MODULAR REPRESENTATIONS OF FINITE GROUPS WITH TRIVIAL RESTRICTION TO SYLOW SUBGROUPS

PAUL BALMER

Abstract. Let 𝕜 be a field of characteristic p. Let G be a finite group of order divisible by p and P a p-Sylow subgroup of G. We describe the kernel of the restriction homomorphism T (G) → T (P), for T (−) the group of endotrivial representations. Our description involves functions G → 𝕜^{×} that we call weak P -homomorphisms. These are generalizations to possibly non-normal P ≤ G of the classical homomorphisms G/P → 𝕜^{×} appearing in the normal case.

1. Introduction

Let 𝕜 be a field of characteristic p > 0, not necessarily algebraically closed. Let G be a finite group of order divisible by p. Although we are chiefly interested in the restriction to the p-Sylow subgroup, we can equally well describe the case of any subgroup H ≤ G whose index [G : H] is invertible in 𝕜, i.e. such that H contains a p-Sylow subgroup of G. Consider the kernel of restriction

T (G, H) := Ker ( Res_{H}^{G} : T (G) ⟶ T (H)) .

where we denote by T (G) = T_{𝕜}(G) the abelian group of endotrivial 𝕜G-modules. (See Remark 3.2.) Equivalently, T (G, H) is the group of stable isomorphism classes of those 𝕜G-modules whose restriction to H is isomorphic to the trivial representation 𝕜, up to projective summands (for such modules are necessarily endotrivial).

Endotrivial modules M are important for various reasons, the most obvious one being that, by definition, the functor M ⊗ − provides an auto-equivalence on the stable category 𝕜G - stab = \frac{𝕜G - mod}{𝕜G - proj}. But there are further reasons to study them, as well as the larger class of so-called endopermutation modules, for instance as Green sources of simple modules. We refer the reader to Thévenaz [16] for a survey and more motivation. Let us simply indicate that the study of such modules has played a major role in the development of modular representation theory over the last decades. To borrow Alperin’s words [1], a “triumph in finite group theory” has been their complete classification over p-groups, by work of Carlson-Thévenaz [7, 8] for endotrivial modules and Bouc [2] for endopermutation modules.

With this triumphant classification in mind, the natural question for a general group G becomes to compare T (G) to T (P) for a p-Sylow subgroup P ≤ G. This explains the importance of the kernel T (G, P) in general. An extensive literature has recently flourished around this question, see for instance [3, 4, 5, 6, 11, 12, 13, 15], usually with the objective of describing T (G, P) for specific classes of groups in very explicit terms (e.g. by generators and relations).

Date: April 24, 2012.

1991 Mathematics Subject Classification. 20C20.

Key words and phrases. Endotrivial module, trivial restriction to Sylow, weak homomorphism.

Research supported by NSF grant DMS-0969644.

The goal of the present paper is to give a description of this kernel T (G, H), valid for all G and H, in purely elementary terms, notably not using stable categories, nor representations, but essentially only the action of G by conjugation on the lattice of its p-subgroups.

In case the subgroup H ◁G is normal, it is well-known that T (G, H) amounts to one-dimensional representations of the quotient G/H, that is, to group homomorphisms G → 𝕜^{×} which are trivial on H. (Note that these coincide with all group homomorphisms G → 𝕜^{×} if H is the Sylow subgroup, since 𝕜 has no nontrivial p^{th} root of unity.) Our description of T (G, H) for arbitrary, not necessarily normal H ≤ G involves a generalization of these homomorphisms, that we call “weak Hhomomorphisms” from G to 𝕜^{×} (Definition 2.2). These are functions u : G → 𝕜^{×} which are constant on left and right H-cosets, which are trivial on H, and almost behave like group homomorphisms but not entirely. In fact, the relation u(g_{2} g_{1}) = u(g_{2}) u(g_{1}) only holds for some pairs g_{1}, g_{2} of elements of G. The deep reason why some of those relations are “lost” is that the stable category of the corresponding subgroup H ∩ H^{g_{1}} ∩ H^{g_{2}g_{1}} vanishes. This happens exactly when that subgroup has order prime to the characteristic p. We come back to this phenomenon in Remark 4.11.

We shall construct explicit isomorphisms, in both directions, between the kernel T (G, H) and the group A(G, H) of weak H-homomorphisms G → 𝕜^{×}.

It is high time we should give some precise definitions.

Beyond this introduction, the paper is organized as follows. In Section 2, we introduce weak H-homomorphisms and state the main theorems. These results are proved in Section 4 after recalling some basic modular representation theory in Section 3. The final Section 5 gives a couple of little corollaries of our description of T (G, H), e.g. about the possible orders of elements in that finite abelian group.

2. Weak H-homomorphisms and T (G, H)

Fix H ≤ G a subgroup of index prime to p, for instance a p-Sylow subgroup. The following simple definition will be important throughout the paper.

2.1. Definition. We say that an element g ∈ G is H-secant if the order |H ∩ H^{g}| is divisible by p, where of course H^{g} = g^{−1}Hg is the conjugate of H. In case H ≤ G is a p-Sylow, an element g ∈ G is H-secant if and only if H ∩ H^{g} is non-trivial.

2.2. Definition. Define a weak H-homomorphism from G to 𝕜^{×}, to be a function u : G → 𝕜^{×} satisfying the following three properties :

(WH1) For every h ∈ H, we have u(h) = 1. (WH2) For every non-H-secant g (i.e. |H ∩ H^{g}| prime to p), we have u(g) = 1. (WH3) For every g_{1}, g_{2} ∈ G such that |H ∩ H^{g_{1}} ∩ H^{g_{2}g_{1}}| is divisible by p, we have

u(g_{2}g_{1}) = u(g_{2}) · u(g_{1}) .

We define A(G, H) to be the abelian group of weak H-homomorphisms from G to 𝕜^{×}, under element-wise multiplication : (u · v)(g) = u(g) · v(g) for every g ∈ G.

2.3. Examples. Here are two extreme cases, where T (G, H) is already well-known.

(1) Suppose that H ◁ G is normal. Then every g ∈ G is H-secant and every pair g_{1}, g_{2} ∈ G satisfies (WH 3). Hence A(G, H) is the group Hom(G/H, 𝕜^{×}) of group homomorphisms from the quotient G/H to 𝕜^{×}.

(2) Suppose that H ≤ G is “strongly p-embedded”, meaning that for every g ∈ G not in H the subgroup H ∩ H^{g} has order prime to p. Then A(G, H) = 1 since (WH 1) and (WH 2) cover all possible g ∈ G and force u(g) = 1 everywhere.

Interestingly, the same group A(G, H) is isomorphic to T (G, H) in general, not only in those special cases. Let us explain how weak H-homomorphisms naturally appear in our problem. For this, it is convenient to use the following notation.

2.4. Remark. Let g ∈ G and let L and K be subgroups of G such that ^{g}L ≤ K. We can combine twisting the action and restriction to a subgroup to obtain a g-twisted restriction functor ^{g}Res_{L}^{K} : 𝕜K - mod ⟶ 𝕜L - mod. It is defined by ^{g}ResM = M as a 𝕜-vector space but with L acting via ℓ · m := (^{g}ℓ) m for all ℓ ∈ L. On morphisms, gRes_{L}^{K}(f) = f as usual. It induces a functor ^{g}Res_{L}^{K} : 𝕜K - stab ⟶ 𝕜L - stab on stable categories.

Here is a first relation between A(G, H) and endotrivial 𝕜G-modules.

2.5. Construction. Let M be an endotrivial 𝕜G-module such that Res_{H}^{G} M ≃ 𝕜 in the stable category 𝕜H - stab, i.e. the isomorphism class [M]_{≃} belongs to our kernel T (G, H). Choose an isomorphism ξ : 𝕜 →^{∼} Res_{H}^{G} M in 𝕜H - stab. Then, for every H-secant element g ∈ G, consider the following subgroup of H

(2.6)  H(g) := H ∩ H^{g},

whose order is divisible by p by the assumption that g is H-secant. Consider the two restrictions of M to H(g), namely the plain one Res_{H(g)}^{G} M and the g-twisted one gRes_{H(g)}^{G}M as in Remark 2.4. Note that m ↦ g m gives an H(g)-linear isomorphism Res_{H(g)}^{G} M →^{∼} ^{g}Res_{H(g)}^{G}M, simply denoted “ g· ”. Note also that ^{g}(H(g)) ≤ H and that ^{g}Res_{H(g)}^{H}𝕜 = Res_{H(g)}^{H} 𝕜 = 𝕜. Note finally that the group of automorphisms of 𝕜 in 𝕜H(g) - stab is exactly 𝕜^{×}, via multiplication. This is where we use the assumption that g is H-secant. Otherwise the stable category 𝕜H(g) - stab would be trivial. So, there exists a unique scalar, that we call u(g) ∈ 𝕜^{×}, which makes the following diagram commute in 𝕜H(g) - stab :

(2.7)  [figure: Res_{H(g)}^{G} M; g·; ≃; ^{g}Res_{H(g)}^{G} M; 𝕜; Res_{H(g)}^{H} ξ; ≃; ∃! u(g); 𝕜 .; _{g}Res_{H(g)}^{H}ξ; ≃]

We shall see that the scalar u(g) does not depend on the choice of ξ, nor on the isomorphism class of M. Extending u to non-H-secant g by setting u(g) = 1, we shall see that u : G → 𝕜^{×} is a weak H-homomorphism in the sense of Definition 2.2. We denote this weak H-homomorphism u by υ(M).

This construction actually gives us everything :

2.8. Theorem. Construction 2.5 induces a well-defined isomorphism

υ : Ker (T (G) → T (H)) ⟶^{∼} A(G, H).

The proof is given in Section 4. This first construction explains how weak Hhomomorphisms enter the picture. Let us now give the announced homomorphism A(G, H) ⟶ T (G) more concretely. That is, let us describe what is the endotrivial module corresponding to a weak H-homomorphism. This will yield an inverse to υ.

For g ∈ G, we denote by [g] the class gH of g in the quotient G/H. We shall see that every weak H-homomorphism u : G → 𝕜^{×} is constant on H-classes hence u([g]) := u(g) is well-defined on G/H. See Remark 4.2 (2).

2.9. Theorem. Consider the 𝕜G-module 𝕜(G/H) with usual left G-action on its 𝕜-basis G/H. Let u : G → 𝕜^{×} be a weak H-homomorphism (Definition 2.2). Define a map e_{u} : 𝕜(G/H) → 𝕜(G/H), depending on u, by the formula

(2.10)  e_{u}([g]) = \frac{1}{[G : H]} ∑ ; d ∈ G/H u(d)^{−1} · g · d

for every [g] ∈ G/H, extended 𝕜-linearly as always. Then we have :

(i) The homomorphism e_{u} : 𝕜(G/H) → 𝕜(G/H) is well-defined and 𝕜G-linear. Moreover, it is an idempotent e_{u} ∘ e_{u} = e_{u} in the stable category 𝕜G - stab.

(ii) Since the category 𝕜G - stab is idempotent complete (Remark 3.1), there exists a unique decomposition 𝕜(G/H) ≅ M_{u} ⊕ N_{u} in 𝕜G - stab, such that e_{u} is the projection on M_{u} along N_{u}; in other words, e_{u} becomes ( \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} ) on M_{u} ⊕ N_{u}.

(iii) The object M_{u} is endotrivial and its restriction to H is trivial. (iv) The construction u ↦ [M_{u}]_{≃} described above yields a well-defined group ho-

momorphism A(G, H) ⟶ T (G) which gives an isomorphism

α : A(G, H) ⟶^{∼} Ker (T (G) → T (H)) ,

inverse to the isomorphism υ of Theorem 2.8.

This Theorem is proven simultaneously with Theorem 2.8, in Section 4.

2.11. Remark. We can try to reduce the amount of information involved in describing a weak H-homomorphism u : G → 𝕜^{×}. Here is an alternate formulation which might be interesting for subgroups H ≤ G with small double quotient H \G/H. First note that g ∈ G being H-secant is a well-defined property of the class of g in G/H or even in H \G/H. Of course, we call such classes H-secant as well. We already mentioned that u ∈ A(G, H) is constant on left and right H-cosets (see Remark 4.2). It follows that we could describe our weak H-homomorphisms u ∈ A(G, H) as functions u : H \G/H ⟶ 𝕜^{×} such that u(H) = 1, u(c) = 1 if c is not H-secant and u(c_{3}) = u(c_{2}) · u(c_{1}) each time c_{1}, c_{2} and c_{3} are the classes of some elements g_{1}, g_{2} and g_{2}g_{1} for which H ∩ H^{g_{1}} ∩ H^{g_{2}g_{1}} has order divisible by p. The latter condition, however, seems to depend on the choice of g_{i} ∈ c_{i} for i = 1, 2. This is why we prefer formulation (WH 3), to avoid confusion.

3. Basics

We recall some standard facts about modular representation theory of finite groups and fix some notation. In this section, H ≤ G can be any subgroup, not necessarily of index prime to p.

We denote by 𝕜G - mod the category of finitely generated left 𝕜G-modules and by 𝕜G - stab = 𝕜G - mod /𝕜G - proj the stable category obtained as the additive quotient of the Frobenius abelian category 𝕜G - mod by its subcategory of projective (=injective) modules. See Happel [10] for details. It is a triangulated category but, sadly enough, we shall not use this fact in this paper. The usual tensor product of representations, M ⊗ N = M ⊗_{𝕜} N with diagonal G-action, passes to the stable category 𝕜G - stab.

3.1. Remark. An additive category is idempotent complete (a. k. a. karoubian or pseudo-abelian) if every idempotent endomorphism e = e^{2} : A → A yields a decomposition A = im(e) ⊕ ker(e), that is, a decomposition under which e becomes ( \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} ). Such decomposition is unique up to unique isomorphism.

The stable category 𝕜G - stab is idempotent complete. This fact is folklore, e.g. since 𝕜G - stab coincides with the thick subcategory of compact objects in the big stable category 𝕜G - Stab of all, not necessarily finitely generated, 𝕜G-modules modulo projectives and since 𝕜G - Stab has infinite coproducts hence is idempotent complete (use [14, Prop. 1.6.8]).

Alternatively, one can sketch a direct proof as follows. Let M be a finitely generated 𝕜G-module that we can assume without projective summand. The latter assumption implies that any endomorphism of M which vanishes in 𝕜G - stab is nilpotent (using nilpotence of the Jacobson radical). Hence, if e : M → M is an endomorphism in 𝕜G - mod such that e^{2} = e in 𝕜G - stab, then h = e^{2}−e is nilpotent in 𝕜G - mod. The usual lifting of idempotents modulo nilpotents yields a correction ẽ of e such that ẽ^{2} = ẽ in 𝕜G - mod already, with the same ẽ = e in 𝕜G - stab. (By induction, reduce to the case h^{2} = 0 and verify that ẽ = e + h − 2 eh will do in that case; note that eh = he.) Then we have M = im(ẽ) ⊕ ker(ẽ) in 𝕜G - mod, inducing the wanted decomposition in 𝕜G - stab.

3.2. Remark. Recall that a 𝕜G-module M is endotrivial if M is ⊗-invertible M ⊗ M^{∗} ≃ 𝟙 in the stable category 𝕜G - stab, or, in eponymic terms, if its module of endomorphisms is the trivial 𝕜G-module, up to projective : End_{𝕜}(M) ≃ 𝕜 ⊕ (proj). Being endotrivial can be tested on the elementary abelian p-subgroups of G by Chouinard’s Theorem [9]. In particular, it suffices that Res_{P}^{G} M be endotrivial on the p-Sylow P ≤ G or any subgroup in between P ≤ H ≤ G. (See also Remark 3.4.)

3.3. Notation. For a subgroup H ≤ G, we denote the class gH ∈ G/H by [g]_{H} or just [g] when H is obvious from the context. Here, 𝕜(G/H) will always be a left 𝕜G-module via g · [x] = [gx]. We have the usual restriction-induction adjunction

Res_{H}^{G} : 𝕜G - stab ⇆ 𝕜H - stab : Ind_{H}^{G} = 𝕜G ⊗_{𝕜H} −

whose unit η_{M} : M → 𝕜G ⊗_{𝕜H} M is given by m ↦ ∑_{[g] ∈ G/H} g ⊗ g^{−1}m.

3.4. Remark. Assume that a subgroup H ≤ G has index [G: H] prime to p. Then the above unit η : Id → Ind_{H}^{G} Res_{H}^{G} has a retraction π : Ind_{H}^{G} Res_{H}^{G} → Id, namely π_{M} : 𝕜G ⊗_{𝕜H} M → M defined by g ⊗ m ↦ \frac{1}{[G:H]} g m. It follows that the functor Res_{H}^{G} : 𝕜G - stab → 𝕜H - stab is faithful. Indeed, if Res(f) = 0 then f = π ∘ η ∘ f = π ∘ Ind Res(f) ∘ η = 0 by naturality of η. (Of course, Res is usually not full.) Then Res_{H}^{G} : 𝕜G - stab → 𝕜H - stab detects vanishing of objects. Applying this property with the cone (cokernel) of the obvious morphism 𝕜 → End_{𝕜}(M) ≅ M ⊗ M^{∗}, we see that if Res_{H}^{G}(M) is ⊗-invertible then so is M. Hence the functor Res_{H}^{G} detects endotriviality, as already mentioned. More generally, applying faithfulness to the cone of any morphism in 𝕜G - stab shows that Res_{H}^{G} detects isomorphisms.

3.5. Remark (Mackey formulas). Let K, L ≤ H be two subgroups of the same group H. We shall use a couple of Mackey bijections between some left K-sets and some Mackey isomorphisms between left 𝕜K-modules. We therefore recall them together beforehand. Let T ⊂ H be a set of representatives of K\H/L. For every t ∈ T, consider the morphism of left H-sets

mack_{t} : H/^{t}L ⟶ H/L  defined by  mack_{t} ([ h]_{^{t}L}) = [ ht]_{L} .

One instance of the Mackey formula tells us that the map obtained by restricting these maps to the subsets K/(K ∩^{t}L) of H/^{t}L and taking their coproduct over t ∈ T yields a bijection of left K-sets :

(3.6) mack : ∐_{t∈T} K/(K ∩^{t}L) ⟶^{∼} H/L ,  [k]_{K∩^{t}L} ⟼ mack_{t}([k]) = [kt]_{L} .

Similarly, for every 𝕜L-module N, there is a Mackey isomorphism of 𝕜K-modules, ⨁_{t∈T} Ind_{K∩^{t}L}^{K} Res^{t^{−1}}_{K∩^{t}L}^{L} N ⟶^{∼} Res_{K}^{H} Ind_{L}^{H}(N), still denoted mack and given by

(3.7)  [figure: mack : ⨁_{t∈T} 𝕜K ⊗_{𝕜(K∩^{t}L)} N; ≃; 𝕜H ⊗_{𝕜L} N; x ⊗ y; xt ⊗ y .]

Recall from Remark 2.4 that by definition of the twisted restriction Res^{t^{−1}}_{K∩^{t}L}^{L} N, each subgroup K ∩^{t}L acts on the corresponding factor N appearing on the left-hand side of (3.7) via k · n = k^{t} n, observing that k^{t} ∈ L for every k ∈ K ∩^{t}L.

Let H be a finite group H, let X be a left H-set and let x ∈ X. Then we denote by H_{x} = { h ∈ H | hx = x } the stabilizer of x in H. The following result will be essential for our computations in stable categories.

3.8. Lemma. Let H be a finite group and let X, Y be two finite left H-sets. Let f : 𝕜X → 𝕜Y be a 𝕜H-linear homomorphism. It is given by scalars a_{x,y} ∈ 𝕜 indexed by x ∈ X and y ∈ Y such that f(x) = ∑_{y∈Y} a_{x,y} y for every x ∈ X.

If a_{x,y} = 0 for every x ∈ X and y ∈ Y such that p divides |H_{x} ∩ H_{y}| then the morphism f is zero in 𝕜H - stab, i.e. it factors via a projective 𝕜H-module.

Proof. The 𝕜H-linearity of f gives us for every h ∈ H, x ∈ X and y ∈ Y that

(3.9)  a_{hx,hy} = a_{x,y} .

Now, consider the diagonal action of H on X × Y and note that the property that |H_{x} ∩H_{y}| is prime to p is constant on the H-orbit of (x, y) ∈ X ×Y. Let S ⊂ X ×Y be a set of representatives of only those H-orbits in X × Y on which |H_{x} ∩ H_{y}| is prime to p. Consider the free 𝕜H-module Ind_{1}^{H} Res_{1}^{H}(𝕜Y) = 𝕜H ⊗_{k} 𝕜Y (that is with H-action only on the left factor) and the morphisms f_{1} and f_{2} as follows :

[figure: x ∈ X; 𝕜X; f; f_{1}; ∑_{h∈H} ∑_{{(x_{0},y_{0})∈S \atop s.t. x=h x_{0}}} |H_{x_{0}} ∩ H_{y_{0}}|^{−1} a_{x_{0},y_{0}} · h ⊗ y_{0}; 𝕜H ⊗_{k} 𝕜Y; f_{2}; 𝕜Y; h ⊗ y; h y .]

The 𝕜H-linearity of f_{2} is immediate and that of f_{1} is easy by a standard change of variables on the summation index h ∈ H. It now suffices to check that f_{2} ∘ f_{1} = f. Let x ∈ X and let b_{x,y} ∈ 𝕜 for all y ∈ Y be such that f_{2} ∘ f_{1}(x) = ∑_{y∈Y} b_{x,y} y. By the above construction, we have

b_{x,y} = ∑ ; h ∈ H ∑ ; {(x_{0}, y_{0}) ∈ S \atop x = h x_{0} and y = h y_{0}} |H_{x_{0}} ∩ H_{y_{0}}|^{−1} a_{x_{0},y_{0}} .

We want to prove that b_{x,y} = a_{x,y} for all y ∈ Y. If p divides |H_{x} ∩ H_{y}|, there is no (x_{0}, y_{0}) ∈ S with (x, y) = (h x_{0} , h y_{0}) by choice of S, hence the above summation is empty in that case and we get b_{x,y} = 0 which coincides with a_{x,y} by hypothesis. Now, suppose that |H_{x} ∩ H_{y}| is prime to p, then the index (x_{0}, y_{0}) ∈ S of the above sum is unique. However, there are many h ∈ H with the property that (h x_{0}, h y_{0}) = (x, y), namely there are |H_{x_{0}} ∩ H_{y_{0}}| of them. In that case, we obtain b_{x,y} = a_{x_{0},y_{0}} but the latter is also equal to a_{x,y} by (3.9) since (x, y) = (hx_{0}, hy_{0}). □

4. Proof of the Theorems

As in Section 2, H ≤ G is a subgroup of index prime to p. Let us denote this index by n := [G: H] for short. So, \frac{1}{n} exists in our field 𝕜. Recall that g ∈ G is H-secant if p divides the order of H(g) := H ∩ H^{g}.

It might be reassuring to start with the following.

4.1. Example. Let u ≡ 1 be the trivial weak H-homomorphism from G to 𝕜^{×}. Then, under the identification 𝕜(G/H) ≅ Ind_{H}^{G} Res_{H}^{G}(𝕜) = 𝕜G ⊗_{𝕜H} 𝕜, given by [g] ↦ g ⊗ 1, the endomorphism e_{u} of 𝕜(G/H), see (2.10), coincides with the idempotent η_{𝕜} ∘ π_{𝕜} of Ind Res(𝕜), see Remark 3.4. Indeed, for every g ∈ G, we have e_{1}([g]) = \frac{1}{n} ∑_{d∈G/H} gd = \frac{1}{n} ∑_{d′∈G/H} d′ = η_{𝕜}(\frac{1}{n}) = η_{𝕜}(π_{𝕜}([g])). So this idempotent corresponds to the trivial module 𝕜 appearing as a direct summands of 𝕜(G/H) via η_{𝕜}.

4.2. Remarks. Let u : G → 𝕜^{×} be a weak H-homomorphism (Definition 2.2). We shall repeatedly use the following facts, often without mention.

(1) We have u(g^{−1}) = u(g)^{−1} for every g ∈ G. Indeed, by (WH 2), we can assume that g, or equivalently g^{−1}, is H-secant. Then 1 = u(1) = u(g^{−1}) u(g) follows by (WH 1) and (WH 3) since p divides |H ∩ H^{g} ∩ H^{1}|.

(2) For every g ∈ G and h ∈ H, we have u(hg) = u(g) = u(gh). To see this, note that g, hg and gh are simultaneously H-secant. By (WH 2), we can assume that they are all H-secant. In that case, the groups H ∩ H^{g} ∩ H^{hg} = H ∩ H^{g} and H ∩H^{h} ∩H^{gh} = H ∩H^{gh} have order divisible by p, and the relations follow from (WH 3) and the fact that u(h) = 1 by (WH 1).

4.3. Proposition. Let u : G → 𝕜^{×} be a weak H-homomorphism (Definition 2.2). Then the endomorphism e_{u} : 𝕜(G/H) → 𝕜(G/H) given in (2.10) is well-defined and 𝕜G-linear.

Proof. Let g ∈ G and consider the well-defined element

e_{u}(g) = \frac{1}{n} ∑_{d ∈ G/H} u(d)^{−1} · g · d

in 𝕜(G/H), as in (2.10). We want to show that e_{u}(g) = e_{u}(gh) for every h ∈ H:

e_{u}(gh) = \frac{1}{n} ∑_{d ∈ G/H} u(d)^{−1} g h d = \frac{1}{n} ∑_{d′ ∈ G/H} u(d′)^{−1} g d′ = e_{u}(g)

using the change of variables d′ = h d on G/H which preserves the value of the scalar u(d′) = u(hd) = u(d) by Remark 4.2 (2). Hence, e_{u}(g) only depends on [g]_{H} ∈ G/H, which means that e_{u} is well-defined. It is clearly 𝕜G-linear by definition of the action of G on G/H, which appears on the left of g (and of course commutes with the scalar u(d)^{−1} ∈ 𝕜). □

4.4. Main Lemma. Let u ∈ A(G, H) and \underline{u} : Res_{H}^{G}(𝕜(G/H)) → 𝕜 its 𝕜-linear extension, i.e. mapping every basis element c ∈ G/H to u(c). It is 𝕜H-linear. Consider the 𝕜H-linear homomorphism w : 𝕜 → Res_{H}^{G}(𝕜(G/H)) given by

(4.5)  w(1) = \frac{1}{n} ∑_{d∈G/H} u(d)^{−1} · d .

Then we have \underline{u} ∘ w = id_{𝕜} and w ∘ \underline{u} = Res_{H}^{G}(e_{u}) in 𝕜H - stab.

Proof. It is easy to verify that both \underline{u} and w are indeed 𝕜H-linear (see Remark 4.2 (2) if necessary). Let us also observe right away that \underline{u} ∘ w = id_{𝕜}:

\underline{u}(w(1)) = \frac{1}{n} ∑_{d∈G/H} u(d)^{−1} · u(d) = \frac{|G/H|}{n} = 1 .

So, let us prove that w ∘ \underline{u} = e_{u} in 𝕜H - stab. We are going to use Lemma 3.8 for the left H-set X = Y = G/H = Res_{H}^{G}(G/H) and the morphism f = w ∘ \underline{u} − e_{u} : 𝕜X → 𝕜Y, which we claim is zero in 𝕜H - stab. For every x ∈ G/H, we have

w ∘ \underline{u} (x) = \frac{1}{n} ∑_{y∈Y} u(x) u(y)^{−1} · y .

On the other hand, let us choose g_{1} ∈ G such that x = [g_{1}^{−1}]_{H}. Then we have

e_{u}(x) = \frac{1}{n} ∑_{d ∈ G/H} u(d)^{−1} · g_{1}^{−1} · d = \frac{1}{n} ∑_{y∈Y} u(g_{1} y)^{−1} · y

using the change of variables d = g_{1} y on Y = G/H. By Lemma 3.8, in the above expressions for w∘\underline{u}(x) and e_{u}(x), it suffices to identify the coefficients of only those y ∈ Y such that p divides |H_{x} ∩ H_{y}|. But here the stabilizers are H_{x} = H ∩^{x}H and H_{y} = H ∩^{y}H. So we can assume that y ∈ Y is such that p divides |H ∩^{x}H ∩^{y}H|. Choose g_{2} ∈ G be such that [g_{2}^{−1}]_{H} = g_{1} y. Note that then [(g_{2}g_{1})^{−1}]_{H} = y. Since p divides |H ∩^{x}H ∩^{y}H| = |H ∩ H^{g_{1}} ∩ H^{g_{2}g_{1}}|, property (WH 3) of u together with Remark 4.2 give us

u(y)^{−1} = u(g_{2}g_{1}) = u(g_{2}) u(g_{1}) = u(g_{1}y)^{−1}u(x)^{−1}

hence the wanted u(x) u(y)^{−1} = u(g_{1} y)^{−1}. □

4.6. Proposition. For every weak H-homomorphism u : G → 𝕜^{×}, we have an idempotent e_{u} ∘ e_{u} = e_{u} on the object 𝕜(G/H) of 𝕜G - stab. Moreover, the corresponding direct summand M_{u} of 𝕜(G/H) restricts to 𝕜 on 𝕜H - stab, that is Res_{H}^{G}(M_{u}) ≃ 𝕜. Finally M_{u} is an endotrivial 𝕜G-module.

Proof. Since the functor Res_{H}^{G} : 𝕜G - stab → 𝕜H - stab is faithful (Remark 3.4), it suffices to verify (Res(e_{u}))^{2} = Res(e_{u}) in 𝕜H - stab, where it follows directly from our Main Lemma 4.4 since Res(e_{u}) = w ∘ \underline{u} and \underline{u} ∘ w = id_{𝕜} in 𝕜H - stab :

[figure: 𝕜; w; \underline{u}∘w=id; 𝕜(G/H); \underline{u}; w∘\underline{u}=e_{u}]

The direct summand of the object Res_{H}^{G}(𝕜(G/H)) = 𝕜(G/H) of 𝕜H - stab corresponding to the idempotent w ∘ \underline{u} is simply 𝕜. This proves the second claim (see Remark 3.1). Finally, Res_{H}^{G} being faithful, it detects endotriviality by Remark 3.4 (or Chouinard’s Theorem [9]). So, M_{u} is an endotrivial 𝕜G-module. □

4.7. Remark. Here is another approach to the idempotent e_{u}. By (WH 2), the value u(g) is only interesting when g ∈ G is H-secant. Similarly, the only g_{1}, g_{2} satisfying the hypothesis of (WH 3) must be H-secant, and so must be g_{2}g_{1} (easy exercise). Continuing in this vein, we can use Lemma 3.8 and Remark 3.4 to show that the endomorphism e_{u} of 𝕜(G/H) is equal in 𝕜G - stab to

[g]_{H} ↦ \frac{1}{[G : H]} ∑ ; {d ∈ G/H \atop H-secant} u(d)^{−1} · g · d .

This formula makes it apparent that only the values u(d) for H-secant d are relevant.

We now turn to the other side of the game, namely the construction of the homomorphism υ : T (G, H) → A(G, H).

4.8. Remark. For g ∈ G and subgroups K, L ≤ G such that ^{g}L ≤ K, recall the twisted restriction ^{g}Res_{L}^{K} : 𝕜K - stab → 𝕜L - stab as in Remark 2.4. When ^{g_{1}}L ≤ K and ^{g_{2}}K ≤ H then ^{g_{2}g_{1}}L ≤ H and it is easy to check that we have an equality of functors ^{g_{2}g_{1}}Res_{L}^{H} = ^{g_{1}}Res_{L}^{K} ∘ ^{g_{2}}Res_{K}^{H}. (Watch the order of the g_{i}!) The latter equality will often be used tacitly in the sequel.

Recall Construction 2.5, which associates a function u : G → 𝕜^{×} to every 𝕜Gmodule M coming with an isomorphism ξ : 𝕜 →^{∼} Res_{H}^{G} M in 𝕜H - stab. For convenience, we repeat diagram (2.7) whose commutativity defines u(g) for every H-secant g ∈ G (recall that H(g) stands for H ∩ H^{g}) :

(4.9)  \xymatrix{Res_{H(g)}^{G} M \ar[r]^{g·}_{≃} & ^{g}Res_{H(g)}^{G} M \\ 𝕜 \ar[u]^{Res_{H(g)}^{H} ξ}_{≃} \ar[r]_{u(g)} & 𝕜 \ar[u]_{_{g}Res_{H(g)}^{H}ξ}^{≃}}

For non-H-secant g ∈ G, we defined u(g) = 1. Note however that (4.9) also trivially commutes in that case since 𝕜H(g) - stab = 0 for non-H-secant g.

4.10. Lemma. The scalar u(g) as in (4.9) does not depend on the choice of the isomorphism ξ. Moreover, u : G → 𝕜^{×} is a weak H-homomorphism.

Proof. For the first part, we can assume that g is H-secant, for otherwise u(g) = 1 anyway. Let ξ′ : 𝕜 →^{∼} Res_{H}^{G}(M) be another isomorphism and u′(g) the associated scalar, i.e. u′(g) = (^{g}Res(ξ′))^{−1} ∘ (g·) ∘ Res(ξ′). Then ξ^{−1}ξ′ : 𝕜 →^{∼} 𝕜 is an automorphism of 𝕜 in 𝕜H - stab, hence it is given by an invertible scalar v ∈ 𝕜^{×}. So, we have ξ′ = v · ξ = ξ · v. Hence v and v^{−1} cancel out in u′(g) giving u′(g) = u(g).

Let us now check that u is a weak H-homomorphism as in Definition 2.2. To check (WH 1), it suffices to use that ξ is H-linear and that 𝕜 has trivial action. Indeed, for every h ∈ H (necessarily H-secant) we have a commutative square

\xymatrix{Res_{H(h)}^{G} M \ar[r]^{h·}_{≃} & ^{H}Res_{H(h)}^{G} M \\ 𝕜 \ar[u]^{Res_{H(h)}^{H} ξ}_{≃} \ar[r]^{h·}_{=id} & 𝕜 . \ar[u]_{_{h}Res_{H(h)}^{H}ξ}^{≃}}

This proves u(h) = 1. Property (WH 2) holds by construction. Let us verify (WH 3). Suppose that g_{1} and g_{2} are such that the subgroup L := H∩H^{g_{1}} ∩H^{g_{2}g_{1}} has order divisible by p. This means that 𝕜L - stab is not the zero category and endomorphisms of 𝕜 in that category identify with 𝕜 (under multiplication, as usual). Consider the following commutative diagram in 𝕜L - stab :

[figure: Res_{L}^{G} M; g_{1}·; ≃; (g_{2}g_{1})·; ≃; ^{g_{1}}Res_{L}^{G} M; g_{2}·; ≃; ^{g_{2}g_{1}}Res_{L}^{G} M; 𝕜; Res_{L}^{H} ξ; ≃; u(g_{1}); ≃; u(g_{2}g_{1}); ≃; 𝕜; _{g_{1}} Res_{L}^{H}ξ; ≃; u(g_{2}); ≃; 𝕜 .; _{g_{2}g_{1}} Res_{L}^{H}ξ; ≃]

The left-hand central square commutes by (4.9) for g_{1}, restricted to L. The righthand one commutes by (4.9) for g_{2} after applying ^{g_{1}}Res_{L}^{H(g_{2})} to it, using that g_{1}L ≤ H ∩ H^{g_{2}} = H(g_{2}) and the relation ^{g_{2}g_{1}}Res_{L}^{H} = ^{g_{1}}Res_{L}^{H(g_{2})} ∘ ^{g_{2}}Res_{H(g_{2})}^{H} already seen in Remark 4.8. The outside square commutes by (4.9) again but now for g_{2} g_{1}. Hence the lower “triangle” gives us u(g_{2}g_{1}) = u(g_{2}) u(g_{1}) as wanted. □

4.11. Remark. In the above proof, it is essential that L = H ∩H^{g_{1}} ∩H^{g_{2}g_{1}} has order divisible by p to deduce from the relation u(g_{2}g_{1}) = u(g_{2}) u(g_{1}) in End_{𝕜L - stab}(𝕜) that the same relation holds in 𝕜. When L has order prime to p this stable endomorphism ring is trivial and 𝕜 → End_{𝕜L - stab}(𝕜) is not injective. This is why the “homomorphism property of u”, u(g_{2}g_{1}) = u(g_{2}) u(g_{1}), does not hold for general g_{1} and g_{2} and why we are left with weak homomorphisms as in Definition 2.2 (WH 3).

4.12. Proposition. The assignment M ↦ u as in Construction 2.5 yields a welldefined group homomorphism υ : T (G, H) ⟶ A(G, H).

Proof. Suppose that M′ is isomorphic to M in 𝕜G - stab and let u = υ(M) and u′ = υ(M′). For g non-H-secant, we have u(g) = 1 = u′(g) by definition. So, let g ∈ G be H-secant. There exists a 𝕜G-linear morphism f : M → M′ such that f is an isomorphism in 𝕜G - stab, hence also in 𝕜H - stab after restriction. We can then create the following commutative cube in 𝕜H(g) - stab :

[figure: Res_{H(g)}^{G} M; g·; ≃; Res(f)=f; ≃; ^{g}Res_{H(g)}^{G} M; _{g}Res(f)=f; ≃; Res_{H(g)}^{G} M′; g·; ≃; _{g}Res_{H(g)}^{G} M′; 𝕜; Res_{H(g)}^{H} ξ; ≃; u(g); 𝕜; _{g}Res_{H(g)}^{H}ξ; ≃; 𝕜; Res_{H(g)}^{H} ξ′; ≃; u′(g); 𝕜; _{g}Res_{H(g)}^{H}ξ′; ≃]

whose back and front squares are (4.9) for M and M′ respectively. To compute u′(g), we choose ξ′ : 𝕜 →^{∼} Res_{H}^{G} M′ in 𝕜H - stab to be Res(f) ∘ ξ, which is allowed by Lemma 4.10. This makes the side squares commute. The top square commutes by 𝕜G-linearity of f. Hence the bottom square commutes, which shows that u is independent of the isomorphism class of M in 𝕜G - stab.

Finally, for i = 1, 2, let M_{i} be a 𝕜G-module, let ξ_{i} : 𝕜 →^{∼} Res_{H}^{G} M_{i} be an isomorphism in 𝕜H - stab and let u_{i} := υ(M_{i}). To compute υ(M_{1} ⊗M_{2}), we can use the isomorphism ξ_{1} ⊗ξ_{2} : 𝕜 →^{∼} Res_{H}^{G}(M_{1} ⊗M_{2}) in 𝕜H - stab. Again, we can assume that g ∈ G is H-secant. Tensoring the two commutative squares (4.9) defining u_{1}(g) and u_{2}(g), we get the following commutative diagram in 𝕜H(g) - stab :

\xymatrix{Res_{H(g)}^{G} M_{1} ⊗ M_{2} \ar[r]^{(g·)⊗(g·)}_{≃} & ^{g}Res_{H(g)}^{G} M_{1} ⊗ M_{2} \\ 𝕜 \ar[u]^{Res_{H(g)}^{H} ξ_{1}⊗ξ_{2}}_{≃} \ar[r]_{u_{1}(g)·u_{2}(g)} & 𝕜 . \ar[u]_{_{g}Res_{H(g)}^{H}ξ_{1}⊗ξ_{2}}^{≃}}

Now observe that the multiplication g· on the 𝕜G-module M_{1} ⊗ M_{2} → M_{1} ⊗ M_{2} is precisely given by m_{1} ⊗ m_{2} ↦ (g m_{1}) ⊗ (g m_{2}). So, it coincides with the top morphism in the above square. Hence u_{1}(g)·u_{2}(g) must be equal to υ(M_{1} ⊗M_{2}) (g) by (4.9) again, but this time applied to M_{1} ⊗ M_{2}. □

We now need a small result, which palliates the lack of multiplicativity of weak H-homomorphisms.

4.13. Lemma. Let u : G → 𝕜^{×} be a weak H-homomorphism. Let g ∈ G be H-secant and let s ∈ G. Let us abbreviate H(g, s) := H ∩ H^{g} ∩^{s}H. Then, in 𝕜

[H(g) : H(g, s)] · u(gs) = [H(g) : H(g, s)] · u(g) · u(s) .

Proof. The result is trivial if p divides the number [H(g) : H(g, s)] which appears on both sides. So, we can assume that p does not divide that index. But since g is H-secant, p divides |H(g)|. These two facts force p to divide |H(g, s)| = |H ∩ H^{g} ∩ H^{s^{−1}}| = |H^{s} ∩ H^{gs} ∩ H|. Hence, by (WH 3), for g_{2} = g and g_{1} = s, we have u(gs) = u(g) · u(s) and the result also holds in that case. □

4.14. Proposition. Let u : G → 𝕜^{×} be a weak H-homomorphism and M_{u} the associated endotrivial 𝕜G-module, as in Proposition 4.6. Then υ(M_{u}) = u.

Proof. By construction of M_{u}, there exist a 𝕜G-module N_{u} and a 𝕜G-linear homomorphism f : M_{u} ⊕ N_{u} → 𝕜(G/H) which is an isomorphism in 𝕜G - stab and such that the idempotent e_{u} on 𝕜(G/H) becomes ( \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} ) on M_{u} ⊕ N_{u}, again in 𝕜G - stab. After restriction to H, we also know that the idempotent e_{u} corresponds to the direct summand 𝕜 embedding in Res(𝕜(G/H)) via w : 𝕜 → 𝕜(G/H), as in (4.5). Since such a decomposition is unique (Remark 3.1), we can choose an isomorphism ξ : 𝕜 →^{∼} Res_{H}^{G}(M_{u}) in 𝕜H - stab such that the following diagram commutes :

[figure: 𝕜; ξ; ≃; w; Res_{H}^{G}(M_{u}); _{(\begin{matrix}1 \\ 0 \end{matrix})}; Res_{H}^{G}(𝕜(G/H)); Res_{H}^{G}(M_{u}) ⊕ Res_{H}^{G}(N_{u}); Res f; ≃]

in 𝕜H - stab. For every H-secant g ∈ G, consider the following commutative diagram in 𝕜H(g) - stab :

[figure: Res_{H(g)}^{G} 𝕜(G/H); g·; ≃; ^{g}Res_{H(g)}^{G} 𝕜(G/H); Res_{H(g)}^{G} (M_{u} ⊕ N_{u}); g·; ≃; Res f; ≃; _{g}Res_{H(g)}^{G} (M_{u} ⊕ N_{u}); _{g}Resf; ≃; Res_{H(g)}^{G} M_{u}; g·; ≃; _{( \begin{matrix}1 \\ 0 \end{matrix} )}; _{g}Res_{H(g)}^{G} M_{u}; _{( \begin{matrix}1 \\ 0 \end{matrix} )}; 𝕜; Res_{H(g)}^{H} ξ; Res_{H(g)}^{H} w; (υ(M_{u}))(g); 𝕜; _{g}Res_{H(g)}^{H}ξ; _{g}Res_{H(g)}^{H}w]

where the unit (υ(M_{u}))(g) at the bottom is the one of Construction 2.5 associated to M_{u}, which makes the lower square commute. The middle and upper squares commute by 𝕜G-linearity of the decomposition 𝕜(G/H) ≃ M_{u} ⊕ N_{u}. Finally, the two lateral pieces commute by the above discussion. By Lemma 4.4, the morphism w on the very right is retracted by \underline{u} : 𝕜(G/H) → 𝕜. So, in order to check that υ(M_{u})(g) is indeed our u(g), it suffices to establish the following equality in 𝕜:

gRes_{H(g)}^{H}(\underline{u}) ∘ (g·) ∘ Res_{H(g)}^{H}(w)(1) = u(g).

Unfolding the definition of w from (4.5), the above left-hand side becomes

\underline{u}(g · w(1)) = \frac{1}{n} ∑_{d∈G/H} u(d)^{−1}u(g d) .

Let us use a Mackey formula. Let S ⊂ G be a set of representatives of H(g)\G/H and recall the Mackey bijection (3.6) of left H(g)-sets

(4.15)  ∐_{s∈S} H(g)/H(g, s) →^{∼} G/H

given by [x]_{H(g,s)} ↦ [xs]_{H} where H(g, s) := H(g) ∩^{s}H = H ∩ H^{g} ∩^{s}H. Using this change of variables d = [xs]_{H} in the above sum, we get

\underline{u}(g · w(1)) = \frac{1}{n} ∑_{s∈S} ∑_{[x]∈H(g)/H(g,s)} u(s)^{−1} · u(gxs) since x ∈ H

= \frac{1}{n} ∑_{s∈S} u(s)^{−1} · ∑_{[x]∈H(g)/H(g,s)} u(gs)  for gxs = ^{g}xgs and ^{g}x ∈ H

= \frac{1}{n} ∑_{s∈S} u(s)^{−1} · [H(g) : H(g, s)] · u(gs)

= \frac{1}{n} ∑_{s∈S} [H(g) : H(g, s)] · u(g)  by Lemma 4.13

= \frac{1}{n} · |G/H| · u(g) = u(g) .

The penultimate equality uses again the same Mackey bijection (4.15). □

4.16. Proposition. Let M be an endotrivial 𝕜G-module in T (G, H). Suppose that υ(M) = 1 in A(G, H). Then M ≃ 𝕜 in 𝕜G - stab.

Proof. Let ξ : 𝕜 → Res_{H}^{G} M be a 𝕜H-linear homomorphism which is an isomorphism in 𝕜H - stab. The assumption about υ(M) = 1 implies that for every H-secant element g ∈ G, the following diagram commutes in 𝕜H(g) - stab :

(4.17)  [figure: Res_{H(g)}^{G} M; g·; ≃; ^{g}Res_{H(g)}^{G} M; 𝕜; Res_{H(g)}^{H} ξ; ≃; _{g}Res_{H(g)}^{H}ξ; ≃]

On the other hand, if g is not H-secant, the same diagram trivially commutes in 𝕜H(g) - stab = 0. Therefore, diagram (4.17) commutes in 𝕜H(g) - stab for all g ∈ G. Let us now define a 𝕜G-linear homomorphism ξ̂: 𝕜 → M by the composition

[figure: 𝕜; η_{𝕜}; ξ̂:= π_{M} ∘ Ind_{H}^{G}(ξ) ∘ η_{𝕜}; Ind_{H}^{G} 𝕜; Ind_{H}^{G}(ξ); ≃; Ind_{H}^{G} Res_{H}^{G} M; π_{M}; M]

where η : Id → Ind Res is the unit of the adjunction and π : Ind Res → Id is the retraction of η described in Remark 3.4. We claim that ξ̂ is an isomorphism in 𝕜G - stab. By Remark 3.4, it suffices to see that its restriction Res_{H}^{G}(ξ̂) is an isomorphism in 𝕜H - stab. We claim more precisely that Res_{H}^{G}(ξ̂) = ξ in 𝕜H - stab. This is not a mere property of the adjunction but will require (4.17) above. Applying Res_{H}^{G} to the last diagram, we get the upper part of the following commutative diagram in 𝕜H - stab :

[figure: 𝕜; Res η_{𝕜}; ^{Res ξ̂}; η′; Res_{H}^{G} Ind_{H}^{G} 𝕜; Res Ind(ξ); ≃; Res_{H}^{G} Ind_{H}^{G} Res_{H}^{G} M; Res π_{M}; Res M; ⨁_{t∈T} Ind_{H∩^{t}H}^{H} 𝕜; ≃; mack; ⊕_{t} Ind Res^{t^{−1}}(ξ); ≃; ⨁_{t∈T} Ind_{H∩^{t}H}^{H} Res^{t^{−1}}_{H∩^{t}H}^{G}M; ≃; mack]

in which the lower part is constructed as follows. The vertical isomorphisms mack : ⨁_{t∈T} Ind_{H∩^{t}H}^{H} Res^{t^{−1}}_{H∩^{t}H}^{H} ⟶ Res_{H}^{G} Ind_{H}^{G} are Mackey isomorphisms (3.7), for

any choice of a set T ⊂ G of representatives of H \G/H. Explicitly, the component mack : 𝕜H ⊗_{𝕜(H∩^{t}H)} N → 𝕜G ⊗_{𝕜H} N is given by mack(x ⊗ y) = (xt) ⊗ y. They are applied to the 𝕜H-modules N = 𝕜 and N = Res_{H}^{G} M respectively. Finally, η′ : 𝕜 → ⊕_{t∈T} Ind_{H∩^{t}H}^{H} 𝕜 = ⊕_{t∈T} 𝕜H ⊗_{𝕜(H∩^{t}H)}𝕜 is defined by 1 ↦ ∑_{t∈T} ∑_{x∈H/(H∩^{t}H)} x⊗1, that is η′ = mack^{−1} ∘ Res(η_{𝕜}). Hence the left-hand triangle commutes.

In each term of the bottom morphism, we can use (4.17) and replace every Res^{t^{−1}}(ξ) by (t^{−1}·) ∘ Res(ξ), since that relation holds in 𝕜(H ∩ ^{t}H) - stab and can then be induced to H. Therefore, in 𝕜H - stab, the morphism Res(ξ̂) is equal to

Res(π_{M}) ∘ mack ∘( ⊕_{t} Ind((t^{−1}·) ∘ Res(ξ))) ∘ η′.

We claim that the latter composition is simply ξ in 𝕜H - mod already. We compute the image of 1 ∈ 𝕜 under this morphism and get in M the equalities

Res(π_{M}) ∘ mack ∘( ⊕_{t} Ind((t^{−1}·) ∘ Res(ξ))) ∘ η′(1) =

= \frac{1}{n} ∑_{t∈T} ∑_{[x]∈H/(H∩^{t}H)} xt · t^{−1} · ξ(1)  unfolding the definitions
= \frac{1}{n} ∑_{t∈T} ∑_{[x]∈H/(H∩^{t}H)} ξ(x · 1)  by H-linearity of ξ
= \frac{1}{n} ∑_{t∈T} ∑_{[x]∈H/(H∩^{t}H)} ξ(1)  for the module 𝕜 is trivial

= \frac{1}{n}( ∑_{t∈T} |H/(H ∩^{t}H)|) · ξ(1) = \frac{|G/H|}{n} ξ(1) = ξ(1).

The penultimate equality uses again the same Mackey bijection (3.6). □

Everything is now in place to wrap it up :

Proof of Theorems 2.8 and 2.9. We have the well-defined maps α : A(G, H) → T (G, P) of Proposition 4.6 and υ : T (G, P) → A(G, H) of Proposition 4.12, where we also saw that υ is a group homomorphism. Proposition 4.14 tells us that υ ∘ α is the identity, hence υ is surjective. Proposition 4.16 shows that υ is injective. □

4.18. Remark. It follows that α is a group homomorphism, which was not obvious from its definition.

We also have naturality with respect to restriction to subgroups :

4.19. Proposition. Let H ≤ G′ ≤ G be an intermediate subgroup. Then every weak H-homomorphism from G to 𝕜^{×} restricts to a weak H-homomorphism from G′ to 𝕜^{×}. The induced homomorphism A(G, H) → A(G′, H) is compatible with the restriction Res_{G′}^{G} : T (G, H) → T (G′, H), via the isomorphisms υ and α.

Proof. The properties that p divides |H ∩ H^{g}| or |H ∩ H^{g_{1}} ∩ H^{g_{2}g_{1}}| are unchanged if we consider elements g, g_{1}, g_{2} ∈ G′ as elements of G. Consequently the restriction of u : G → 𝕜^{×} to G′ will satisfy conditions (WH 1-3) for G′. So, Res_{G′}^{G} : A(G, H) → A(G′, H) is well-defined. It is easy to check that the following diagram commutes

\xymatrix{T (G, H) \ar[r]^{υ} \ar[d]_{_{Res{G \atop G′}}} & A(G, H) \ar[d]^{_{Res{G \atop G′}}} \\ T (G′, H) \ar[r]^{υ} & A(G′, H)}

since we can use the same isomorphism ξ : 𝕜 →^{∼} Res_{H}^{G}(M) = Res_{H}^{G′}(M′) in 𝕜H - stab, for M and for M′ := Res_{G′}^{G}(M). It follows that α = υ^{−1} is also compatible with restriction, although this was maybe less obvious a priori. □

Let us finish with the two extreme cases of Example 2.3.

4.20. Example. Suppose that H◁G is a normal subgroup. Then every element g ∈ G is H-secant and we observed in Example 2.3 (1) that A(G, H) ≅ Hom(G/H, 𝕜^{×}). Given ρ ∈ Hom(G/H, 𝕜^{×}), let us write 𝕜_{ρ} for the one-dimensional representation g · x = ρ([g]) x of G. It clearly belongs to T (G, H). The associated weak Hhomomorphism u = υ(𝕜_{ρ}) of Construction (2.5) is characterized by Diagram (2.7) for M = 𝕜 and ξ = id, from which it follows that u(g) = ρ(g). In other words, υ(𝕜_{ρ}) = ρ, as one would of course expect. It also follows that α(ρ) = 𝕜_{ρ}, i.e. M_{ρ} is isomorphic to 𝕜_{ρ} in the stable category 𝕜G - stab. This last fact is less evident but can also be checked directly from the definition of M_{ρ} with the idempotent e_{ρ}.

4.21. Example. Suppose that H ≤ G is strongly p-embedded, as in Example 2.3 (2), where we saw that A(G, H) = 1. Then T (G, H) = 1. This is a well-known fact, which trivially follows from the underlying property that the restriction functor Res_{H}^{G} : 𝕜G - stab →^{∼} 𝕜H - stab is an equivalence of tensor categories in that case.

5. Some Corollaries

The following corollary is known in case O_{p}(G) ≠ 1, by [13, Lemma 2.6].

5.1. Corollary. Let H ≤ G be a subgroup of index prime to p. Suppose that H contains some subgroup K ◁ G, normal in G and of order divisible by p. Then the kernel Ker (T (G) → T (H)) consists only of one-dimensional representations, i.e. it is isomorphic to { ρ ∈ Hom_{gps}(G, 𝕜^{×}) | ρ(H) = 1 }.

Proof. For all g_{1}, g_{2} ∈ G, the subgroup H ∩ H^{g_{1}} ∩ H^{g_{2}g_{1}} contains K, hence (WH 3) holds. So every weak H-homomorphism u : G → 𝕜^{×} is a group homomorphism. □

5.2. Remark. It is well-known that the abelian group T (G, H) is finite. Indeed, every M ∈ T (G, H) appears as a direct summand of 𝕜(G/H) (see Theorem 2.9) and finiteness follows by the Krull-Schmidt Theorem. This gives an upper bound |T (G, H)| ≤ [G : H]. So, T (G, H) is a torsion abelian group and we can discuss the order of its elements. The following observation can be deduced from combining [13, Lemma 2.6] and [4, Prop. 2.6]. Our proof is direct.

5.3. Corollary. Let H ≤ G be a subgroup of index prime to p. Then there is no ptorsion in Ker (T (G) → T (H)). That is, if M is a 𝕜G-module such that M^{⊗p} ≃ 𝕜 in 𝕜G - stab and Res_{H}^{G}(M) ≃ 𝕜 in 𝕜H - stab then M ≃ 𝕜 in 𝕜G - stab already.

Proof. Let u : G → 𝕜^{×} be a weak H-homomorphism such that u^{p} = 1. Since the field 𝕜 has characteristic p, the relation u(g)^{p} = 1 forces u(g) = 1 for all g ∈ G. □

5.4. Remark. Actually, the orders of elements of T (G, H) are related to the coefficient field 𝕜. So, let us write T_{𝕜}(G, H) and A_{𝕜}(G, H) when we want to emphasize the choice of 𝕜. As in Proposition 4.19, we can show that υ : T_{𝕜}(G, H) → A_{𝕜}(G, H) and α : A_{𝕜}(G, H) → T_{𝕜}(G, H) are natural in 𝕜.

Interestingly, for a p-group P, the group T_{𝕜}(P) almost never depends on the field 𝕜, except sometimes in characteristic 2, with quaternion groups. See [16, § 2]. As we shall now see, the kernel T_{𝕜}(G, P) does depend on 𝕜, although in a nice way.

5.5. Corollary. Let H ≤ G be a subgroup of index prime to p.

(a) For every field extension 𝕜′/𝕜, the homomorphism T_{𝕜}(G, H) → T_{𝕜′}(G, H) is injective. Moreover, an element M ∈ T_{𝕜′}(G, H) of order d belongs to the image of this homomorphism (i.e. is defined over 𝕜) if and only if 𝕜 admits a primitive d^{th} root of unity.

(b) Let 𝕜′/𝕜 be a field extension and suppose that for every d ≤ [G : H], every d^{th} root of unity in 𝕜′ already belongs to 𝕜, e.g. if 𝕜 and 𝕜′ are both algebraically closed. Then T_{𝕜}(G, H) = T_{𝕜′}(G, H).

(c) Let q be a power of p and let 𝕜 be a field containing the finite field 𝔽_{q}. An element M ∈ T_{𝕜}(G, H) of order d is defined over 𝔽_{q} if and only d divides q − 1.

Proof. All these properties are easy to verify for A_{𝕜}(G, H), hence can be transported to T_{𝕜}(G, H). Indeed, a weak H-homomorphism u ∈ A_{𝕜}(G, H) has order d if and only if every u(g) is a d^{th} root of unity and some u(g) is a primitive one. Details are easily left to the reader. □

5.6. Example. Let H ≤ G be a subgroup of odd index and 𝔽_{2} the field with two elements. Then restriction T_{𝔽_{2}}(G) → T_{𝔽_{2}}(H) is injective. Indeed (𝔽_{2})^{×} = 1.

Acknowledgments : I am thankful to Nadia Mazza, Serge Bouc, Jon Carlson and Jacques Thévenaz for numerous discussions, big or small, around the theme of this paper and to the referee for elegantly revamping Lemma 3.8 and the proof of Lemma 4.4.

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Paul Balmer, Mathematics Department, UCLA, Los Angeles, CA 90095-1555, USA

E-mail address: balmer@math.ucla.edu
URL: http://www.math.ucla.edu/∼balmer