Diagonal p-permutation functors

Serge Bouc, Deniz Yılmaz^{1,∗}

CNRS-LAMFA Université de Picardie-Jules Verne, 33, rue St Leu, 80039, Amiens Cedex 01 - France

University of California, Santa Cruz Department of Mathematics CA 95064 USA

Abstract

Let k be an algebraically closed field of positive characteristic p, and 𝔽 be an algebraically closed field of characteristic 0. We consider the 𝔽-linear category 𝔽pp_{k}^{Δ} of finite groups, in which the set of morphisms from G to H is the 𝔽-linear extension 𝔽T^{Δ}(H, G) of the Grothendieck group T^{Δ}(H, G) of p-permutation (kH, kG)-bimodules with (twisted) diagonal vertices. The 𝔽-linear functors from 𝔽pp_{k}^{Δ} to 𝔽-Mod are called diagonal p-permutation functors. They form an abelian category F_{pp_{k}}^{Δ}.

We study in particular the functor 𝔽T^{Δ} sending a finite group G to the Grothendieck group 𝔽T (G) of p-permutation kG-modules, and show that 𝔽T^{Δ} is a semisimple object of F_{pp_{k}}^{Δ}, equal to the direct sum of specific simple functors parametrized by isomorphism classes of pairs (P, s) of a finite p-group P and a generator s of a p′-subgroup acting faithfully on P. This leads to a precise description of the evaluations of these simple functors. In particular, we show that the simple functor indexed by the trivial pair (1, 1) is isomorphic to the functor sending a finite group G to 𝔽K_{0}(kG), where K_{0}(kG) is the group of projective kG-modules.

Keywords: biset functors, p-permutation, twisted diagonal. 2010 MSC: 18B99, 20J15, 16W99

1. Introduction

Let p be a prime number. Throughout we denote by 𝔽 an algebraically closed field of characteristic zero, and by k an algebraically closed field of characteristic p. The p-permutation modules play a crucial role in the study of modular representation theory of finite groups. A splendid Rickard equivalence, introduced by Rickard [8], between blocks of finite group algebras is given by a chain complex consisting of p-permutation bimodules. Also a p-permutation equivalence, introduced by Boltje

∗The second author is supported by the Chateaubriand Fellowship of the Office of Science and Technology of the Embassy of France in the United States.

1The second author is thankful to LAMFA for their hospitality during the visit.

Preprint submitted to Journal of Algebra June 30, 2019

and Xu [1], and studied extensively later by Boltje and Perepelitsky [7], is an element in the Grothendieck group of p-permutation bimodules.

In [5], Ducellier studied p-permutation functors: Consider the category 𝔽pp_{k} where the objects are finite groups and the morphisms between groups G and H are given by the Grothendieck group 𝔽⊗_{ℤ}T (H, G) of p-permutation (kH, kG)-bimodules. A p-permutation functor is an 𝔽-linear functor from 𝔽pp_{k} to 𝔽-Mod. The indecomposable direct summands of the bimodules that appears in a p-permutation equivalence between blocks of finite group algebras have twisted diagonal vertices. Therefore, inspired by the work of Ducellier, we consider a category with less morphisms: Let 𝔽pp_{k}^{Δ} be a category where the objects are finite groups and the morphisms between groups G and H are given by the Grothendieck group 𝔽⊗_{ℤ}T^{Δ}(H, G) of p-permutation (kH, kG)-bimodules whose indecomposable direct summands have twisted diagonal vertices. An 𝔽-linear functor from 𝔽pp_{k}^{Δ} to 𝔽-Mod is called a diagonal p-permutation functor.

By [2], the simple diagonal p-permutation functors are parametrized by the pairs (G, V) of a finite group G and a simple module V of the essential algebra E^{Δ}(G) = End_{𝔽pp_{k}^{Δ}}(G)/I at G, where I is the ideal generated by the morphisms that factor through groups of smaller order. We show that the essential algebra E^{Δ}(G) is isomorphic to the essential algebra studied in [5]. As a result this implies that the essential algebra E^{Δ}(G) is non-zero if and only if the group G is of the form P ⋊ ⟨s⟩ where P is a p-group and s is a generator of a p′-cyclic group acting faithfully on P. Moreover in that case there is an algebra isomorphism E^{Δ}(G) ≅ (𝔽[X]/Φ_{n}[X]) ⋊ Out(G) where n is the order of s. See Theorem 3.3.

We also study the functor 𝔽T^{Δ} that sends a finite group G to the Grothendieck group 𝔽T (G) of p-permutation kG-modules. We describe the subfunctor lattice (Theorem 5.11) and simple quotients (Proposition 5.15) of 𝔽T^{Δ}. We also give a description for the 𝔽-dimension of the evaluations of simple quotients of 𝔽T^{Δ} at a finite group G (Theorem 5.18). Moreover we prove that the simple functor S_{1,1} that corresponds to the pair (1, 1) is isomorphic to the functor that sends a finite group G to the 𝔽-linear extension 𝔽K_{0}(kG) of the Grothendieck group of projective kG-modules (Theorem 5.20).

2. Preliminaries

Let G and H be finite groups. We denote by p_{1} : G×H → G and p_{2} : G× H → H the canonical projections. Let X ⩽ G × H be a subgroup. We define the subgroups k_{1}(X) := p_{1}(X ∩ ker(p_{2})) and k_{2}(X) := p_{2}(X ∩ ker(p_{1})) of p_{1}(X) and p_{2}(X), respectively. Note that k_{1}(X) × k_{2}(X) is a normal subgroup of X. Moreover, k_{i}(X) is a normal subgroup of p_{i}(X) and one has a canonical isomorphism X/(k_{1}(X) × k_{2}(X)) → p_{i}(X)/k_{i}(X) induced by the projection map p_{i} for i = 1, 2.

Let ϕ : P → Q be an isomorphism between subgroups P ⩽ G and Q ⩽ H. Then {(ϕ(x), x) : x ∈ P } is a subgroup of H × G and a subgroup of that form is called a twisted diagonal subgroup of H × G. Note that a subgroup X ⩽ H × G is a twisted diagonal subgroup if and only if k_{1}(X) = 1 and k_{2}(X) = 1.

Let P be a subgroup of G and M be a kG-module. We denote by M^{P} the k-vector space of P -fixed points of M. If Q ⩽ P is a subgroup, then the map Tr_{Q}^{P} : M^{Q} → M^{P} defined by Tr(m) = ∑_{x∈[P/Q]} x · m is called the relative trace map. The quotient

M[P] := M^{P} / ∑_{Q<P} Tr_{Q}^{P}(M^{Q})

is called the Brauer quotient of M at P. Note that M[P] is a k\overline{N}_{G}(P)-module, where \overline{N}_{G}(P) := N_{G}(P)/P. We have M[P] = 0 if P is not a p-group.

A (kG, kH)-bimodule M can be viewed as a k(G × H)-module via (g, h) · m := gmh^{−1}, for (g, h) ∈ G × H and m ∈ M. Similarly a k(G × H)-module can be viewed as a (kG, kH)-bimodule. We will usually switch between these two points of views.

Definition 2.1. Let G be a finite group. A kG-module M is called a permutation module, if M has a G-stable k-basis. A p-permutation kG-module is a kG-module M such that Res_{S}^{G}M is a permutation kS-module for a Sylow p-subgroup S of G.

For a finite group G we denote by T (G) the Grothendieck group of p-permutation kG-modules with respect to direct sum decompositions. If M is a p-permutation kGmodule, then the class of M in T (G) will be abusively denoted by M. The group T (G) has a ring structure induced by the tensor product of modules over k, and T (G) will be called the ring of p-permutation modules of G, for short. If H is another finite group, we set T (G, H) := T (G × H). We denote by T^{Δ}(G, H) the subgroup of T (G, H) spanned by p-permutation k(G × H)-modules whose indecomposable direct summands have twisted diagonal vertices.

Let P_{G,p} denote the set of pairs (P, E) where P is a p-subgroup of G and E is a projective indecomposable k\overline{N}_{G}(P)-module. The group G acts on the set P_{G,p} via conjugation and we denote by [P_{G,p}] a set of representatives of G-orbits of P_{G,p}. For (P, E) ∈ P_{G,p}, let M_{P,E} denote the unique (up to isomorphism) indecomposable p-permutation kG-module with the property that M_{P,E}[P] ≅ E. Note that M_{P,E} has the group P as a vertex [4, Theorem 3.2]. We denote by P_{G×H,p}^{Δ} the set of pairs (P, E) ∈ P_{G×H,p} where P is a twisted diagonal p-subgroup of G × H.

Remark 2.2. The isomorphism classes of the modules M_{P,E} where (P, E) ∈ P_{G×H,p}^{Δ} form a ℤ-basis for T^{Δ}(G, H).

Definition 2.3. [5, Definition 2.3.1] Let (P, s) be a pair where P is a p-group and s is a generator of a p′-cyclic group acting on P. We denote the semidirect product P ⋊ ⟨s⟩ by ⟨P s⟩. Let (Q, t) be another such pair. We say that the pairs (P, s) and (Q, t) are isomorphic if there are group isomorphisms ϕ : P → Q and ψ : ⟨s⟩ → ⟨t⟩ such that ψ(s) = q · t for some q ∈ Q and ϕ(s · u) = ψ(s) · ϕ(u) for all u ∈ P. In that case we write (P, s) ≃ (Q, t).

Lemma 2.4. [5, Proposition 2.3.3] Let (P, s) and (Q, t) be two pairs. Then (P, s) ≃ (Q, t) if and only if there is a group isomorphism f : ⟨P s⟩ → ⟨Qt⟩ such that f(s) is conjugate to t.

Let Q_{G,p} denote the set of pairs (P, s) where P is a p-subgroup of G and s ∈ N_{G}(P) is a p′-element. In that case ⟨P s⟩ denotes the semidirect product P ⋊ ⟨s⟩ where the action of ⟨s⟩ on P is induced by conjugation. The group G acts on the set Q_{G,p} and we denote by [Q_{G,p}] a set of representatives of G-orbits. We denote by Q_{G×H,p}^{Δ} the set of pairs (P, s) ∈ Q_{G×H,p} where P is a twisted diagonal p-subgroup of G × H.

For any pair (P, s) ∈ Q_{G,p} let τ_{P,s}^{G} denote the additive map T (G) → 𝔽 that sends a p-permutation kG-module M to the value of the Brauer character of M[P] at s. The map τ_{P,s}^{G} is a ring homomorphism and it extends to an 𝔽-algebra homomorphism τ_{P,s}^{G} : 𝔽 ⊗_{ℤ} T (G) → 𝔽. The set {τ_{P,s}^{G} : (P, s) ∈ [Q_{G,p}]} is the set of all species from 𝔽T (G) := 𝔽 ⊗_{ℤ} T (G) to 𝔽 [3, Proposition 2.18].

The algebra 𝔽T (G) is split semisimple and its primitive idempotents F_{P,s}^{G} are indexed by pairs (P, s) ∈ [Q_{G,p}] [3, Corollary 2.19]. If ϕ : ⟨s⟩ → k^{×} is a group homomorphism, we denote by k_{ϕ} the k⟨s⟩-module k on which the element s acts as multiplication by ϕ(s). Let ˆ⟨s⟩ = Hom(⟨s⟩, k^{×}) denote the set of group homomorphisms. By [3, Theorem 4.12] we have the idempotent formula

F_{P,s}^{G} = \frac{1}{|P ||s||C_{\overline{N}_{G}(P)}(s)|} ∑_{\begin{matrix}φ∈ˆ⟨s⟩ \\ L⩽⟨P s⟩ \\ P L=⟨P s⟩ \end{matrix}} 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}, and φ̃ is the Brauer character of k_{φ}.

By [5, Proposition 2.7.8] we have another formula

F_{P,s}^{G} = \frac{1}{|C_{N_{G}(P)}(s)|} ∑_{\begin{matrix}φ∈ˆ⟨s⟩ \\ L⩽P \\ L^{s}=L \end{matrix}} φ̃(s^{−1})|C_{L}(s)|μ((L, P)^{s})Ind_{⟨Ls⟩}^{G}k_{⟨Ls⟩,φ}^{⟨P s⟩}.

Here μ((−, −)^{s}) is the Möbius function of the poset of s-stable subgroups of P.

Lemma 2.5. For finite groups G and H, the set {F_{P,s}^{G×H} : (P, s) ∈ [Q_{G×H,p}^{Δ}]} of primitive idempotents form an 𝔽-basis for the split semisimple algebra 𝔽T^{Δ}(G, H).

Proof. First we will show that we have F_{P,s}^{G×H} ∈ 𝔽T^{Δ}(G, H) whenever (P, s) ∈ [Q_{G×H,p}^{Δ}]. Let φ ∈ ˆ⟨s⟩ and L ⩽ ⟨P s⟩. It suffices to show that Ind_{L}^{G}k_{L,φ}^{⟨P s⟩} ∈ 𝔽T^{Δ}(G, H). Since P acts trivially on Inf_{⟨s⟩}^{⟨P s⟩}k_{φ}, the subgroup P is contained in a vertex of k_{φ} considered as a k⟨P s⟩-module. But since P is the Sylow p-subgroup of ⟨P s⟩, it follows that P is the vertex of k_{φ}. Therefore the module k_{L,φ}^{⟨P s⟩} = Res_{L}^{⟨P s⟩}Inf_{⟨s⟩}^{⟨P s⟩}k_{φ} has a vertex contained in L ∩ ^{x}P ⩽ P for some x ∈ ⟨P s⟩. Since a subgroup of twisted diagonal subgroup is again twisted diagonal, this means that k_{L,φ}^{⟨P s⟩} has twisted diagonal vertices. This shows that Ind_{L}^{G}k_{L,φ}^{⟨P s⟩} ∈ 𝔽T^{Δ}(G, H) as desired. Now since the 𝔽-dimension of 𝔽T^{Δ}(G, H) is equal to the cardinality of [P_{G×H,p}^{Δ}], which is equal to the cardinality of [Q_{G×H,p}^{Δ}], it follows that the set {F_{P,s}^{G×H} : (P, s) ∈ [Q_{G×H,p}^{Δ}]} of primitive idempotents form an 𝔽-basis for 𝔽T^{Δ}(G, H).

Let G, H and L be finite groups. If X is a (kG, kH)-bimodule and Y is a (kH, kL)- bimodule, then X ∘ Y := X ⊗_{kH} Y is a (kG, kL)-bimodule. Extending this product by 𝔽-bilinearity, we get a map

𝔽T (G, H) ∘ 𝔽T (H, L) → 𝔽T (G, L).

Note that this induces a map

𝔽T^{Δ}(G, H) ∘ 𝔽T^{Δ}(H, L) → 𝔽T^{Δ}(G, L)

which is used to define the composition of morphisms in the following category.

Definition 2.6. Let 𝔽pp_{k}^{Δ} be the category with

• objects: finite groups

• Mor_{𝔽pp_{k}^{Δ}}(G, H) = 𝔽 ⊗_{ℤ} T^{Δ}(H, G) = 𝔽T^{Δ}(H, G).

An 𝔽-linear functor from 𝔽pp_{k}^{Δ} to 𝔽-Mod is called a diagonal p-permutation functor. Diagonal p-permutation functors form an abelian category F_{pp_{k}}^{Δ}.

3. The Essential Algebra

For a finite group G, the quotient algebra

E^{Δ}(G) := 𝔽T^{Δ}(G, G)/( ∑_{|H|<|G|} 𝔽T^{Δ}(G, H) ∘ 𝔽T^{Δ}(H, G))

is called the essential algebra of G.

By [5, Proposition 4.1.2 and Theorem 4.1.12] the algebra

E(G) := 𝔽T (G, G)/( ∑_{|H|<|G|} 𝔽T (G, H) ∘ 𝔽T (H, G))

is non-zero if and only if there exists a pair (P, s) in G such that G = ⟨P s⟩ and C_{⟨s⟩}(P) = 1. In that case, we also have an algebra isomorphism

E(G) ≅ (𝔽[X]/Φ_{n}[X]) ⋊ Out(G)

where n is the order of s [5, Theorem 4.1.12].

Note that the inclusion map 𝔽T^{Δ}(G, G) ↪ 𝔽T (G, G) induces a map

Θ : E^{Δ}(G) → E(G).

We will show that this map is an algebra isomorphism.

Let φ ∈ Aut(G) be an automorphism and λ : G/O_{p}(G) → k^{×} be a character, where O_{p}(G) denotes the largest normal p-subgroup of G. We define a (kG, kG)- bimodule structure on kG, denoted by kG_{φ,λ}, via

a · g · b := λ(b)agφ(b)

for a, b, g ∈ G.

Let ⟨Rt⟩ be a twisted diagonal subgroup of G × G with p_{1}(⟨Rt⟩) = G and p_{2}(⟨Rt⟩) = G. Let also η : p_{1}(⟨Rt⟩) → p_{2}(⟨Rt⟩) be the canonical isomorphism. Then by [5, Section 4.1.2] we have an isomorphism

Ind_{⟨Rt⟩}^{G×G}k_{⟨Rt⟩,φ}^{⟨Rt⟩} ≅ kG_{η^{−1},φ^{−1}}

of (kG, kG)-bimodules. Again by [5, Section 4.1.2] the algebra E(G) is generated by the images of kG_{φ,λ}.

Proposition 3.1. If the essential algebra E^{Δ}(G) of a finite group G is non-zero, then there exists a pair (P, s) in G such that G = ⟨P s⟩ and C_{⟨s⟩}(P) = 1.

Proof. Let (Q, t) be a pair contained in G × G such that Q is a twisted diagonal subgroup and recall the idempotent formula

F_{Q,t}^{G×G} = \frac{1}{|C_{N_{G×G(Q)}(t)}|} ∑_{\begin{matrix}φ∈ˆ⟨t⟩ \\ L⩽Q \\ L^{t}=L \end{matrix}} φ̃(t^{−1})|C_{L}(t)|μ((L, Q)^{t})Ind_{⟨Lt⟩}^{G×G}k_{L,φ}^{⟨Qt⟩}.

By [5, Lemma 2.5.9] we have an isomorphism

Ind_{⟨Lt⟩}^{G×G}k_{L,φ}^{⟨Qt⟩} ≅ Ind_{p_{1}(⟨Lt⟩)}^{G} ⊗_{p_{1}(⟨Lt⟩)} Ind_{⟨Lt⟩}^{p_{1}(⟨Lt⟩)×p_{2}(⟨Lt⟩)}(k_{L,φ}^{⟨Qt⟩}) ⊗_{p_{2}(⟨Lt⟩)} Res_{p_{2}(⟨Lt⟩)}^{G}
≅ kG ⊗_{p_{1}(⟨Lt⟩)} Ind_{⟨Lt⟩}^{p_{1}(⟨Lt⟩)×p_{2}(⟨Lt⟩)}(k_{L,φ}^{⟨Qt⟩}) ⊗_{p_{2}(⟨Lt⟩)} kG

of (kG, kG)-bimodules. As (kG, kG)-bimodule, we have the isomorphism kG ≅ Ind_{ΔG}^{G×G}k. Thus as (kG, kp_{1}(⟨Lt⟩))-bimodule we have,

Res_{G×p_{1}(⟨Lt⟩)}^{G×G}kG ≅ Res_{G×p_{1}(⟨Lt⟩)}^{G×G}Ind_{ΔG}^{G×G}k ≅ Ind_{Δ(p_{1}(⟨Lt⟩))}^{G×p_{1}(⟨Lt⟩)}Res_{Δ(p_{1}(⟨Lt⟩))}^{Δ(G)}k.

Therefore as (kG × p_{1}(⟨Lt⟩))-module, the indecomposable direct summands of kG have vertices contained in Δ(p_{1}(⟨Lt⟩)). Similary, one can show that the indecomposable direct summands of kG as k(p_{2}(⟨Lt⟩) × G)-module, have vertices contained in Δ(p_{2}(⟨Lt⟩)). We also know that the module k_{L,φ}^{⟨Qt⟩}, and hence the indecomposable direct summands of Ind_{⟨Lt⟩}^{p_{1}(⟨Lt⟩)×p_{2}(⟨Lt⟩)}(k_{L,φ}^{⟨Qt⟩}), have twisted diagonal vertices. Now suppose E^{Δ}(G) is non-zero. Then there is an idempotent F_{Q,t}^{G×G} whose image in E^{Δ}(G) is non-zero. Therefore the argument above shows that there is a pair (Q, t) in G × G such that p_{1}(⟨Qt⟩) = G and p_{2}(⟨Qt⟩) = G. This implies that there is a p-subgroup P of G and a p′-element s of G that normalises P such that G = ⟨P s⟩. Now we will show that in that case we have C_{⟨s⟩}(P) = 1. Let \overline{G} := G/C_{⟨s⟩}(P), Q := \{(u, \overline{u} : u ∈ P \} ⩽ G × \overline{G} and Q′ := \{(\overline{u}, u) : u ∈ P \} ⩽ \overline{G} × G. Then by [5, Proof of Proposition 4.1.2] we have an isomorphism between kG and

⨁_{i} Indinf_{\overline{N}_{G×\overline{G}}(Q)}^{G×\overline{G}}(kC_{G}(P)/C_{⟨s⟩}(P)⊗_{k}k_{α_{i}})⊗_{k\overline{G}}Indinf_{\overline{N}_{\overline{G}×G}(Q′)}^{\overline{G}×G}(kC_{G}(P)/C_{⟨s⟩}(P)⊗_{k}k_{α′_{i}})

as (kG, kG)-bimodules, where Indinf_{\overline{N}_{G×\overline{G}}(Q)}^{G×\overline{G}} = Ind_{N_{G×\overline{G}}(Q)}^{G×\overline{G}} ∘ Inf_{\overline{N}_{G×\overline{G}}(Q)}^{N_{G×\overline{G}}(Q)}. Here α_{i} and α′_{i} run over the irreducible characters of ⟨s⟩. Again by [5, Proof of Proposition 4.1.2] for each i, the modules kC_{G}(P)/C_{⟨s⟩}(P)⊗_{k} k_{α_{i}} and kC_{G}(P)/C_{⟨s⟩}(P)⊗_{k} k_{α′_{i}} are projective indecomposable k\overline{N}_{G×\overline{G}}(Q)-modules and k\overline{N}_{\overline{G}×G}(Q′)-modules respectively. Now since kC_{G}(P)/C_{⟨s⟩}(P) ⊗_{k} k_{α_{i}} is projective indecomposable, it has the trivial group as vertex. So Inf_{\overline{N}_{G×\overline{G}}(Q)}^{N_{G×\overline{G}}(Q)}(kC_{G}(P)/C_{⟨s⟩}(P) ⊗_{k} k_{α_{i}}) has the group Q as a vertex. Note that the group Q is twisted diagonal. Therefore indecomposable direct summands of Indinf_{\overline{N}_{G×\overline{G}}(Q)}^{G×\overline{G}}(kC_{G}(P)/C_{⟨s⟩}(P) ⊗_{k} k_{α_{i}}) have twisted diagonal

vertices, i.e. Indinf_{\overline{N}_{G×\overline{G}}(Q)}^{G×\overline{G}}(kC_{G}(P)/C_{⟨s⟩}(P) ⊗_{k} k_{α_{i}}) ∈ 𝔽T^{Δ}(G, \overline{G}). Similarly, we

have Indinf_{\overline{N}_{\overline{G}×G}(Q′)}^{\overline{G}×G}(kC_{G}(P)/C_{⟨s⟩}(P) ⊗_{k} k_{α′_{i}}) ∈ 𝔽T^{Δ}(\overline{G}, G). Now since E^{Δ}(G) ≠ 0,

the image of identity element kG ∈ 𝔽T^{Δ}(G, G) in E^{Δ}(G) is non-zero. Hence we have \overline{G} = G, i.e. C_{⟨s⟩}(P) = 1.

Suppose we have G = ⟨P s⟩ and C_{⟨s⟩}(P) = 1. The essential algebra E^{Δ}(G) is generated by the images of the primitive idempotents

F_{Q,t}^{G×G} = \frac{1}{| C_{N_{G×G(Q)}(t)} |} ∑_{\begin{matrix}φ∈ˆ⟨t⟩ \\ L⩽Q \\ L^{t}=L \end{matrix}} φ̃(t^{−1})|C_{L}(t)|μ((L, Q)^{t})Ind_{⟨Lt⟩}^{G×G}k_{L,φ}^{⟨Qt⟩}

where Q is a twisted diagonal subgroup of G × G. By [5, Lemma 2.5.9], if the image of Ind_{⟨Lt⟩}^{G×G}k_{L,φ}^{⟨Qt⟩} is non-zero, then we must have that p_{1}(⟨Lt⟩) = G = p_{2}(⟨Lt⟩). Write t = (u, v). Then p_{1}(⟨Lt⟩) = ⟨p_{1}(L)u⟩ and p_{2}(⟨Lt⟩) = ⟨p_{2}(L)v⟩. Therefore we have |u| = |v| = |s|. Being a subgroup of twisted diagonal subgroup Q, the group L itself is also twisted diagonal. Since k_{1}(L) = k_{2}(L) = 1 and |u| = |v| = |s|, we have k_{1}(⟨Lt⟩) = k_{2}(⟨Lt⟩) = 1. This shows that the subgroup ⟨Lt⟩ is twisted diagonal and p_{1}(⟨Lt⟩) = G = p_{2}(⟨Lt⟩). Since the images of Ind_{⟨Lt⟩}^{G×G}k_{L,φ}^{⟨Qt⟩} in E(G) with ⟨Lt⟩ satisfying these properties, generate the non-zero algebra E(G), this shows that the algebra E^{Δ}(G) is also non-zero and the map Θ : E^{Δ}(G) → E(G) is surjective. Thus we have proved the following:

Proposition 3.2. The essential algebra E^{Δ}(G) is non-zero if and only if there is a pair (P, s) in G such that G = ⟨P s⟩ and C_{⟨s⟩}(P) = 1. Moreover the map Θ : E^{Δ}(G) → E(G) is surjective.

Suppose we have G = ⟨P s⟩ for some pair and C_{⟨s⟩}(P) = 1. We will show that the map Θ : E^{Δ}(G) → E(G) is also injective. Suppose an element ∑ \overline{r_{φ,α}kG_{φ,α}} ∈ E^{Δ}(G) is mapped to zero by Θ. Then the element ∑ \overline{r_{φ,α}kG_{φ,α}} of E(G) is zero. Write

∑ r_{φ,α}kG_{φ,α} = ∑_{|H|<|G|} t_{H,U_{H},V_{H}}U_{H} ⊗_{kH} V_{H}

for some (kG, kH)-bimodule U_{H} and (kH, kG)-bimodule V_{H} and some constants t_{H,U_{H},V_{H}} ∈ 𝔽. Suppose the coefficient t_{H,U_{H},V_{H}} is non-zero for some group H. Then as in [5] we can assume that H = ⟨Rt⟩ for some pair (R, t) and that the modules U_{H} and V_{H} are indecomposable. By [5, Section 4.1] one has

U_{H} ⊗_{kH} V_{H} ≅ Indinf_{\overline{N}_{G×G}(Δ(P))}^{G×G} ⨁_{i} (kZ(P) ⊗ k_{λ_{i}})^{n_{i}}

where λ_{i} is a character of ⟨s⟩ and n_{i} ∈ ℕ. Again by [5, Section 4.1] each summand kZ(P) ⊗ k_{λ_{i}} is a projective indecomposable k\overline{N}_{G×G}(Δ(P))-module. This shows that if the the coefficient t_{H,U_{H},V_{H}} is non-zero, then the indecomposable direct summands of the bimodule U_{H} ⊗_{kH} V_{H} have twisted diagonal vertices. Therefore the element ∑ \overline{r_{φ,α}kG_{φ,α}} is zero in E^{Δ}(G). This proves that the map Θ : E^{Δ}(G) → E(G) is injective. We summarise our results as a theorem below.

Theorem 3.3. The essential algebra E^{Δ}(G) is non-zero if and only if there is a pair (P, s) in G such that G = ⟨P s⟩ and C_{⟨s⟩}(P) = 1. In that case, the algebra E^{Δ}(G) is isomorphic to the algebra (𝔽[X]/Φ_{n}[X]) ⋊ Out(G) where n is the order of s.

4. D^{Δ}-pairs

Let H ⩽ G be a subgroup. The (kG, kH)-bimodule kG is denoted by Ind_{H}^{G} and (kH, kG)-bimodule kG is denoted by Res_{H}^{G}. Similarly, if N ⊴ G is a normal subgroup, the (kG/N, kG)-bimodule kG/N is denoted by Def_{G/N}^{G} and (kG, kG/N)- bimodule kG/N is denoted by Inf_{G/N}^{G}. This notation is consistent with our previous use of induction, restriction, inflation and deflation symbols, in the sense that for example, if M is a kH-module, then the induced module Ind_{H}^{G}M is isomorphic to Ind_{H}^{G} ⊗_{kH} M.

We have the following lemma due to [3] and [5].

Lemma 4.1. (i) Let (P, s) ∈ Q_{G,p} be a pair and H ⩽ G be a subgroup. Then we have

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

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

(ii) Let (Q, t) ∈ Q_{H,p} be a pair and H ⩽ G be a subgroup. Then we have

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

(iii) Let N ⊴ G and (P, s) ∈ Q_{G/N,p}. Then

Inf_{G/N}^{G}F_{P,s}^{G/N} = ∑_{Q,t} F_{Q,t}^{G}

where (Q, t) runs over a set of representatives of G-conjugacy classes of pairs in Q_{G,p} such that QN/N = \overline{g}P and \overline{t} = ^{g}s for some \overline{g} ∈ G/N.

(iv) Let N ⊴ G and (P, s) ∈ Q_{G,p}. Then

Def_{G/N}^{G}F_{P,s}^{G} = m_{P,s,N} · F_{Q,t}^{G/N}

for some pair (Q, t) ∈ Q_{G/N,p} and a constant m_{P,s,N} ∈ 𝔽.

If G = ⟨P s⟩ then

Def_{G/N}^{G}F_{P,s}^{G} = m_{P,s,N} · F_{P N/N,\overline{s}}^{G/N}.

Proof. See [3, Proposition 3.1. and Proposition 3.2.] for (i) and (ii), [5, Proposition 3.1.3] for (iii) and [5, Lemma 3.1.4 and Proposition 3.1.5] for (iv).

Lemma 4.2. Let N ⊴ G be a normal subgroup of G.

(i) We have Def_{G/N}^{G} ∈ 𝔽T^{Δ}(G/N, G) if and only if N is a p′-group.

(ii) We have Inf_{G/N}^{G} ∈ 𝔽T^{Δ}(G, G/N) if and only if N is a p′-group.

Proof. (i) Let Q ⩽ (G/N)×G be a maximal vertex of an indecomposable direct summand of the (kG/N, kG)-bimodule kG/N. Equivalently Q is a maximal p-subgroup having a fixed point on the set G/N. Suppose (aN, b) ∈ Q stabilises a basis element gN of kG/N. Then we have (aN)gNb^{−1} = gN which implies that a^{g} · b^{−1} ∈ N. Since the vertices of an indecomposable module are conjugate, we may assume that g = 1. Thus, up to conjugacy, Q is a Sylow p-subgroup of

H = {(aN, b) : ab^{−1} ∈ N} ⩽ (G/N) × G.

Note that k_{1}(Q) = k_{1}(H) = 1 and k_{2}(Q) is a Sylow p-subgroup of N. Hence Q is twisted diagonal if and only if N is a p′-group. The result follows. (ii) Similar.

Let (P, s) be a pair and suppose G = ⟨P s⟩. Then by [5, Corollary 3.1.9] for any normal subgroup N of G, we have the following formula for the constant m_{P,s,N}:

m_{P,s,N} = \frac{|s|}{|N ∩ ⟨s⟩||C_{G}(s)|} ∑_{\begin{matrix}Q⩽P \\ Q^{s}=Q \\ ⟨Qs⟩N=G \end{matrix}} |C_{Q}(s)|μ((Q, P)^{s}).

Lemma 4.3. Let (P, s) be a pair and suppose G = ⟨P s⟩. Then for any normal p′-subgroup N of G we have

m_{P,s,N} = \frac{1}{|N|}·

Proof. First observe that since N is a p′-group, we have N ⩽ C_{⟨s⟩}(P). For any subgroup Q of P the condition ⟨Qs⟩N = ⟨P s⟩ implies that |Q| = |P | and hence Q = P. Therefore the formula above becomes

m_{P,s,N} = \frac{|s||C_{P} (s)|}{|N||C_{G}(s)|} = \frac{1}{|N|}·

Definition 4.4. A pair (P, s) is called D^{Δ}-pair if Def_{⟨P s⟩/N}^{⟨P s⟩}F_{P,s}^{⟨P s⟩} = 0 for any nontrivial normal p′-subgroup N of ⟨P s⟩.

Lemma 4.5. Let (P, s) be a pair. Then (P, s) is a D^{Δ}-pair if and only if the group ⟨P s⟩ does not have any nontrivial normal p′-subgroup, that is, if and only if C_{⟨s⟩}(P) = 1.

Proof. By Lemma 4.3, for any normal p′-subgroup N ⊴⟨P s⟩ we have m_{P,s,N} = 1/|N|. Therefore (P, s) is a D^{Δ}-pair if and only if the group ⟨P s⟩ does not have any nontrivial normal p′-subgroup. The result follows.

5. The functor 𝔽T^{Δ}

By [2], the simple diagonal p-permutation functors are parametrized by the pairs (G, V) where G is a finite group and V is a simple E^{Δ}(G)-module. Note that this implies E^{Δ}(G) ≠ 0.

For a simple E^{Δ}(G)-module V, we define two functors in 𝔽pp_{k}^{Δ} by:

L_{G,V} (H) := 𝔽T^{Δ}(H, G) ⊗_{E^{Δ}(G)} V

and

J_{G,V} (H) := {∑_{i} ϕ_{i} ⊗ v_{i} ∈ L_{G,V} : ∀ψ ∈ 𝔽T^{Δ}(G, H), ∑_{i}(ψ ∘ ϕ_{i}) · v_{i} = 0} ,

for any finite group H. The action of morphisms in 𝔽pp_{k}^{Δ} on these evaluations is given by left composition. The functor J_{G,V} is the unique maximal subfunctor of L_{G,V}, so the quotient

S_{G,V} := L_{G,V} /J_{G,V}

is a simple functor [2].

Let 𝔽T^{Δ} : 𝔽pp_{k}^{Δ} → 𝔽-Mod be the functor given by

• 𝔽T^{Δ}(G) := 𝔽 ⊗_{ℤ} T (G) = 𝔽T (G),

• 𝔽T^{Δ}(X) : 𝔽T (G) → 𝔽T (H), M ↦ X ⊗_{kH} M for any X ∈ 𝔽T^{Δ}(H, G).

For any kG-module X, we denote by X̃ the (kG, kG)-bimodule k(G × X) where the action of kG-kG is given by

a · (g, x) · b^{−1} := (agb, b^{−1}x)

for all a, b, g ∈ G and x ∈ X. We have an isomorphism of (kG, kG)-bimodules

X̃ ≅ Ind_{δ(G)}^{G×G^{op}}Iso(δ)(X)

where δ : G → G × G^{op}, g ↦ (g, g^{−1}). See [5, Definition 2.5.17]. Note that the image δ(G) of G in G × G^{op} is a twisted diagonal subgroup. If X is an indecomposable p-permutation kG-module with a vertex Q, then any vertex of an indecomposable direct summand of X̃ is contained in δ(Q), up to conjugation. Therefore for any X ∈ 𝔽T (G) we have X̃ ∈ 𝔽T^{Δ}(G, G).

Lemma 5.1. Let F be a subfunctor of 𝔽T^{Δ}. Then for any finite group G, the 𝔽-vector space F (G) is an ideal of the algebra 𝔽T^{Δ}(G) of p-permutation modules.

Proof. Let Y ∈ F (G) and assume X is a p-permutation kG-module. By [5, Proposition 2.5.18] we have an isomorphism X ⊗_{k} Y ≅ X̃ ⊗_{kG} Y of kG-modules. Since X̃ ∈ 𝔽T^{Δ}(G, G) and F is a functor, we have X̃ ⊗_{kG} Y ∈ F (G). This shows that F (G) is an ideal of 𝔽T^{Δ}(G).

Definition 5.2. For any pair (P, s) let e_{P,s} denote the subfunctor of 𝔽T^{Δ} generated by the idempotent F_{P,s}^{⟨P s⟩} ∈ 𝔽T^{Δ}(⟨P s⟩).

Proposition 5.3. Let F be a subfunctor of 𝔽T^{Δ}. Then we have

F = ∑_{e_{P,s}⩽F} e_{P,s}.

Proof. Since F is a subfunctor, we have

∑_{e_{P,s}⩽F} e_{P,s} ⩽ F.

Now let G be a finite group, and u = ∑_{(P,s)} λ_{P,s}F_{P,s}^{G}, where (P, s) runs in a set of representatives of G-conjugacy classes of Q_{G,p}, and λ_{P,s} ∈ 𝔽. Then F_{P,s}^{G} · u = λ_{P,s}F_{P,s}^{G} ∈ F (G), since F (G) is an ideal of 𝔽T^{Δ}(G). Hence F_{P,s}^{G} ∈ F (G) if λ_{P,s} ≠ 0. In this case we have Res_{⟨P s⟩}^{G}F_{P,s}^{G} ∈ F (⟨P s⟩), which implies by Lemma 4.1 that F_{P,s}^{⟨P s⟩} ∈ F (⟨P s⟩). This shows that e_{P,s} ⩽ F. By Lemma 4.1 again, F_{P,s}^{G} is a non zero scalar multiple of Ind_{⟨P s⟩}^{G}F_{P,s}^{⟨P s⟩}, so F_{P,s}^{G} ∈ e_{P,s}(G), which gives finally

u ∈ ∑_{e_{P,s}⩽F} e_{P,s}(G).

Therefore we have

F = ∑_{e_{P,s}⩽F} e_{P,s}

as desired.

Proposition 5.4. Let (P_{i}, s_{i})_{i∈I} be a set of pairs for an indexing set I. Then for any pair (Q, t) we have e_{Q,t} ⩽ ∑_{i∈I} e_{P_{i},s_{i}} if and only if e_{Q,t} ⩽ e_{P_{i},s_{i}} for some i ∈ I.

Proof. If e_{Q,t} ⩽ e_{P_{i},s_{i}} for some i ∈ I, then we obviously have e_{Q,t} ⩽ ∑_{i∈I} e_{P_{i},s_{i}}. Conversely assume we have e_{Q,t} ⩽ ∑_{i∈I} e_{P_{i},s_{i}}. Then e_{Q,t}(⟨Qt⟩) ⩽ ∑_{i∈I} e_{P_{i},s_{i}}(⟨Qt⟩) and so F_{Q,t}^{⟨Qt⟩} ∈ ∑_{i∈I} e_{P_{i},s_{i}}(⟨Qt⟩). Since F_{Q,t}^{⟨Qt⟩} is a primitive idempotent and since e_{P_{i},s_{i}}(⟨Qt⟩) is an ideal of 𝔽T^{Δ}(⟨Qt⟩) it follows that we have F_{Q,t}^{⟨Qt⟩} ∈ e_{P_{i},s_{i}}(⟨Qt⟩) for some i ∈ I and hence e_{Q,t} ⩽ e_{P_{i},s_{i}}.

Let G be a finite group and (P, s) ∈ Q_{G,p} be a pair such that G = ⟨P s⟩. Let also (Q, t) ∈ Q_{H×G,p}^{Δ} for a finite group H. Suppose that η : p_{1}(Q) → p_{2}(Q) is the canonical isomorphism. Up to conjugation in H × G, we can assume t = (u, s^{j}). By [5, Section 3.2] if p_{2}(⟨Qt⟩) ≠ G, then the product F_{Q,t}^{H×G} ⊗_{kG} F_{P,s}^{G} is zero. So assume that we have p_{2}(⟨Qt⟩) = G. This implies that we have p_{2}(Q) = P and |s^{j}| = |s|.

Then since k_{1}(Q) = k_{2}(Q) = 1, this implies that we have p_{1}(Q) ≅ P. Since the group Q is t-stable, the isomorphism η : p_{1}(Q) → P commutes with conjugations by u and s^{j}. Now [5, Equation (3.3), Section 3.2] implies that as kH-module the product F_{Q,t}^{H×G} ⊗_{kG} F_{P,s}^{G} is equal to

\frac{1}{|C_{N_{H×G}}(Q)(t)||C_{G}(s)|} ∑_{\begin{matrix}φ∈\overline{⟨t⟩} \\ ψ∈\overline{⟨s⟩} \\ φ^{|u|}ψ^{j|u|}=1 \end{matrix}} φ̃(t)^{−1}ψ̃(s)^{−1}|C_{Q}(t)| ∑_{\begin{matrix}J⩽p_{1}(Q) \\ J^{u}=J \end{matrix}} σ(J)Ind_{⟨Ju⟩}^{H}(k_{⟨Ju⟩,ϕ}^{⟨p_{1}(Q)u⟩})

where σ(J) := ∑\begin{matrix}L⩽P \\ L^{s}=L \\ η(J)=L \end{matrix} |C_{L}(s)|μ((L, P)^{s}) and ϕ(u) := φ(u, s^{j})ψ(s)^{j}.

Suppose we have H = ⟨P′s′⟩ for a pair (P′, s′). Then by [5, Lemma 2.7.6] if τ_{P′,s′}^{H}(F_{Q,t}^{H×G} ⊗_{kG} F_{P,s}^{G}) ≠ 0, then we must have p_{1}(Q) = P′ and |u| = |s′|. This implies in particular that we must have P′ ≅ P. Moreover again by [5, Lemma 2.7.6] we have τ_{P′,s′}^{H}(Ind_{⟨Ju⟩}^{H}(k_{⟨Ju⟩,ϕ}^{⟨p_{1}(Q)u⟩})) = 0 if J ≠ P′. Therefore if we have P′ ≅ P then τ_{P′,s′}^{H}(F_{Q,t}^{H×G} ⊗_{kG} F_{P,s}^{G}) is equal to

\frac{1}{|C_{N_{H×G}}(Q)(t)||C_{G}(s)|} ∑_{\begin{matrix}φ∈\overline{⟨t⟩} \\ ψ∈\overline{⟨s⟩} \\ φ^{|u|}ψ^{j|u|}=1 \end{matrix}} φ̃(t)^{−1}ψ̃(s)^{−1}|C_{Q}(t)||C_{P} (s)|ϕ̃(s′).

This shows that if we have 𝔽T^{Δ}(⟨P′s′⟩, ⟨P s⟩) ⊗_{k⟨P s⟩} F_{P,s}^{⟨P s⟩} ≠ 0, then there is an isomorphism η : P′ → P and a p′-element (u, s^{j}) ∈ ⟨P′s′⟩ × ⟨P s⟩ such that η ∘c_{u} = c_{s^{j}} ∘η and |u| = |s′|, |s^{j}| = |s|. In that case, assume further that C_{⟨s⟩}(P) = 1. Then we have |c_{s}| = |s| and |c_{s^{j}}| = |s^{j}|. Since we have η ∘ c_{u} = c_{s^{j}} ∘ η it follows that |c_{u}| = |c_{s^{j}}|. Therefore we have |s| | |s′|. But then [5, Proposition 2.3.6] implies that there is a surjective group homomorphism \overline{η} : ⟨P′s′⟩ → ⟨P s⟩ that induces an isomorphism of pairs (P′ ker(\overline{η})/ ker(\overline{η}), s′ ker(\overline{η})) ≃ (P, s). Note that since |P′| = |P | the order of ker(\overline{η}) is coprime to p. We have the following:

Lemma 5.5. Let (P, s) be a pair with C_{⟨s⟩}(P) = 1 and set G := ⟨P s⟩. Let H be a finite group. The following statements are equivalent:

(i) 𝔽T^{Δ}(H, G) ⊗_{kG} F_{P,s}^{G} ≠ 0.

(ii) There exists a pair (P′, s′) contained in H such that the pair (P, s) is isomorphic to a p′-quotient of the pair (P′, s′), that is, there exists a normal p′-subgroup K of ⟨P′s′⟩ such that (P, s) ≃ (P′K/K, s′K).

Proof. (i) ⇒ (ii) Suppose we have 𝔽T^{Δ}(H, G) ⊗_{kG} F_{P,s}^{G} ≠ 0. Then there exists a pair (P′, s′) in H such that

F_{P′,s′}^{H} ∈ 𝔽T^{Δ}(H, G) ⊗_{kG} F_{P,s}^{G}.

Via the restriction map this implies that we have

F_{P′,s′}^{⟨P′s′⟩} ∈ 𝔽T^{Δ}(⟨P′s′⟩, G) ⊗_{kG} F_{P,s}^{G}.

Therefore by the argument above we have an isomorphism (P′K/K, s′K) ≃ (P, s) of pairs where K is a normal p′-subgroup of ⟨P′s′⟩. (ii) ⇒ (i) Suppose Φ : (P′K/K, s′K) → (P, s) is an isomorphism of pairs where K is a normal p′-subgroup of ⟨P′s′⟩. Then we have

Ind_{⟨P′s′⟩}^{H}Inf_{⟨P′s′⟩/K}^{⟨P′s′⟩}Iso(Φ)F_{P,s}^{G} ≠ 0.

This shows (i).

Proposition 5.6. Let (P, s) be a pair. The following are equivalent:

(i) (P, s) is a D^{Δ}-pair.

(ii) For any finite group H with |H| < |⟨P s⟩|, we have e_{P,s}(H) = {0}.

(iii) If H is a finite group with e_{P,s}(H) ≠ {0}, then the pair (P, s) is isomorphic to a p′-quotient of a pair (P′, s′) contained in H.

(iv) For any nontrivial normal p′-subgroup N of ⟨P s⟩, we have Def_{⟨P s⟩/N}^{⟨P s⟩}F_{P,s}^{⟨P s⟩} = 0.

(v) The group ⟨P s⟩ does not have any nontrivial normal p′-subgroup.

(vi) We have C_{⟨s⟩}(P) = 1.

Proof. (vi)⇔(v)⇔(i) : This follows from Lemma 4.5. (iv)⇔ (i): This follows from the definition of D^{Δ}-pairs. (i)⇒ (iii): Since (P, s) is a D^{Δ}-pair, we have C_{⟨s⟩}(P) = 1. So (iii) follows from Lemma 5.5. (iii)⇒ (ii): Assume that (iii) holds and e_{P,s}(H) ≠ 0 where H is a finite group with |H| < |⟨P s⟩|. Then by the assumption, we have |H| ≥ |⟨P′s′⟩| ≥ |⟨P s⟩|. Contradiction. (ii)⇒ (iv): Clear.

Proposition 5.7. Let (P, s) and (Q, t) be two pairs.

(i) If the pair (Q, t) is isomorphic to a p′-quotient of the pair (P, s), then we have e_{P,s} = e_{Q,t}.

(ii) If (Q, t) is a D^{Δ}-pair, and if e_{P,s} ⩽ e_{Q,t}, then (Q, t) is isomorphic to a p′- quotient of (P, s).

Proof. (i) Assume we have an isomorphism ϕ : (P K/K, sK) → (Q, t) of pairs for some normal p′-subgroup K of ⟨P s⟩. Then we have

F_{P,s}^{⟨P s⟩} ⊗_{k} Inf_{⟨P s⟩/K}^{⟨P s⟩}Iso(ϕ^{−1})F_{Q,t}^{⟨Qt⟩} ≠ 0.

Therefore we have F_{P,s}^{⟨P s⟩} ∈ e_{Q,t}(⟨P s⟩) which implies that e_{P,s} ⩽ e_{Q,t}.

Now we also have

F_{Q,t}^{⟨Qt⟩} ⊗_{k} Iso(ϕ)Def_{⟨P s⟩/K}^{⟨P s⟩}F_{P,s}^{⟨P s⟩} ≠ 0

which implies that F_{Q,t}^{⟨Qt⟩} ∈ e_{P,s}(⟨Qt⟩). Therefore we have e_{Q,t} ⩽ e_{P,s} and so e_{Q,t} = e_{P,s} as desired. (ii) Since e_{P,s} ⩽ e_{Q,t}, we have F_{P,s}^{⟨P s⟩} ∈ e_{Q,t}(⟨P s⟩). Since (Q, t) is a D^{Δ}-pair, by the proof of Lemma 5.5, there exists a normal p′-subgroup K of ⟨P s⟩ such that (Q, t) ≃ (P K/K, sK).

Proposition 5.8. Let F be a nonzero subfunctor of 𝔽T^{Δ}. If H is a minimal group of F, then H = ⟨Qt⟩ for some D^{Δ}-pair (Q, t). Moreover we have

F (H) ⩽ ⨁_{\begin{matrix}(Q′,t′),D^{Δ}−pair \\ ⟨Q′t′⟩=H \end{matrix}} 𝔽F_{Q′,t′}^{H}

and e_{Q,t} ⩽ F.

In particular, if F = e_{Q,t} for some D^{Δ}-pair (Q, t), then we have

e_{Q,t}(⟨Qt⟩) = ⨁_{\begin{matrix}(Q′,t′)≃(Q,t) \\ ⟨Q′t′⟩=⟨Qt⟩ \end{matrix}} 𝔽F_{Q′,t′}^{H}.

Proof. Let F be a nonzero subfunctor of 𝔽T^{Δ} and assume H is a minimal group of F. Since F (H) ≠ 0, there exists a pair (Q, t) ∈ Q_{H,p} such that F_{Q,t}^{H} ∈ F (H). This implies, via the restriction map, that we have F_{Q,t}^{⟨Qt⟩} ∈ F (⟨Qt⟩). Since H is a minimal group, this implies that we have H = ⟨Qt⟩. Now if N is a normal p′-subgroup of ⟨Qt⟩, then Def_{⟨Qt⟩/N}^{⟨Qt⟩}F_{Q,t}^{⟨Qt⟩} = \frac{1}{|N|}F_{QN/N,tN}^{⟨Qt⟩/N} ≠ 0. Again since H is a minimal group this means that N is trivial and hence the pair (Q, t) is a D^{Δ}-pair. It follows moreover that

F (H) ⩽ ⨁_{\begin{matrix}(Q′,t′),D^{Δ}−pair \\ ⟨Q′t′⟩=H \end{matrix}} 𝔽F_{Q′,t′}^{H}.

For the last part, consider the functor e_{Q,t} for some D^{Δ}-pair (Q, t). If F_{Q′,t′}^{⟨Qt⟩} ∈ e_{Q,t}(⟨Qt⟩) for some D^{Δ}-pair (Q′, t′), then by the second part of Proposition 5.7, the pair (Q, t) is isomorphic to a p′-quotient of the pair (Q′, t′). But the pair (Q′, t′) is contained in ⟨Qt⟩. Thus we have (Q′, t′) ≃ (Q, t). Conversely, if the pairs (Q′, t′) and (Q, t) are isomorphic via a map ϕ, then we have F_{Q′,t′}^{⟨Qt⟩} = Iso(ϕ)F_{Q,t}^{⟨Qt⟩}. Therefore we have

e_{Q,t}(⟨Qt⟩) = ⨁_{\begin{matrix}(Q′,t′)≃(Q,t) \\ ⟨Q′t′⟩=⟨Qt⟩ \end{matrix}} 𝔽F_{Q′,t′}^{H}.

Let (P, s) be a pair and N a normal p′-subgroup of ⟨P s⟩. Then the pair (P N/N, sN) is a p′-quotient of the pair (P, s) and so by Proposition 5.7 we have e_{P,s} = e_{P N/N,sN}.

Proposition 5.9. Let (P, s) be a pair. Then the group ⟨P s⟩/C_{⟨s⟩}(P) is the unique, up to isomorphism, minimal group of the functor e_{P,s}. Moreover there is a unique isomorphism class of D^{Δ}-pairs (P′, s′) such that ⟨P′s′⟩ ≅ ⟨P s⟩/C_{⟨s⟩}(P) and we have e_{P′,s′} = e_{P,s}. Furthermore we have (P′, s′) ≃ (P C_{⟨s⟩}(P)/C_{⟨s⟩}(P), sC_{⟨s⟩}(P)).

Proof. Let (P′, s′) be a D^{Δ}-pair such that ⟨P′s′⟩ is a minimal group of the functor e_{P,s}. By Proposition 5.8, we have e_{P′,s′} ⩽ e_{P,s}. Let N := C_{⟨s⟩}(P). Then the pair (P N/N, sN) is a D^{Δ}-pair, and we have e_{P,s} = e_{P N/N,sN}. Since (P N/N, sN) is a D^{Δ}-pair, by Proposition 5.7 there exists a normal p′-subgroup K of ⟨P′s′⟩ such that (P′K/K, s′K) ≃ (P N/N, sN). This means that the idempotent F_{P′K/K,s′K}^{⟨P′s′⟩/K} is in the evaluation at ⟨P′s′⟩/K of the functor e_{P N/N,sN} = e_{P,s}. Since the group ⟨P′s′⟩ is a minimal group of e_{P,s} it follows that we must have K = 1. Thus we have (P′, s′) ≃ (P N/N, sN). Therefore we have e_{P′,s′} = e_{P N/N,sN} = e_{P,s}.

Now we will show the uniqueness of the isomorphism class of the minimal groups of e_{P,s}. Let H be a minimal group of e_{P,s}. It suffices to show that H is isomorphic to ⟨P′s′⟩. By Proposition 5.8 the group H is of the form H = ⟨Qt⟩ for some D^{Δ}-pair (Q, t). By the first part of the proof we have e_{Q,t} = e_{P,s} = e_{P′,s′}. Since both (Q, t) and (P, s) are D^{Δ}-pairs, the equality e_{Q,t} = e_{P′,s′} implies that (Q, t) is isomorphic to a p′-quotient of (P, s), and vice versa. Therefore we have (Q, t) ≃ (P′, s′) which implies that H = ⟨Qt⟩ ≅ ⟨P′s′⟩ as desired.

For any pair (P, s) we denote by (P̃, s̃) a representative of the isomorphism class of the pair (P C_{⟨s⟩}(P)/C_{⟨s⟩}(P), sC_{⟨s⟩}(P)).

Theorem 5.10. Let (P, s) be a pair.

(i) If (Q, t) is isomorphic to a p′-quotient of (P, s) and if (Q, t) is a D^{Δ}-pair, then (Q, t) is isomorphic to the pair (P̃, s̃). In particular, for any normal p′- subgroup N ⊴ ⟨P s⟩, we have (P N/N, sN) ≃ (P̃, s̃) if and only if (P N/N, sN) is a D^{Δ}-pair.

(ii) Let N ⊴ ⟨P s⟩ be a normal p′-subgroup. Then the pair (P̃, s̃) is isomorphic to a p′-quotient of (P N/N, sN) and we have (P̃, s̃) ≃ ({ \atop P N/N}, s̃N).

Proof. (i) Since the pair (Q, t) is isomorphic to a p′-quotient of the pair (P, s), by Proposition 5.7, we have e_{P̃,s̃} = e_{P,s} ⩽ e_{Q,t}. Since (Q, t) is a D^{Δ}-pair, again by Proposition 5.7, the pair (Q, t) is isomorphic to a p′-quotient of (P̃, s̃). But since the pair (P̃, s̃) is a D^{Δ}-pair, it follows that the pair (Q, t) is isomorphic to the pair (P̃, s̃).

(ii) Since the constant m_{P,s,N} is non-zero, we have F_{P N/N,sN}^{⟨P s⟩/N} ∈ e_{P,s}(⟨P s⟩/N) = e_{P̃,s̃}(⟨P s⟩/N). Therefore we have e_{P N/N,sN} ⩽ e_{P̃,s̃} and since (P̃, s̃) is a D^{Δ}-pair, by Proposition 5.7, (P̃, s̃) is isomorphic to a p′-quotient of (P N/N, sN). Again since the pair (P̃, s̃) is a D^{Δ}-pair, by part (i), it is isomorphic to the pair ({ \atop P N/N}, s̃N).

Let [D^{Δ}-pair] denote a set of isomorphism classes of D^{Δ}-pairs. Then the subfunctor lattice of the functor 𝔽T^{Δ} is isomorphic to the lattice of subsets of the set [D^{Δ}-pair] ordered by inclusion.

Theorem 5.11. Let S be the lattice of subfunctors of 𝔽T^{Δ} ordered by inclusion of subfunctors. Let T be the lattice of subsets of [D^{Δ}-pair] ordered by inclusion of subsets. Then the map

Θ : S → T

that sends a subfunctor F to the set {(P, s) ∈ [D^{Δ}-pair] : e_{P,s} ⩽ F }, is an isomorphism of lattices with inverse

Ψ : T → S

that sends a subset A to the functor ∑_{(P,s)∈A} e_{P,s}.

Proof. We need to show that the maps Θ and Ψ are inverse of each other. Let F ∈ S be a subfunctor. By Proposition 5.3 we have

F = ∑_{\begin{matrix}(P,s)∈Γ \\ e_{P,s}⩽F \end{matrix}} e_{P,s}

where Γ is a set of representatives of the isomorphism classes of pairs. But for any pair (P, s) we have e_{P,s} = e_{P̃,s̃} and (P̃, s̃) is a D^{Δ}-pair. Therefore we have

F = ∑_{\begin{matrix}(P,s)∈[D^{Δ}-pair] \\ e_{P,s}⩽F \end{matrix}} e_{P,s}.

This shows that Ψ(Θ(F)) = F.

Now let A ∈ T be a subset and let (Q, t) ∈ Θ(Ψ(A)) be a D^{Δ}-pair. Then we have e_{Q,t} ⩽ ∑_{(P,s)∈A} e_{P,s} and so by Proposition 5.4 this implies that we have e_{Q,t} ⩽ e_{P,s} for some (P, s) ∈ A. Since both (P, s) and (Q, t) are D^{Δ}-pairs, it follows that (P, s) ≃ (Q, t) and hence (Q, t) ∈ A. This shows that Θ(Ψ(A)) ⊆ A. The inclusion A ⊆ Θ(Ψ(A)) is trivial. Therefore we have Θ(Ψ(A)) = A.

The following corollary follows immediately from Theorem 5.11.

Corollary 5.12. We have 𝔽T^{Δ} = ⨁_{(P,s)∈[D^{Δ}-pair]} e_{P,s}.

The first statement of Proposition 5.8 can also be made stronger.

Corollary 5.13. Let F be a nonzero subfunctor of 𝔽T^{Δ}. If H is a minimal group of F, then H = ⟨Qt⟩ for some D^{Δ}-pair (Q, t) and we have

F (H) = ⨁_{\begin{matrix}(Q′,t′)≃(Q,t) \\ ⟨Q′t′⟩=⟨Qt⟩ \end{matrix}} 𝔽F_{Q′,t′}^{H}.

Proof. Since H is a minimal group of F, by Proposition 5.8 it follows that H = ⟨Qt⟩ for some D^{Δ}-pair with the property that e_{Q,t} ⩽ F. By Theorem 5.11 we have

F = ∑_{\begin{matrix}(Q,t)∈[D^{Δ}-pair] \\ e_{Q,t}⩽F \end{matrix}} e_{Q,t}.

Therefore by Proposition 5.8 again we have

F (H) = e_{Q,t}(H) = ⨁_{\begin{matrix}(Q′,t′)≃(Q,t) \\ ⟨Q′t′⟩=⟨Qt⟩ \end{matrix}} 𝔽F_{Q′,t′}^{H}

as desired.

Corollary 5.14. Let (P, s) be a D^{Δ}-pair. Then the subfunctor e_{P,s} of 𝔽T^{Δ} is iso-

morphic to the simple functor S_{⟨P s⟩,W_{P,s}} where W_{P,s} = ⊕\begin{matrix}(Q,t)≃(P,s) \\ ⟨Qt⟩=⟨P s⟩ \end{matrix}𝔽F_{P,s}^{⟨P s⟩}.

Proof. By Theorem 5.11 the lattice of subfunctors of e_{P,s} is isomorphic to the lattice of subsets of the set Θ(e_{P,s}) = {(Q, t) ∈ [D^{Δ}-pair] : e_{Q,t} ⩽ e_{P,s}} = {(P, s)}. Therefore the subfunctor e_{P,s} is simple. By Proposition 5.9 the group ⟨P s⟩ is a minimal group of the functor e_{P,s}. By Proposition 5.8 we have e_{P,s}(⟨P s⟩) = W_{P,s}. Moreover, by [5, Theorem 4.2.5], the module W_{P,s} is a simple module for the essential algebra E^{Δ}(⟨P s⟩). Thus we have e_{P,s} ≃ S_{⟨P s⟩,W_{P,s}} as desired.

Proposition 5.15. If F ⩽ F′ are subfunctors of 𝔽T^{Δ} such that F′/F is simple, then there exists a unique D^{Δ}-pair (P, s) ∈ [D^{Δ}-pair] such that e_{P,s} ⩽ F′ and e_{P,s} ⩽̸ F. In particular, we have e_{P,s} + F = F′, e_{P,s} ∩ F = {0}, and F′/F ≃ S_{⟨P s⟩,W_{P,s}}

Proof. The existence of a pair (P, s) with the property that e_{P,s} ⩽ F′ and e_{P,s} ⩽̸ F is clear. Suppose (P′, s′) is another pair with these properties. Since F′/F is simple, we have

(P′, s′) ∈ Θ(F) ∪ {(P, s)}.

Thus (P′, s′) ≃ (P, s) as (P′, s′) ∉ Θ(F). Now since e_{P,s} ⩽̸ F and F′/F is simple, we have e_{P,s} + F = F′. Thus the quotient e_{P,s}/(e_{P,s} ∩ F) ≃ F′/F is simple and so e_{P,s} ∩ F = {0}. Therefore we have F′/F ≃ S_{⟨P s⟩,W_{P,s}}.

Proposition 5.16. Let F ⩽ F′ be subfunctors of 𝔽T^{Δ} such that F′/F is simple. Let H (respectively H′) be a finite group and W (respectively W′) be a simple E^{Δ}(H)- mod (respectively E^{Δ}(H′)-mod) such that S_{H,W} ≃ S_{H′,W′} ≃ F′/F. Then H ≅ H′. Moreover W ≅ W′, after identification of H and H′ via the previous isomorphism.

Proof. By Proposition 5.15 there exists a unique D^{Δ}-pair (P, s) such that F′/F ≃ S_{⟨P s⟩,W_{P,s}}. Therefore it suffices to prove that H ≅ ⟨P s⟩. Since (F′/F)(H) ≠ 0 there exists a pair (Q, t) contained in H such that F_{Q,t}^{H} ∈ F′(H) \ F (H). Since H is a minimal group of F′/F, it follows that H = ⟨Qt⟩ and (Q, t) is a D^{Δ}-pair. Moreover we have e_{Q,t} ⩽ F′ and e_{Q,t} ⩽̸ F. But the pair (P, s) is the unique D^{Δ}-pair with these properties. Therefore we have (Q, t) ≃ (P, s). Thus H ≅ ⟨P s⟩ as desired. The last assertion follows from the fact that S_{H,W} (H) ≅ W.

Proposition 5.17. Let (P, s) be a pair. Then for any finite group H, the 𝔽-vector space e_{P,s}(H) is the subspace of 𝔽T (H) generated by the set of primitive idempotents F_{Q,t}^{H} where (Q, t) runs over a set of conjugacy classes of pairs in H with the property that (P, s) is isomorphic to a p′-quotient of (Q, t).

Proof. Since the pair (P̃, s̃) is isomorphic to a p′-quotient of the pair (P, s) and since e_{P,s} = e_{P̃,s̃}, we may assume that the pair (P, s) is a D^{Δ}-pair. Since e_{P,s}(H) is an ideal of 𝔽T (H), it has a 𝔽-basis consisting of a set of primitive idempotents F_{Q,t}^{H}. If F_{Q,t}^{H} ∈ e_{P,s}(H), then F_{Q,t}^{⟨Qt⟩} ∈ e_{P,s}(⟨Qt⟩) and so e_{Q,t} ⩽ e_{P,s}. Since (P, s) is a D^{Δ}-pair, by Proposition 5.7, it is isomorphic to a p′-quotient of the pair (Q, t). Conversely, if (P, s) is isomorphic to a p′-quotient of the pair (Q, t), then again by Proposition 5.7, we have e_{Q,t} ⩽ e_{P,s}. So we have F_{Q,t}^{⟨Qt⟩} ∈ e_{P,s}(⟨Qt⟩) and hence F_{Q,t}^{H} ∈ e_{P,s}(H). The result follows.

Theorem 5.18. Let (P, s) be a D^{Δ}-pair. Then for any finite group H, the 𝔽dimension of S_{⟨P s⟩,W_{P,s}}(H) is equal to the number of conjugacy classes of pairs (Q, t) in H such that (Q̃, t̃) ≃ (P, s).

Proof. By Proposition 5.17 e_{P,s}(H) is generated by the idempotents F_{Q,t}^{H} where (Q, t) is a pair in H with the property that the pair (P̃, s̃) ≃ (P, s) is isomorphic to a p′- quotient of the pair (Q, t). Since (P, s) is a D^{Δ}-pair, Theorem 5.10 implies that (Q̃, t̃) ≃ (P, s). The result follows.

Corollary 5.19. Let H be a finite group. The 𝔽-dimension of S_{1,𝔽}(H) is equal to the number of isomorphism classes of simple kH-modules.

Proof. By Theorem 5.18, dim_{𝔽}S_{1,𝔽}(H) is equal to the number of conjugacy classes of pairs (Q, t) in H such that (Q̃, t̃) ≃ (1, 1). Suppose (Q, t) is a pair with (Q̃, t̃) ≃ (1, 1). Then we have Q̃ = 1 and t̃ = 1. So there exists a normal p′-subgroup N of ⟨Qt⟩ such that (QN/N, tN) ≃ (1, 1). Since |Q| and |N| are coprime, this implies that Q = 1. We also have t ∈ N. But then N ⊴ ⟨t⟩ implies that N = ⟨t⟩. Therefore the number of conjugacy classes of pairs (Q, t) in H such that (Q̃, t̃) ≃ (1, 1) is equal to the number of conjugacy classes of p′-elements in H. The result follows.

Theorem 5.20. The functor S_{1,𝔽} is isomorphic to the functor that sends a finite group H to the subspace 𝔽K_{0}(kH) of 𝔽T^{Δ}(H) generated by the projective indecomposable kH-modules.

Proof. Let H be a finite group. We have

S_{1,𝔽}(H) = (𝔽T^{Δ}(H, 1) ⊗_{𝔽} 𝔽)/J_{1,𝔽}(H) ≅ 𝔽T^{Δ}(H, 1)/J_{1,𝔽}(H)

where J_{1,𝔽}(H) = {ϕ ∈ 𝔽T^{Δ}(H, 1) : ∀ψ ∈ 𝔽T^{Δ}(1, H), (ψ ∘ ϕ) · 1 = 0}. Now 𝔽T^{Δ}(H, 1) is isomorphic to the subspace 𝔽K_{0}(kH) of 𝔽T (H) generated by the isomorphism classes of projective indecomposable kH-modules. Similarly any W ∈ 𝔽T^{Δ}(1, H) can be identified with W^{∗} ∈ 𝔽K_{0}(kH). As in [6] we have the following: For any p-permutation kH-modules V and W we have

(W^{∗} ⊗_{kH} V) · 1 = dim_{k}(W^{∗} ⊗_{kH} V) = dim_{k}(Hom_{kH}(W, V)).

Therefore J_{1,𝔽}(H) is the right kernel of the bilinear form

< −, − >: 𝔽K_{0}(kH) → 𝔽

defined as < W, V >:= dim_{k}(Hom_{kH}(W, V)). But the matrix that represents this bilinear form is the Cartan matrix of kH. Since the Cartan matrix of a group algebra is non-degenerate, it follows that J_{1,𝔽}(H) = 0. Therefore we have

S_{1,𝔽}(H) = 𝔽T^{Δ}(H, 1) ⊗_{𝔽} 𝔽 ≅ 𝔽T^{Δ}(H, 1) ≅ 𝔽K_{0}(kH).

Note that both of these isomorphisms are functorial in H. The result follows.

[1] Boltje R., Xu B., “On p-permutation equivalences: between Rickard equivalences and isotypies”, Trans. Amer. Math. Soc. 360(10): 5067-5087, 2008.

[2] Bouc S., “Foncteurs d’ensembles munis d’une double action”, J. Algebra 183 (3): 664-736, 1996.

[3] Bouc S., Thévenaz J., “The primitive idempotents of the p-permutation ring”, J. Algebra 323(10): 2905-2915, 2010.

[4] Broué M., “On Scott modules and p-permutation modules: An approach through the Brauer morphism”, Proc. Amer. Math. Soc. 93(3): 401-408, 1985.

[5] Ducellier M., “Foncteurs de p-permutation” PhD Diss., http://www.lamfa.upicardie.fr/bouc/These-Ducellier.pdf, 2015.

[6] Ducellier M., “A study of a simple p-permutation functor”, J. Algebra 447: 367-382, 2016.

[7] Perepelitsky P., “p-permutation equivalences between blocks of finite groups” ProQuest LLC, Ann Arbor, MI, 2014. Thesis (Ph.D.) University of California, Santa Cruz.

[8] Rickard J., “Splendid equivalences: derived categories and permutation modules”, Proc. London Math Soc. (3) 72(2): 331-358, 1996.