{H} is a (cyclic) p′-group and Q_{H} is a p-group. It follows that u ∈ ∑_{H∈S} Ind_{H}^{G}F (H) for any u ∈ F (G), so

F (G) = ∑_{H∈S} Ind_{H}^{G}F (H).

Now by Proposition 2.2, for each H ∈ S, the identity (kH, kH)-bimodule kH splits as

kH ≅ ⊕_{V ∈Irr_{k}(L_{H})} (\overrightarrow{V} ⊗_{kQ_{H}} \overleftarrow{V}).

It follows that

F (H) = ∑_{V ∈Irr_{k}(L_{H})} F (\overrightarrow{V})(F (Q_{H})).

Now F (Q_{H}) ⊆ kR_{k}(Q_{H}) = k, since Q_{H} is a p-group. So F (Q_{H}) is either zero or k. It is non zero exactly when k_{Q_{H}} ∈ F (Q_{H}). So F (H) = 0 or

F (H) = ∑_{V ∈Irr_{k}(L_{H})} F (\overrightarrow{V})(F (Q_{H})) = ⟨k_{Q_{H}} ⟩(H),

and F (H) is non-zero. It follows that

F (G) = ∑_{\begin{matrix}H∈S \\ k_{Q_{H}} ∈F (Q_{H}) \end{matrix}} Ind_{H}^{G}⟨k_{Q_{H}} ⟩(H) ⊆ ∑_{\begin{matrix}P ∈P, P ≤G \\ k_{P} ∈F (P) \end{matrix}}⟨k_{P} ⟩(G),

as was to be shown.

Corollary 3.3: Let F be a subfunctor of kR_{k}. Then

F = ∑_{\begin{matrix}P ∈P \\ k_{P} ∈F (P) \end{matrix}}⟨k_{P} ⟩.

Corollary 3.4: Let G be a finite group, and P be a Sylow p-subgroup of G. Then:

1. kR_{k}(G) = ⟨k_{P} ⟩(G).

2. ⟨k_{G}⟩ = ⟨k_{P} ⟩.

Proof: 1. Indeed kR_{k}(G) = ∑_{Q∈P, Q≤G}⟨k_{Q}⟩(G), by Theorem 3.1 applied to

the subfunctor F = kR_{k}. But if Q ∈ P and Q ≤ G, then Q ↪ P, hence ⟨k_{Q}⟩ ≤ ⟨k_{P} ⟩ by Lemma 2.9. Hence ⟨k_{Q}⟩(G) ≤ ⟨k_{P} ⟩(G), and kR_{k}(G) = ⟨k_{P} ⟩(G).

2. Since k_{G} ∈ ⟨k_{G}⟩(G), we have k_{G} ∈ ⟨k_{P} ⟩(G) by Assertion 1, that is ⟨k_{G}⟩ ≤ ⟨k_{P} ⟩. Conversely k_{P} = Res_{P}^{G}k_{G}, so k_{P} ∈ ⟨k_{G}⟩(P), i.e. ⟨k_{P} ⟩ ≤ ⟨k_{G}⟩. Hence ⟨k_{G}⟩ = ⟨k_{P} ⟩.

4 Proof of Theorem 1.3

We first recall the statement:

Theorem: Let L denote the lattice of subfunctors of kR_{k}, ordered by inclusion of subfunctors. Let F denote the lattice of closed subsets of P, ordered by inclusion of subsets. Then the maps

Ψ : F ∈ L ↦ Ψ(F) = {[P] ∈ P | k_{P} ∈ F (P)} ∈ F, and
Θ : C ∈ F ↦ Θ(C) = ∑_{[P]∈C}⟨k_{P} ⟩ ∈ L,

are well defined isomorphisms of lattices, inverse to each other.

Proof: We first check that Ψ is well defined, i.e. that Ψ(F) is a closed subset of P, for any F ∈ L. This follows from Lemma 2.9: Saying that P ∈ Ψ(F) amounts to saying that ⟨k_{P} ⟩ ≤ F. Now if Q ↪ P ∈ Ψ(F), Lemma 2.9 shows that ⟨k_{Q}⟩ ≤ ⟨k_{P} ⟩ ≤ F, so Q ∈ Ψ(F), as was to be shown.

The maps Ψ and Θ are clearly maps of posets. Moreover Corollary 3.3 shows that Θ ∘ Ψ(F) = F, for any F ∈ L. Conversely, let C be a closed subset of P. Then

Ψ ∘ Θ(C) = {[P] ∈ P | k_{P} ∈ ∑_{Q∈C}⟨k_{Q}⟩(P)}.

So clearly C ⊆ Ψ ∘ Θ(C), since k_{Q} ∈ ⟨k_{Q}⟩(Q). Conversely, if [P] ∈ Ψ ∘ Θ(C), that is if k_{P} belongs to ∑_{Q∈C}⟨k_{Q}⟩(P), there is some Q ∈ C such that ⟨k_{Q}⟩(P)

is non zero. Then P ↪ Q, by Lemma 2.9, so P ∈ C since C is closed. So Ψ ∘ Θ(C) = C, which completes the proof of Theorem 1.3.

Recall (see [8]) that a lattice (L, ≤, ∨, ∧) is called complete if any subset A of L admits a supremum (denoted as the join ∨_{a∈A} a) and an infimum (denoted

as the meet ∧_{a∈A} a). A complete lattice is called completely distributive if arbitrary joins distribute over arbitrary meets. An element l of of a lattice L is called join irreducible if for any finite subset A of L, the equality l = ∨_{a∈A} a implies l ∈ A, and completely join irreducible if L is complete and the same condition holds for arbitrary subsets A of L. An element l of a lattice L is called join prime if for any finite subset A of L, the inequality l ≤ ∨_{a∈A} a implies l ≤ a for some a ∈ A, and completely join prime if L is complete and the same condition holds for arbitrary subsets A of L.

Corollary 4.1:

1. The lattice L is completely distributive.

2. Let F be a subfunctor of kR_{k}. The following are equivalent in the lattice L:

(a) F is completely join irreducible.

(b) F is completely join prime.

(c) There is a finite p-group P such that F = ⟨k_{P} ⟩.

Moreover in (c), the p-group P is unique up to isomorphism.

Proof: 1. Indeed, the lattice L is isomorphic to F, which is clearly completely distributive: its join is union of closed subsets, and its meet is intersection. Now if C and (C_{i})_{i∈I} are closed subsets of P, then C ∩ ∪_{i∈I} C_{i} =

∪_{i∈I}(C ∩ C_{i}).

2. Clearly (b) implies (a). Moreover (c) implies (b): if P is a p-group, then by Lemma 2.9

Ψ(⟨k_{P} ⟩) = {Q ∈ P | Q ↪ P }.

Hence if (C_{i})_{i∈I} is a set of closed subsets of P, then Ψ(⟨k_{P} ⟩) ⊆ ∪_{i∈I} C_{i} if and

only if there exists some i ∈ I such that P ∈ C_{i}, i.e. Ψ(⟨k_{P} ⟩) ⊆ C_{i}. So Ψ(⟨k_{P} ⟩) is completely join prime in F, and ⟨k_{P} ⟩ is completely join prime in L.

Finally (a) implies (c): if F is completely join irreducible in L, since F = ∑_{k_{P} ∈F (P)}⟨k_{P} ⟩, there is a p-group P such that F = ⟨k_{P} ⟩. Moreover P is

unique up to isomorphism, by Lemma 2.9.

Remark 4.2: There are join irreducible elements of L, or equivalently of F, which are not completely join irreducible: Let for example C be the subset of P consisting of cyclic p-groups. Then C is closed, and one checks easily that C is join irreducible in F, but not completely join irreducible.

5 Some simple diagonal p-permutation functors

Proposition 5.1: Let P be a finite p-group. Then ⟨k_{P} ⟩ has a unique (proper) maximal subfunctor J_{P}, defined for a finite group G by

J_{P} (G) = {u ∈ ⟨k_{P} ⟩(G) | kT^{Δ}(P, G)(u) = 0}.

Moreover J_{P} = ∑_{\begin{matrix}Q↪P \\ Q≇P \end{matrix}}⟨k_{Q}⟩.

Proof: First it is easy to check that the assignment G ↦ J_{P} (G) defines a subfunctor of kR_{k}, hence of ⟨k_{P} ⟩. Moreover J_{P} (P) = 0. Now Lemma 2.9 implies that ⟨k_{P} ⟩(P) = k, so J_{P} is a proper subfunctor of ⟨k_{P} ⟩. If F is a subfunctor of ⟨k_{P} ⟩, there are two possibilities: Either F (P) = k, and then k_{P} ∈ F (P), so F = ⟨k_{P} ⟩. Or F (P) = 0, and then for any finite group G, we have kT^{Δ}(P, G)(F (G)) ≤ F (P) = 0, that is F ≤ J_{P}.

For the last assertion, denote by ]., P] the subset of P consisting of pgroups isomorphic to a subgroup of P, and by ]., P [ the subset of ]., P] consisting of p-groups isomorphic to a proper subgroup of P. Then clearly ]., P] ∈ F, and Θ(]., P]) = ⟨k_{P} ⟩, by Lemma 2.9. Also ]., P [∈ F, and Θ(]., P [) = ∑_{\begin{matrix}Q↪P \\ Q≇P \end{matrix}}⟨k_{Q}⟩, by definition of Θ. Now]., P [ is clearly the unique

maximal proper closed subset of ]., P], so Θ(]., P [) = J_{P}, by Theorem 1.3. This completes the proof.

Notation 5.2: For a finite p-group P, we denote by

S_{P} = ⟨k_{P} ⟩/J_{P}

the unique simple quotient of ⟨k_{P} ⟩.

Lemma 5.3: Let G be a finite group, and P be a finite p-group.

1. If S_{P} (G) ≠ 0, then P ↪ G.

2. If Q is a finite p-group, then

S_{P} (Q) = \{ \begin{matrix}k & if Q ≅ P \\ 0 & otherwise. \end{matrix}

In particular S_{P} ≅ S_{Q} if and only if P ≅ Q.

Proof: 1. By Theorem 3.1, we have

⟨k_{P} ⟩(G) = ∑_{\begin{matrix}Q∈P, Q≤G \\ k_{Q}∈⟨k_{P} ⟩(Q) \end{matrix}}⟨k_{Q}⟩(G).

Now k_{Q} ∈ ⟨k_{P} ⟩(Q) if and only if ⟨k_{Q}⟩ ≤ ⟨k_{P} ⟩, i.e. Q ↪ P, by Lemma 2.9. So if P is not isomorphic to a subgroup of G, then Q ≤ G and Q ↪ P implies that Q is isomorphic to a proper subgroup of P, and then ⟨k_{Q}⟩(G) ≤ J_{P} (G), by Proposition 5.1. It follows that ⟨k_{P} ⟩(G) ≤ J_{P} (G), so S_{P} (G) = 0.

2. Indeed, if S_{P} (Q) ≠ 0, then in particular ⟨k_{P} ⟩(Q) ≠ 0, so Q ↪ P by Lemma 2.9. But also P ↪ Q by Assertion 1. So S_{P} (Q) = 0 unless Q ≅ P. Moreover S_{P} (P) = ⟨k_{P} ⟩(P)/J_{P} (P) = k/{0} ≅ k, which completes the proof.

Theorem 5.4:

1. Let F_{2} < F_{1} be subfunctors of kR_{k} such that F_{1}/F_{2} is a simple functor. Then there exists a (unique, up to isomorphism) finite p-group P such that F_{1}/F_{2} ≅ S_{P}.

2. There exists a filtration

0 = F_{0} < F_{1} < … < F_{n} < F_{n+1} < …

of kR_{k} by subfunctors F_{i}, for i ∈ ℕ, such that:

(a) ⋃_{i=0}^{∞} F_{i} = kR_{k}.

(b) For i > 0, the functor F_{i}/F_{i−1} is simple, isomorphic to S_{P_{i}}, for a finite p-group P_{i}.

(c) For every finite p-group P, there is exactly one integer i > 0 such that P_{i} ≅ P.

Proof: 1. Set C_{i} = Ψ(F_{i}), for i ∈ {1, 2}. Then C_{1} and C_{2} are closed subsets of P, and C_{2} ⊂ C_{1}. Since F_{1}/F_{2} is simple, any subfunctor F of kR_{k} such that F_{2} ≤ F ≤ F_{1} is equal either to F_{1} or F_{2}. Then any closed subset C of P such that C_{2} ⊆ C ⊆ C_{1} is equal either to C_{1} or C_{2}. If P ∈ C_{1} − C_{2}, then C_{2}∪]., P] is closed, different from C_{2}, and contained in C_{1}. So C_{2}∪]., P] = C_{1}. Now if P′ ∈ C_{1} − C_{2}, then P′ ∈]., P], and P ∈]., P′], by exchanging the roles of P and P′. It follows that P ≅ P′, so C_{1} − C_{2} = {[P]}. Now

F_{1} = Θ(C_{1}) = Θ(C_{2}) + ⟨k_{P} ⟩ = F_{2} + ⟨k_{P} ⟩.

It follows that F_{1}/F_{2} ≅ ⟨k_{P} ⟩/(⟨k_{P} ⟩ ∩ F_{2}) is a simple quotient of ⟨k_{P} ⟩, so F_{1}/F_{2} ≅ S_{P}. The uniqueness of P (up to isomorphism) with this property follows from Lemma 5.3.

2. Choose an enumeration P_{1}, P_{2}, …, P_{n}, … of P such that for any indices i and j, P_{i} ↪ P_{j} implies i ≤ j. This can be achieved starting with P_{1} = 1, P_{2} = C_{p}, and then enumerating all the p-groups of order p^{2}, then the groups of order p^{3}, and so on. With such a numbering, set C_{0} = ∅ and C_{n} = ∪_{i≤n}]., P_{i}] for n > 0, and then set F_{n} = Θ(C_{n}) for n ∈ ℕ. In particular F_{0} = 0.

Since C_{n} = C_{n−1}∪]., P_{n}] for n > 0, the sequence (C_{n})_{n∈ℕ} is increasing. Moreover P_{n} ∈ C_{n} − C_{n−1}, for otherwise P_{n} ∈]., P_{i}] for some i < n, meaning that P_{n} ↪ P_{i}, which implies n ≤ i, a contradiction. So the sequence (C_{n})_{n∈ℕ} is stricly increasing, and its union is the whole of P. In other words, the sequence (F_{n})_{n∈ℕ} is stricly increasing and its union is equal to kR_{k}, which proves (a).

Since C_{i} = C_{i−1}∪]., P_{i}] for i > 0, and since ]., P_{i}[⊆ C_{i−1} by our choice of the numbering of P, it follows that C_{i} = C_{i−1} ∪{P_{i}}, and then F_{i}/F_{i−1} ≅ S_{P_{i}} as in the proof of Assertion 1. This proves (b). Now (c) is clear, since any finite p-group P appears exactly once in our enumeration.

Remark 5.5: Theorem 5.4 shows that kR_{k} admits a “composition series”, where the “composition factors” are the simple functors S_{P}, for P ∈ P, and each simple functor S_{P} appears exactly once. The quote signs in the previous sentence indicate that one should be careful with the notions of composition factors and composition series for diagonal p-permutation functors.

Remark 5.6: Proposition 5.1 and Lemma 5.3 show that P is a minimal group for S_{P}: More precisely S_{P} (P) is one dimensional, and S_{P} (G) = {0} for any group G of order smaller than |P |. Moreover P is unique (up to isomorphism) with this property. In addition S_{P} (P) is generated by the image of the trivial module k_{P}, and in particular the group Out(P) of outer automorphisms of P acts trivially on S_{P} (P). These two facts show that with the notation of Theorem 5.25 of [4], the functor S_{P} is isomorphic to the simple functor S_{P,1,k}.

6 The simple functor S_{1}

We consider first the case where the p-group P is trivial^{2}. In this case S_{P} = S_{1} = ⟨k_{1}⟩ ≤ kR_{k}, so for a finite group G

S_{1}(G) = kT^{Δ}(G, 1)(k_{1}).

Now T^{Δ}(G, 1) is the group P_{k}(G) of projective kG-modules, and we set set kP_{k}(G) = k ⊗_{ℤ} P_{k}(G). So S_{1}(G) = ⟨k_{1}⟩(G) is equal to the image of the map kc^{G} : kP_{k}(G) → kR_{k}(G) linearly extending the Cartan map c^{G} : P_{k}(G) → R_{k}(G) sending a projective kG-module to the sum of its composition factors.

Remark 6.1: The functor S_{1} is the only simple subfunctor of kR_{k}: indeed, any non-empty closed subset of P must contain the trivial group, so any non-zero subfunctor of kR_{k} must contain ⟨k_{1}⟩ = S_{1}.

As there are non-empty closed subsets of P different from {1}, this shows that S_{1} is a proper subfunctor of kR_{k}, so kR_{k} is not a semisimple diagonal p-permutation functor. It follows that in contrast to [3], the category of diagonal p-permutation functors over a field k of characteristic p is not semisimple.

2The content of the present section 6 is essentially the same as Subsection 6.5 of [4].

We include it here for the reader’s convenience.

We choose a p-modular system (K, O, k), and we assume that K is big enough for the group G. If S is a simple kG-module, we denote by P_{S} its projective cover as kG-module, and by Φ_{S} : G_{p′} → O the modular character of P_{S}, where G_{p′} is the set of p-regular elements of G. If v = ∑_{S∈Irr_{k}(G)} ω_{S}P_{S},

where ω_{S} ∈ O, is an element of OP_{k}(G) := O ⊗_{ℤ} P_{k}(G), we denote by Φ^{v} the map ∑_{S∈Irr_{k}(G)} ω_{S}Φ_{S} from G_{p′} to O, and we call Φ^{v} the modular character

of v.

Then for a simple kG-module T, the multiplicity of T as a composition factor of P_{S} is equal to the Cartan coefficient

c_{T,S}^{G} = dim_{k} Hom_{kG}(P_{T} , P_{S}) = \frac{1}{|G|} ∑_{x∈G_{p′}} Φ_{T} (x)Φ_{S}(x^{−1}).

In order to describe the image of the map kc^{G}, we want to evaluate the image of this integer under the projection map ρ : O → k. For this, we denote by [G_{p′}] a set of representatives of conjugacy classes of G_{p′}, and we observe that in the field K, we have

c_{T,S}^{G} = \frac{1}{|G|} ∑_{x∈[G_{p′}]} \frac{|G|}{|C_{G}(x)|}Φ_{T} (x)Φ_{S}(x^{−1})

= ∑_{x∈[G_{p′}]} \frac{1}{|C_{G}(x)|} \frac{Φ_{T} (x)}{|C_{G}(x)|_{p}} \frac{Φ_{S}(x^{−1})}{|C_{G}(x)|_{p}}|C_{G}(x)|_{p}^{2}

= ∑_{x∈[G_{p′}]} \frac{(Φ_{T} (x)/|C_{G}(x)|_{p})(Φ_{S}(x^{−1})/|C_{G}(x)|_{p})}{|C_{G}(x)_{p′}|}|C_{G}(x)|_{p}.  (6.2)

But since Φ_{S} and Φ_{T} are modular characters of projective kG-modules, and since C_{G}(x) = C_{G}(x^{−1}), the quotients Φ_{T} (x)/|C_{G}(x)|_{p} and Φ_{S}(x^{−1})/|C_{G}(x)|_{p} are in O, so

∀x ∈ [G_{p′}], \frac{(Φ_{T} (x)/|C_{G}(x)|_{p})(Φ_{S}(x^{−1})/|C_{G}(x)|_{p})}{|C_{G}(x)_{p′}|} ∈ O.

Now it follows from 6.2 that

ρ(c_{T,S}^{G}) = ∑_{x∈[G_{0}]} ρ (\frac{Φ_{T} (x)Φ_{S}(x^{−1})}{|C_{G}(x)|}) ,  (6.3)

where [G_{0}] is a set of representatives of conjugacy classes of the set G_{0} of elements of defect zero of G, i.e. the set of p-regular elements x such that C_{G}(x) is a p′-group.

Notation 6.4: For x ∈ G_{0}, we set

Γ_{G, x} = ∑_{S∈Irr(kG)} \frac{Φ_{S}(x^{−1})}{|C_{G}(x)|}S ∈ OR_{k}(G),

where Irr(kG) is a set of representatives of isomorphism classes of simple kG-modules. We also set

γ_{G, x} = ∑_{S∈Irr(kG)} ρ (\frac{Φ_{S}(x^{−1})}{|C_{G}(x)|}) S ∈ kR_{k}(G),

Remark 6.5: We note that Γ_{G, x} and γ_{G, x} only depend on the conjugacy class of x in G, that is Γ_{G, x} = Γ_{G, x^{g}} and γ_{G, x} = γ_{G, x^{g}} for g ∈ G.

By Section III.16 of [5] (see also Theorem 3.6.32 of [7]), the elementary divisors of the Cartan matrix of G are equal to |C_{G}(x)|_{p}, for x ∈ [G_{p′}]. It follows that the rank of the Cartan matrix modulo p, is equal to the number of conjugacy classes of elements of defect 0 of G, i.e. the cardinality of [G_{0}]. The following can be viewed as an explicit form of this result:

Proposition 6.6:

1. Let T be a simple kG-module. Then, in kR_{k}(G),

kc^{G}(P_{T}) = ∑_{x∈[G_{0}]} ρ(Φ_{T} (x)) γ_{G, x}.

2. The elements γ_{G, x}, for x ∈ [G_{0}], form a basis of S_{1}(G) ≤ kR_{k}(G).

Proof: Throughout the proof, we simply write γ_{x} instead of γ_{G, x}.

1. By 6.3, we have

kc^{G}(P_{T}) = ∑_{S∈Irr(kG)} ρ(c_{T,S}^{G})S = ∑_{S∈Irr(kG)} ∑_{x∈[G_{0}]} ρ (\frac{Φ_{T} (x)Φ_{S}(x^{−1})}{|C_{G}(x)|}) S

= ∑_{x∈[G_{0}]} ∑_{S∈Irr(kG)} ρ (\frac{Φ_{T} (x)Φ_{S}(x^{−1})}{|C_{G}(x)|}) S

= ∑_{x∈[G_{0}]} ρ(Φ_{T} (x)) ∑_{S∈Irr(kG)} ρ (\frac{Φ_{S}(x^{−1})}{|C_{G}(x)|}) S

= ∑_{x∈[G_{0}]} ρ(Φ_{T} (x)) γ_{x}.

2. We first prove that γ_{x} lies in the image of kc^{G}, for any x ∈ G_{0}. So let x ∈ G_{0}, and 1_{x} : ⟨x⟩ → O be the map with value 1 at x and 0 elsewhere. Then |x|1_{x} = ∑_{ζ} ζ(x^{−1})ζ, where ζ runs through the modular characters of the simple k⟨x⟩-modules, i.e. the group homomorphisms ⟨x⟩ → O^{×}, is an element of OP_{k}(⟨x⟩) = OR_{k}(⟨x⟩). Let v_{x} = Ind_{⟨x⟩}^{G}(|x|1_{x}). Then v_{x} ∈ OP_{k}(G), and its modular character evaluated at g ∈ G is equal to

Φ^{v_{x}}(g) = \frac{1}{|x|} ∑_{\begin{matrix}h∈G \\ g^{h}∈⟨x⟩ \end{matrix}} Φ^{|x|1_{x}}(g^{h})

= \frac{1}{|x|} ∑_{\begin{matrix}h∈G \\ g^{h}=x \end{matrix}} |x| = \{ \begin{matrix}|C_{G}(x)| & if g =_{G} x \\ 0 & otherwise, \end{matrix}  (6.7)

where g =_{G} x means that g is conjugate to x in G. Now from Assertion 1, we get that

kc^{G}(v_{x}) = ∑_{y∈[G_{0}]} ρ(Φ^{v_{x}}(y))γ_{y} = |C_{G}(x)|γ_{x},  (6.8)

so γ_{x} is in the image of kc^{G}, since |C_{G}(x)| ≠ 0 in k.

Now by Assertion 1, the elements γ_{x}, for x ∈ [G_{0}], generate the image of kc^{G}, i.e. S_{1}(G). They are moreover linearly independent: Suppose indeed that some linear combination ∑_{x∈[G_{0}]} λ_{x}γ_{x}, where λ_{x} ∈ k, is equal to 0. For

all x ∈ [G_{0}], choose λ̃_{x} ∈ O such that ρ(λ̃_{x}) = λ_{x}. By (6.8), we get an element ∑_{x∈[G_{0}]} λ̃_{x} \frac{v_{x}}{|C_{G}(x)|} of OP_{k}(G) whose modular character has values in

the maximal ideal J(O) of O. But by (6.7), the value at g ∈ G_{p′} of this modular character is equal to

∑_{x∈[G_{0}]} λ̃_{x} \frac{Φ^{v_{x}}(g)}{|C_{G}(x)|},

which is equal to 0 if g ∉ G_{0}, and to λ̃_{x} if g is conjugate to x ∈ [G_{0}] in G. It follows that λ̃_{x} ∈ J(O), hence λ_{x} = ρ(λ̃_{x}) = 0. Since g ∈ G_{p′} was arbitrary, we get that λ_{x} = 0 for any x ∈ [G_{0}], so the elements γ_{x}, for x ∈ [G_{0}], are linearly independent. This completes the proof of Proposition 6.6.

7 The simple functors S_{P}

In this section, we generalize the results of Section 6 to the functor S_{P}, for an arbitrary finite p-group P.

Theorem 7.1: Let P be a finite p-group. Then for any finite group G, the evaluation ⟨k_{P} ⟩(G) is generated by the elements of kR_{k}(G) of the form

Ind_{RC_{G}(R)}^{G}Inf_{RC_{G}(R)/R}^{RC_{G}(R)}kc^{RC_{G}(R)/R}(F),

where R is a subgroup of G such that R ↪ P, and F is an indecomposable projective kRC_{G}(R)/R-module.

Proof: By definition ⟨k_{P} ⟩(G) = kT^{Δ}(G, P)(k_{P}), and T^{Δ}(G, P) is generated by the bimodules of the form Ind_{N_{R,π,Q}}^{G×P}Inf_{\overline{N}_{R,π,Q}}^{N_{R,π,Q}}E, where:

• N_{R,π,Q} is the normalizer in G×P of a diagonal p-subgroup Δ(R, π, Q) = {(π(q), q) | q ∈ Q}, where π : Q → R is a group isomorphism from a subgroup Q of P to a subgroup R of G.

• \overline{N}_{R,π,Q} = N_{R,π,Q}/Δ(R, π, Q).

• E is an indecomposable projective k\overline{N}_{R,π,Q}-module.

Then N_{R,π,Q} = {(a, b) ∈ G × P | ∀q ∈ Q, ^{a}π(q) = π(^{b}q)}. Let Q̂ ≤ P denote the second projection of N_{R,π,Q}. Then \overline{N}_{R,π,Q} fits in a short exact sequence

\xymatrix{1 \ar[r] & C_{G}(R) \ar[r]^{i} & \overline{N}_{R,π,Q} \ar[r]^{s} & Q̂/Q \ar[r] & 1,}

where i is the map x ↦ (x, 1)Δ(R, π, Q) from C_{G}(R) to \overline{N}_{R,π,Q}, and s maps (a, b)Δ(R, π, Q) ∈ \overline{N}_{R,π,Q} to bQ ∈ Q̂/Q. Since Q̂/Q is a p-group and k is algebraically closed, it follows from Corollary 5.12.4 of [6] that there exists an indecomposable projective kC_{G}(R)-module M such that

E ≅ Ind_{C_{G}(R)}^{\overline{N}_{R,π,Q}}M.

Then

Ind_{N_{R,π,Q}}^{G×P}Inf_{\overline{N}_{R,π,Q}}^{N_{R,π,Q}}E ≅ Ind_{N_{R,π,Q}}^{G×P}Inf_{\overline{N}_{R,π,Q}}^{N_{R,π,Q}}Ind_{C_{G}(R)}^{\overline{N}_{R,π,Q}}M
≅ Ind_{N_{R,π,Q}}^{G×P}Ind_{Δ(R,π,Q)(C_{G}(R)×1)}^{N_{R,π,Q}}Inf_{C_{G}(R)}^{Δ(R,π,Q)(C_{G}(R)×1)}M
≅ Ind_{Δ(R,π,Q)(C_{G}(R)×1)}^{G×P}Inf_{C_{G}(R)}^{Δ(R,π,Q)(C_{G}(R)×1)}M.

Now the tensor product T := (Ind_{N_{R,π,Q}}^{G×P}Inf_{\overline{N}_{R,π,Q}}^{N_{R,π,Q}}E) ⊗_{kP} k_{P} can be viewed as

(Ind_{X}^{G×P} Inf_{C_{G}(R)}^{X}M) ⊗_{kP} (Ind_{Y}^{P ×1}k_{P} ) ,

where X = Δ(R, π, Q)(C_{G}(R) × 1) ≤ G × P and Y = P × 1 ≤ P × 1. It follows from [1] that this tensor product is isomorphic to

T ≅ ⨁_{x∈p_{2}(X)\P/p_{1}(Y)} Ind_{X∗^{(x,1)}Y}^{G×1} ((Inf_{C_{G}(R)}^{X}M) ⊗_{k[k_{2}(X)∩^{x}k_{1}(Y)]} ^{x}k_{P} ), (7.2)

where p_{2}(X) is the second projection of X, and p_{1}(Y) the first projection of Y = P × 1. In particular p_{1}(Y) = P, so p_{2}(X)\P/p_{1}(Y) consists of a single double coset, and we can assume x = 1 in (7.2).

Moreover k_{1}(Y) = {y ∈ P | (y, 1) ∈ Y } = P, and

k_{2}(X) = {y ∈ P | (1, y) ∈ X} = {q ∈ Q | π(q) ∈ C_{G}(R)}
= π^{−1}(R ∩ C_{G}(R)) = Z(Q) ≅ Z(R).

Now X ∗ Y = {(a, b) ∈ G × 1 | ∃c ∈ P, (a, c) ∈ X, (x, 1) ∈ Y } is equal to the first projection of X, that is RC_{G}(R). We get finally

T ≅ Ind_{RC_{G}(R)×1}^{G×1} ((Inf_{C_{G}(R)}^{Δ(R,π,Q)(C_{G}(R)×1)}M) ⊗_{kZ(Q)} k_{P} )
≅ Ind_{RC_{G}(R)}^{G}Inf_{C_{G}(R)/Z(R)}^{RC_{G}(R)}M_{Z(R)},

where M_{Z(R)} = M ⊗_{kZ(R)} k, viewed as a kC_{G}(R)/Z(R)-module. The inflation symbol Inf_{C_{G}(R)/Z(R)}^{RC_{G}(R)} stands more precisely for Inf_{RG_{G}(R)/R}^{RC_{G}(R)}Iso_{C_{G}(R)/Z(R)}^{RC_{G}(R)/R}.

To complete the proof of Theorem 7.1, it remains to observe that the construction M ↦ M_{Z(R)} induces a bijection between the set of isomorphism classes of projective indecomposable kC_{G}(R)-modules and the set of isomorphism classes of projective indecomposable kC_{G}(R)/Z(R)-modules. Transporting this module via the isomorphism C_{G}(R)/Z(R) → RC_{G}(R)/R gives the indecomposable projective kRC_{G}(R)/R-module F := Iso_{C_{G}(R)/Z(R)}^{RC_{G}(R)/R}M_{Z(R)}. Moreover F has to be viewed as an element of kR_{k}(RC_{G}(R)/R), that is as kc^{RC_{G}(R)/R}(F). This completes the proof of Theorem 7.1.

Definition 7.3: Let G be a finite group, and x be a p-regular element of G.

1. Let P be a p-subgroup of G. Then x is said to have defect group P in G if P is conjugate in G to a Sylow p-subgroup of C_{G}(x). A conjugacy class of p-regular elements of G is said to have defect group P if some of its elements (or equivalently all of its elements) have defect group P.

2. Let P be a finite p-group. Then x is said to have defect group isomorphic to P if P is isomorphic to a Sylow p-subgroup of C_{G}(x). A conjugacy class of p-regular elements of G is said to have defect group isomorphic to P if some of its elements (or equivalently all of its elements) have defect group isomorphic to P.

Remark 7.4: In both cases, if |P | = p^{d}, recall that x (or its conjugacy class in G) is said to have defect d in G.

Lemma 7.5: Let G be a finite group, and R be a p-subgroup of G.

1. Let xR ∈ (RC_{G}(R)/R)_{p′}. Then xR has defect 0 in RC_{G}(R)/R if and only if there exists an element y ∈ C_{G}(R)_{p′} ∩ xR with defect group R in RC_{G}(R), i.e. such that RC_{G}(R, y)/R is a p′-group.

2. Let y ∈ C_{G}(R)_{p′} such that yR ∈ (RC_{G}(R)/R)_{0}. Then

Ind_{RC_{G}(R)}^{G}Inf_{RC_{G}(R)/R}^{RC_{G}(R)}γ_{RC_{G}(R)/R, yR} = 0 ∈ kR_{k}(G)

unless y has defect group R in G.

Proof: 1. It is well known that since R is a p-group, a p′ element xR of RC_{G}(R)/R can be lifted to a p’-element y ∈ xR of RC_{G}(R). Moreover (RC_{G}(R))_{p′} = C_{G}(R)_{p′}, so y ∈ C_{G}(R)_{p′}, and the centralizer H of yR = xR in RC_{G}(R)/R is the image in RC_{G}(R)/R of the centralizer of y in RC_{G}(R). Thus H = RC_{G}(R, y)/R, and H is a p′-group if and only if R is a Sylow subgroup of RC_{G}(R, y), that is if and only if y has defect group R in RC_{G}(R).

2. The normalizer N_{G}(R) normalizes RC_{G}(R), and it acts on the conjugacy classes of RC_{G}(R)/R by conjugation. Let T_{y} be the stabilizer in N_{G}(R) of the conjugacy class of yR in RC_{G}(R)/R. Then g ∈ T_{y} if and only if gy ∈ ^{c}(yR) = ^{c}yR for some c ∈ RC_{G}(R), that is^{g}y = ^{c}yr, for some element r of R. But y ∈ C_{G}(R), so ^{c}y ∈ C_{G}(^{c}R) = C_{G}(R), hence the p′-element ^{c}y commutes with the p-element r. Then r is the p-part of ^{c}yr = ^{g}y which is a p′-element. This forces r = 1, and ^{g}y = ^{c}y. Then g ∈ cC_{G}(y), hence g ∈ RC_{G}(R)C_{G}(y) ∩ N_{G}(R) = RC_{G}(R)N_{G}(R, y). Thus T_{y} ≤ RC_{G}(R)N_{G}(R, y). Conversely, it is clear that RC_{G}(R)N_{G}(R, y) stabilizes the conjugacy class of yR in RC_{G}(R)/R, so T_{y} = RC_{G}(R)N_{G}(R, y).

Let S be a Sylow p-subgroup of T_{y} containing R. For simplicity, we set W_{y} := Inf_{RC_{G}(R)/R}^{RC_{G}(R)}Γ_{RC_{G}(R)/R, yR} and w_{y} := Inf_{RC_{G}(R)/R}^{RC_{G}(R)}γ_{RC_{G}(R)/R, yR}. Then

Ind_{RC_{G}(R)}^{G}w_{y} = Ind_{SC_{G}(R)}^{G}Ind_{RC_{G}(R)}^{SC_{G}(R)}w_{y},

and we want to show that this is equal to zero unless y has defect group R in G. We will show a little more: We claim that Ind_{RC_{G}(R)}^{SC_{G}(R)}w_{y} = 0 in kR_{k}(SC_{G}(R)) if R is not a Sylow p-subgroup of C_{G}(y).

To prove this claim, we compute the coefficients of Ind_{RC_{G}(R)}^{SC_{G}(R)}w_{y} in the basis of kR_{k}(SC_{G}(R)) consisting of the simple k(SC_{G}(R))-modules. Since w_{y} is the reduction in k of W_{y}, we can use the modular character θ_{y} associated to W_{y} to do this computation. Let U be a simple k(SC_{G}(R))-module. Since RC_{G}(R) is a normal subgroup of SC_{G}(R) with p-power index, it follows from Corollary 5.12.4 of [6] that the projective cover of U is isomorphic to Ind_{RC_{G}(R)}^{SC_{G}(R)}E, where E is a projective kRC_{G}(R)-module. Let Φ^{E} denote the modular character of E. The coefficient of U in the expression of Ind_{RC_{G}(R)}^{SC_{G}(R)}w_{y} in the basis of simple kSC_{G}(R)-modules is then equal to ρ(m_{U}), where

m_{U} = ⟨Ind_{RC_{G}(R)}^{SC_{G}(R)}Φ^{E}, Ind_{RC_{G}(R)}^{SC_{G}(R)}θ_{y}⟩_{SC_{G}(R)}

= ⟨Φ^{E}, Res_{RC_{G}(R)}^{SC_{G}(R)}Ind_{RC_{G}(R)}^{SC_{G}(R)}θ_{y}⟩_{RC_{G}(R)}

= ∑_{g∈SC_{G}(R)/RC_{G}(R)}⟨Φ^{E},^{g}θ_{y}⟩_{RC_{G}(R)}

= |SC_{G}(R)/RC_{G}(R)|⟨Φ^{E}, θ_{y}⟩_{RC_{G}(R)},

since ^{g}θ_{y} = θ_{y} for g ∈ SC_{G}(R), as SC_{G}(R) ≤ T_{y}.

Then ρ(m_{U}) = 0 in k if SC_{G}(R) ≠ RC_{G}(R), or equivalently if p divides |T_{y} : RC_{G}(R)| = |N_{G}(R, y) : RC_{G}(R, y)|, that is, since RC_{G}(R, y)/R is a p′-group by Assertion 1, if p divides |N_{G}(R, y) : R|. Hence if ρ(m_{U}) ≠ 0, then R is a Sylow p-subgroup of N_{G}(R, y) = N_{C_{G}(y)}(R), hence R is a Sylow p-subgroup of C_{G}(y), which proves the claim. This completes the proof of Lemma 7.5.

Corollary 7.6: Let G be a finite group, and P be a finite p-group. Let [G_{↪P}] be a set of representatives of conjugacy classes of the set G_{↪P} of p-regular elements of G with defect group R ↪ P. For x ∈ G_{↪P}, let R_{x} be a chosen Sylow p-subgroup of C_{G}(x). Then the elements

U_{x} = Ind_{R_{x}C_{G}(R_{x})}^{G}Inf_{R_{x}C_{G}(R_{x})/R_{x}}^{R_{x}C_{G}(R_{x})}γ_{R_{x}C_{G}(R_{x})/R_{x}, xR_{x}}

for x ∈ [G_{↪P}], form a basis of ⟨k_{P} ⟩(G).

Proof: By Lemma 7.5, Theorem 7.1, and Proposition 6.6, the elements U_{x}, for x ∈ [G_{↪P}], generate ⟨k_{P} ⟩(G), and all we have to show is that these elements are linearly independent. Let S be a Sylow p-subgroup of G. If x ∈ [G_{↪P}] has defect group R ≤ G, then R ↪ S, so U_{x} ∈ ⟨k_{S}⟩(G). So it is enough to prove that the elements U_{x}, for x ∈ [G_{↪S}], are linearly independent. But ⟨k_{S}⟩(G) = kR_{k}(G) by Corollary 3.4, and |[G_{↪S}]| = |[G_{p′}]| as any element of G_{p′} has defect group R for some R ↪ S. Hence

|[G_{p′}]| = dim_{k} kR_{k}(G) = dim_{k}⟨k_{S}⟩(G) ≤ |[G_{↪S}]| = |[G_{p′}]|,

so dim_{k}⟨k_{S}⟩(G) = |[G_{↪S}]|, which completes the proof.

Remark 7.7: Observe that for x ∈ G_{↪P}, the element U_{x} does not depend on the choice of the Sylow p-subgroup R_{x}. Moreover U_{x} = U_{x^{g}}, for any g ∈ G. So the set {U_{x} | x ∈ [G_{↪P}]} is a canonical basis of ⟨k_{P} ⟩(G).

Theorem 7.8: Let G be a finite group, and P be a finite p-group. Let G_{≅P} denote the set of p-regular elements of G with defect group isomorphic to P, and [G_{≅P}] be a set of representatives of conjugacy classes of elements of G_{≅P}. Then the images of the elements U_{x}, for x ∈ [G_{≅P}], under the projection ⟨k_{P} ⟩(G) ↠ S_{P} (G), form a basis of S_{P} (G).

Proof: Let x ∈ G_{↪P}, and R = R_{x} be a Sylow subgroup of C_{G}(x). Then the element U_{x} defined in Corollary 7.6 lies in ⟨k_{R}⟩(G), so U_{x} is sent to 0 under the projection ⟨k_{P} ⟩(G) ↠ S_{P} (G) if R ≇ P, by Proposition 5.1. It follows that the images of the elements U_{x}, for x ∈ [G_{≅P}], under the projection ⟨k_{P} ⟩(G) ↠ S_{P} (G), generate S_{P} (G). It follows that dim_{k} S_{P} (G) ≤ |[G_{≅P}]|. In particular S_{P} (G) = 0 if P is not isomorphic to a subgroup of G (so Theorem 7.8 holds in this case).

Now, as in the proof of Theorem 5.4, we choose an enumeration P_{1}, P_{2}, … of P with the property that P_{i} ↪ P_{j} implies i ≤ j. This gives a filtration

0 = F_{0} < F_{1} < … < F_{n} < F_{n+1} < …

of kR_{k} by subfunctors F_{i}, for i ∈ ℕ, such that F_{i}/F_{i−1} ≅ S_{P_{i}} for i > 0.

Let n be the unique integer such that P_{n} is isomorphic to a Sylow psubgroup of G. If i > n, then S_{P_{i}}(G) = 0 by Lemma 5.3, since P_{i} is not isomorphic to a subgroup of G: Indeed, if it were, then P_{i} would be isomorphic to a subgroup of P_{n}, which would imply i ≤ n. It follows that

F_{n}(G) = F_{n+1}(G) = … = kR_{k}(G).

So we have a filtration of kR_{k}(G)

0 ≤ F_{1}(G) ≤ … ≤ F_{i−1}(G) ≤ F_{i}(G) ≤ … ≤ F_{n}(G) = kR_{k}(G).

Moreover, if the quotient F_{i}(G)/F_{i−1}(G) ≅ S_{P_{i}}(G) is non zero, then P_{i} ↪ G by Lemma 5.3, i.e. P_{i} ↪ P_{n}, so i ≤ n. Then

|[G_{p′}]| = dim_{k} kR_{k}(G) = ∑_{i=1}^{n} dim_{k} S_{P_{i}}(G)

= ∑_{R↪P_{n}} dim_{k} S_{R}(G) ≤ ∑_{R↪P_{n}} |[G_{≅R}]| = |[G_{p′}]|.

Hence all inequalities dim_{k} S_{R}(G) ≤ |[G_{≅R}]| are equalities, and the theorem follows.

Corollary 7.9: Let G be a finite group, and P be a finite p-group. Then the dimension of S_{P} (G) is equal to the number of conjugacy classes of p-regular elements of G with defect group isomorphic to P.

Remark 7.10: This corollary is consistent with Corollary 6.14 of [4], via Remark 5.6.

Acknowledgements : I wish to thank Deniz Yılmaz for his careful reading of a preliminary version of this work, and his pertinent remarks and suggestions. I also thank the anonymous referee for their very helpful comments.

References

[1] S. Bouc. Bisets as categories, and tensor product of induced bimodules. Applied Categorical Structures, 18(5):517–521, 2010.

[2] S. Bouc and D. Yılmaz. Diagonal p-permutation functors. J. Algebra, 556:1036–1056, 2020.

[3] S. Bouc and D. Yılmaz. Diagonal p-permutation functors, semisimplicity, and functorial equivalence of blocks. Adv. Math., 411(part A): Paper No. 108799, 54, 2022.

[4] S. Bouc and D. Yılmaz. Diagonal p-permutation functors in characteristic p. Preprint, arXiv:2411.05700, 2024.

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

[6] M. Linckelmann. The block theory of finite group algebras. Vol. I, volume 91 of London Mathematical Society Student Texts. Cambridge University Press, Cambridge, 2018.

[7] H. Nagao and Y. Tsushima. Representations of finite groups. Academic Press, Inc., Boston, MA, 1989. Translated from the Japanese.

[8] S. Roman. Lattices and ordered sets. Springer, New York, 2008.

[9] J.-P. Serre. Représentations linéaires des groupes finis. Collection Méthodes. Hermann, 1971.

Serge Bouc, CNRS-LAMFA, Université de Picardie, 33 rue St Leu, 80039, Amiens, France.

serge.bouc@u-picardie.fr