Endo-permutation modules as sources of simple modules

Nadia Mazza^{1} Institut de Mathématiques Université de Lausanne

CH-1015 Lausanne e-mail : Nadia.Mazza@ima.unil.ch

September 30, 2002

1This work is a part of a doctoral thesis in preparation at the University of Lausanne, under the supervision of Prof. Jacques Thévenaz

Abstract

The source of a simple kG-module, for a finite p-solvable group G and an algebraically closed field k of prime characteristic p, is an endo-permutation module (see [Pu1] or [Th]). L. Puig has proved, more precisely, that this source must be isomorphic to the cap of an endo-permutation module of the form ⨂_{Q/R∈S} Ten_{Q}^{P} Inf_{Q/R}^{Q}(M_{Q/R}), where M_{Q/R} is an indecomposable torsion endo-trivial module with vertex Q/R, and S is a set of cyclic, quaternion and semi-dihedral sections of the vertex of the simple kG-module. At present, it is conjectured that, if the source of a simple module is an endo-permutation module, then it should have this shape. In this paper, we are going to give a method that allow us to realize explicitly the cap of any such indecomposable module as the source of a simple module for a finite p-nilpotent group.

1 Introduction

1978, E. Dade introduced the concept of endo-permutation module, in the field of modular representation theory (see [Da1]). The study of these modules occupy at present an important position in representation theory, as they seem to appear very often. In particular, they appear as sources of simple modules for p-solvable groups, as it has been proved in [Pu1] and section 30 in [Th]. This fact raises then the following question: is any indecomposable endo-permutation module the source of a simple module, for some finite group? In other words, given a prime p, a finite p-group P, k an algebraically closed field of characteristic p and an indecomposable endo-permutation module M with vertex P, does there exist a finite group G having P as p-Sylow subgroup and a simple kG-module L, such that M is isomorphic to a direct summand of the restriction of L to P? We can’t answer this question at the moment. However, if we consider a large subfamily of torsion endo-permutation modules, then we are able to realize explicitly these modules as sources, as we will prove later. Let us also point out that, in order to realize explicitly such situations, we took some inspiration from Dade’s methods, used in [Da2].

Let us fix the notations and recall some basic facts we are going to use. Let p be a prime, O a discrete valuation ring, which we assume to be also local and complete, with quotient field K of characteristic 0 and residue field k, algebraically closed, of characteristic p (K is supposed to be ”big enough” for the considered groups). If G is a finite group, H a subgroup of G, M a kG-module and N a kH-module, then we write M↓_{H}^{G} the restriction of M from kG to kH, N↑_{H}^{G} the induction of N from kH to kG and Ten_{H}^{G} N the tensor induction of N from kH to kG. Moreover, if H ◁ G and L is a k[G/H]-module, then we denote by Inf_{G/H}^{G} L the inflation of L from k[G/H] to kG. Recall that if g ∈ G, then ^{g}N, also written g ⊗ N, is the conjugate module of N by g and has a structure of k[^{g}H]-module given by ^{g}h · ^{g}n = ^{g}(h · n), for all h ∈ H and n ∈ N, where · denotes the action of H on N and ^{g}h = ghg^{−1}. If M_{K} and M′_{K} are two KG-modules, then we have a scalar product (over K) defined by

(M_{K}, M′_{K})_{G} = ∑_{g∈G} T r(g, M_{K})T r(g^{−1}, M′_{K}),

where T r(g, M_{K}) is the trace of the K-linear automorphism corresponding to the action of g on M_{K}. Note that we consider only finitely generated modules. In case M_{K} and M′_{K} are simple KG-modules, then the orthogonality relations give M_{K} ≅ M′_{K} ⟺ (M_{K}, M′_{K})_{G} = 1 (else (M_{K}, M′_{K})_{G} = 0). We admit that the reader is familiar with elementary notions of representation theory of finite groups and we come now to endo-permutation modules. A kP -module M is endo-permutation if the P -algebra End_{k} M is a permutation kP -module (i.e. there is a P -invariant k-basis). Moreover, if M is indecomposable, we say that it has vertex P if M is not projective relative to any proper subgroup of P. If M is an endo-permutation kP -module with at least one indecomposable direct summand M_{0} with vertex P, then this summand is unique, up to isomorphism and it is called the cap of M. This allows us to define an equivalence relation on theses modules:

M ∼ N ⟺ M_{0} ≅ N_{0}.

We write D(P) the abelian group formed by the equivalence classes of endo-permutation kP - modules having an indecomposable direct summand of vertex P and we call it the Dade group of P. The composition law is given by [M] + [N] = [M ⊗ N], the neutral element is the class of the trivial kP -module k (containing also all permutation kP -modules having a trivial direct summand) and −[M] = [M^{∗}], where M^{∗} = Hom_{k}(M, k) is the dual of M. We denote by D^{t}(P) the torsion subgroup of D(P) and by T (P) the subgroup of D(P) consisting of equivalence classes of endo-trivial kP -modules, i.e. kP -modules M such that End_{k} M ≅ k ⊕L, for a free kP -module L (notice that equivalence classes of endo-trivial modules are smaller than the corresponding classes in D(P)). For instance, the syzygies Ω_{P}^{n} (k), n ∈ ℤ of thetrivial module are indecomposable endotrivial kP -modules with vertex P. The structure of D(P) is strongly related to the structure of T (P), as it can be noticed in the following theorem, due to Dade (see [Da1]), which classifies all endo-permutation modules in the case of an abelian p-group.

Theorem 1.1 [Dade, [Da]] Let P be an abelian p-group.

1. T (P) is generated by [Ω_{P}^{1} (k)]. Thus T (P) is trivial, if |P | = 2, cyclic of order 2, if P is cyclic of order ≥ 3, and infinite cyclic otherwise.

2. D(P) is isomorphic to the direct sum of the groups T (P/Q), where Q runs over the set of proper subgroups of P.

Now that we have recalled all the notions we are going to use, we can explain how we are going to proceed to realize endo-permutation modules as sources. We start by constructing some examples of p-nilpotent groups G having a given p-subgroup P as p-Sylow subgroup. Then we build, for each group G, a simple kG-module with vertex P and determine its source, that we know to be endo-permutation. Then we prove that, in a sense that will become clear later, the operations of inflation, tensor induction and tensor product provide us a kind of ”tools” that allow us to realize explicitly many other situations from known ones. Finally we note that the previous examples, combined with these ”tools”, lead us directly to the main result:

Theorem Let p be a prime, P a finite p-group and k an algebraically closed field of characteristic p. Then any indecomposable endo-permutation kP -module M with vertex P that is a direct summand of a kP -module of the form ⨂_{Q/R∈S} Ten_{Q}^{P} Inf_{Q/R}^{Q}(M_{Q/R}), where M_{Q/R} is an indecomposable torsion endo-trivial module with vertex Q/R, and S is a set of sections of P that are cyclic, or also quaternion or semi-dihedral, in case p = 2, can be explicitly realized as source of a simple module with vertex P for some finite p-nilpotent group.

To conclude this introductive section, we would try to justify our choice for concentrating on torsion endo-permutation modules. At present, it is conjectured that, if the source of a simple module is an endo-permutation module, then its class should be a torsion element in the Dade group. This result has already been proved for finite p-solvable groups (cf. [Pu1]). Moreover, it is also conjectured that there are no other indecomposable torsion endo-permutation modules as the ones that are isomorphic to the cap of a module of the form ⨂_{Q/R∈S} Ten_{Q}^{P} Inf_{Q/R}^{Q}(M_{Q/R}), as in our main theorem. Hence, if both conjectures hold, then we would have proved that any endo-permutation module that can be the source of a simple module, namely any torsion endopermutation module, can be explicitly realized as the source of a simple module for a finite pnilpotent group. Let us finally point out the fact that, in a talk given at the LMS Durham Symposium on representations of finite groups and related algebras (01-11 July 2002), Jon Carlson proved a result on T (P) which implies that the second conjecture holds for p odd.

2 Construction of finite p-nilpotent groups

Let p be a prime number and P a cyclic, quaternion or semi-dihedral p-group. Following Dade’s ideas, exposed in [Da2], we are going to build a finite p-nilpotent group G = Q ⋊ P, with P as p-Sylow subgroup and Q an extra-special q-group of exponent q and order q^{3}, if P is cyclic or quaternion, and q^{5}, if P is semi-dihedral, for a ”well chosen” prime number q.

Let us recall some useful properties about automorphisms of extra-special q-groups of order q^{3} and exponent q, for an odd prime q. Let Q be generated by two elements x and y of order q satisfying the relation ^{y}x = x[y, x], where [y, x] = yxy^{−1}x^{−1}, and such that z = [y, x] has order q and generates the centerof Q, which coincides with the subgroup Q′ generated by the commutators. The quotient Q/Q′ is a symplectic 𝔽_{q}-vector space of dimension 2, for the symplectic form [ , ] induced by taking the commutator of two elements. In the basis {x̄, ȳ} of Q/Q′ (if g ∈ Q, we

write ḡ its class in Q/Q′), [ , ] corresponds to the matrix (\begin{matrix}0 & 1 \\ −1 & 0 \end{matrix}). Then, the group Aut(Q/Q′)

of automorphisms of Q/Q′ is the symplectic group Sp_{2}(𝔽_{q}), which coincides with SL_{2}(𝔽_{q}). Let Aut_{Q′}(Q) be the subgroup of Aut(Q) formed by the automorphisms of Q which are the identity on restriction to Q′. We have a group homomorphism

π : Aut_{Q′}(Q) ⟶ SL_{2}(𝔽_{q}) , π(φ)(ḡ) = \overline{φ(g̃)}, ∀ g̃ ∈ ḡ ∈ Q/Q′ and ∀ φ ∈ Aut_{Q′}(Q).

Moreover, π is surjective. Indeed, let us consider

\begin{matrix}φ : Q & ⟶ Q & ψ : Q & ⟶ Q \\ x & ⟼ xy and & x & ⟼ x \\ y & ⟼ yz & y & ⟼ xy. \end{matrix}

It is easy to check that φ and ψ belong to Aut_{Q′}(Q) and, as SL_{2}(𝔽_{q}) is generated by e = (\begin{matrix}1 & 0 \\ 1 & 1 \end{matrix})

and f = (\begin{matrix}1 & 1 \\ 0 & 1 \end{matrix}) (see Theorem 2.8.4 in [Go]), the surjectivity of π is immediate.

We also notice that Ker(π) is isomorphic to Q/Q′. Indeed, the two following maps are injective group homomorphisms:

conj : Q/Q′ ⟶ Ker(π)  and  τ : Ker(π) ⟶ Hom_{𝔽_{q}}(Q/Q′, Q′)
ḡ ⟼ conj(ḡ)  φ ⟼ τ(φ),

where conj(ḡ)(h) = g̃h = g̃hg̃^{−1} and τ(φ)(ḡ) = g̃^{−1}φ(g̃), for a representative g̃ ∈ ḡ, ∀ ḡ ∈ Q/Q′, ∀h ∈ Q and ∀ φ ∈ Ker(π). As 𝔽_{q}-vector space, Q′ is isomorphic to 𝔽_{q}. This implies that Hom_{𝔽_{q}}(Q/Q′, Q′) is isomorphic to the dual of Q/Q′ and thus to Q/Q′ as 𝔽_{q}-vector space. In particular, it follows that | Ker(π)| ≤ |Q/Q′| and so conj and τ are isomorphisms. We get then a short exact sequence:

0 ⟶ Q/Q′ ⟶ Aut_{Q′}(Q) ⟶ SL_{2}(𝔽_{q}) ⟶ 0.

Moreover, if p and q are distinct prime numbers and P is a p-subgroup of SL_{2}(𝔽_{q}), then we get another short exact sequence by restriction:

(∗) 0 ⟶ Q/Q′ ⟶ π^{−1}(P) ⟶ P ⟶ 0.

It follows from the Schur-Zassenhaus Theorem (see Corollary 8.40 in [CR]) that this sequence splits, because Q/Q′ and P have coprime orders and Q/Q′ is abelian. So we have an injective map σ : P ⟶ Aut_{Q′}(Q), such that πσ = Id_{P}.

As SL_{2}(𝔽_{q}) has order q(q^{2} − 1) and its 2-Sylow subgroups are quaternion, the following result is immediate.

Lemma 2.1 Let p and q be two distinct prime numbers, P be a finite cyclic p-group or, in case p = 2, cyclic or quaternion, such that |P |||SL_{2}(𝔽_{q})|, and let Q be an extra-special q-group of order q^{3} and exponent q. Then there exists an injective group homomorphism σ : P ⟶ Aut(Q). Moreover, we can choose σ such that the restriction of σ(u) to Q′ is the identity morphism, for all u ∈ P.

□

Following Dade’s ideas (cf. [Da2]), if we choose a prime number q such that q ≡ −1(mod |P |), then |P |||SL_{2}(𝔽_{q})| and so, by the above Lemma, P is identified with a subgroup of Aut(Q), whose elements fix Q′. This defines a semi-direct product G = Q ⋊ P, where P is a p-Sylow subgroup, acting by conjugation on the normal q-subgroup Q and Q′ centralizes P. We can also notice, in case p = 2 and P is quaternion, that we can also consider a 2-nilpotent group, with the same features, if we take a prime number q with q ≡ \frac{|P |}{2} + 1(mod |P |) (then P identifies with a 2-Sylow subgroup of SL_{2}(𝔽_{q})). Let us state some properties of these p-nilpotent groups.

Lemma 2.2 With the same notation as above, G satisfies the following properties.

a) We can choose a section σ : P ⟶ π^{−1}(P) of (∗) such that σ(w) has order p and no fixed point on Q \ Q′, for a generator w of the unique subgroup of P of order p. In other words, w and g ∈ Q commute if and only if g ∈ Q′.

b) The normalizer N of P in G is equal to Q′ × P.

c) P is a trivial intersection subgroup of G.

Proof . a) Assume that P is cyclic of order p^{n}, generated by u and write w = u^{p^{n−1}}. As σ is injective, σ(u) has order p^{n}, and so σ(w) has order p. If σ(w) has a fixed point on Q \ Q′, then π(σ(w)) ∈ SL_{2}(𝔽_{q}) has an eigenvector for the eigenvalue 1 and so is conjugate to a matrix of order

p of the form A = (\begin{matrix}1 & a \\ 0 & 1 \end{matrix}), for an a ∈ 𝔽_{q}. But then, we have A^{p} = Id, if and only if pa = 0, if and

only if a = 0, as p is invertible in 𝔽_{q}. It follows that A = Id, which contradicts the fact that A has order p. Thus σ(w) has no fixed point on Q \ Q′. Now, if p = 2 and P is a quaternion 2-group, then, as before, we can choose σ such that σ(w) has order 2, i.e. σ(w) = − Id. So 1 is not an eigenvalue of π(σ(w)), i.e. σ(w) has no fixed point on Q \ Q′.

b) As G is p-nilpotent, we have N = C_{G}(P)×P and so we just need to verify that C_{G}(P) = Q′. By construction, Q′ centralizes P, so we have C_{G}(P) ≥ Q′. Conversely, if g ∈ C_{G}(P), then, in particular, g and w commute. By a), it implies that g ∈ Q′.

c) Let g ∈ Q and s ∈ ^{g}P ∩ P. There exists s′ ∈ P such that s = ^{g}s′ and so [g, s′] = 1. Indeed,

[g, s′] = ^{g}s′(s′)^{−1} = s(s′)^{−1} ∈ P and [g, s′] = g( g^{s′})^{−1} ∈ Q as Q ◁ G. Thus [g, s′] ∈ P ∩ Q = {1}. It

follows, by a), that g ∈ Q′, if s′ ≠ 1, i.e. s ≠ 1, as w is contained in any non trivial subgroup of P.

□

Now, we are going to construct a finite 2-nilpotent group with a semi-dihedral 2-Sylow subgroup P = <u, v | u^{2^{n−1}} = v^{2} = 1, ^{v}u = u^{2^{n−2}−1}> of order 2^{n}, with n ≥ 4. Consider a prime number q ≡ 2^{n−1} − 1(mod 2^{n}) and

Q = ⟨x_{1}, x_{2}, y_{1}, y_{2} | x_{i}^{q} = y_{i}^{q} = 1, ^{y_{i}}x_{i} = x_{i}[y_{i}, x_{i}], [x_{1}, x_{2}] = [x_{1}, y_{2}] = [x_{i}, y_{j}] = 1, ∀ 1 ≤ i ≠ j ≤ 2⟩

an extra-special q-group of order q^{5}, of exponent q and center Q′ = <z = [y_{1}, x_{1}] = [y_{2}, x_{2}]> cyclic of order q. We can also consider Q as the central product of two extra-special q-groups Q_{1} and Q_{2} of order q^{3}, of exponent q and center Q′, where Q_{i} is the subgroup of Q generated by x_{i} and y_{i}, for i = 1 and 2. There are three maximal subgroups in P, one of which is quaternion, call it R, generated by u^{2} and uv. The previous results allow us to consider a section σ : R ⟶ Aut_{Q′_{1}}(Q_{1}) of the split exact sequence (∗) above. Notice that we get another injective map σ̃ : R ⟶ Aut_{Q′_{2}}(Q_{2}) defined as follows:

if σ : R ⟶ Aut_{Q′_{1}}(Q_{1}) ,  then write  σ̃ : R ⟶ Aut_{Q′_{2}}(Q_{2})
u^{2} ⟼ σ(u^{2})  u^{2} ⟼ σ(u^{2})^{−1}
uv ⟼ σ(uv)  uv ⟼ σ(u^{2})^{2^{n−3}−1}σ(uv).

Indeed, by construction, σ(u^{2}) and σ(uv) generate a quaternion group of order 2^{n−1} and we can easily check that the group generated by σ(u^{2})^{−1} and σ(u^{2})^{2^{n−3}−1}σ(uv) satisfies the relations of a quaternion group of the same order. Consider then

P ⟶ Aut_{Q′}(Q),  where  σ_{u} : Q ⟶ Q  and  σ_{v} : Q ⟶ Q
u ⟼ σ_{u}  x_{1} ⟼ σ̃(uv)(x_{2})  x_{1} ⟼ x_{2}
v ⟼ σ_{v}  x_{2} ⟼ σ(uv)(x_{1})  x_{2} ⟼ x_{1}
y_{1} ⟼ σ̃(uv)(y_{2})  y_{1} ⟼ y_{2}
y_{2} ⟼ σ(uv)(y_{1})  y_{2} ⟼ y_{1}.

We have σ_{u}^{2}(g_{1}) = σ(u^{2})(g_{1}), ∀ g_{1} ∈ Q_{1} and σ_{u}^{2}(g_{2}) = σ̃(u^{2})(g_{2}), ∀ g_{2} ∈ Q_{2} (so σ_{u} has order 2^{n−1} and satisfies σ_{u}^{2^{n−2}}(g) ∈ g^{−1}Q′, ∀ g ∈ Q), σ_{v}^{2} = Id and ^{σ_{v}}σ_{u} = σ_{u}^{2^{n−2}−1}, as σ_{u}σ_{v}(g_{1}) = σ(uv)(g_{1}), ∀ g_{1} ∈ Q_{1} and σ_{u}σ_{v}(g_{2}) = σ̃(uv)(g_{2}), ∀ g_{2} ∈ Q_{2}, i.e. (σ_{u}σ_{v})^{2} is the image of the unique element w of order 2 in R. It follows that the subgroup of Aut_{Q′}(Q) generated by σ_{u} and σ_{v} is semi-dihedral of order 2^{n}. In other words, we get a finite 2-nilpotent group G, which is the semi-direct product of P by Q. Moreover, using the same argument as in the proof of the previous Lemma, we have C_{Q}(P) = Q′ and so N = Q′ × P. The next result shows two differences beetween this construction and the previous one.

Lemma 2.3 P has two conjugacy classes of elements of order 2. Moreover, P is not a trivial intersection subgroup of G.

Proof . P has (at least) a dihedral subgroup of order 8, whose center is <w> and has two other conjugacy classes of elements of order 2. We have <w>◁P, as it is also the center of P, and so form a conjugacy class of size 1. As the two other classes become conjugate in P, the first statement is proved. For the second one, consider the conjugacy morphism by the non-central element v of P of order 2, i.e. the map σ_{v} described above. We have σ_{v}(x_{1}x_{2}) = x_{1}x_{2}, i.e. v ∈ ^{x_{1}x_{2}}P ∩ P, but x_{1}x_{2} ∉ N.

□

Notice that if ^{g}P ∩P ≠ {1}, with g ∉ N, then ^{g}P ∩P is conjugate to <v>. Indeed, any non-trivial subgroup of P which does not contain <w> has order 2 and must then be a conjugate of <v>. As <w> ≤ ^{g}P ∩ P, if and only if g ∈ N, we deduce that ^{g}P ∩ P is conjugate to <v>.

Let us summarize the situation. We have build p-nilpotent groups of the form Q ⋊ P, where Q is an extra-special q-group of exponent q, for a prime number q satisfying the following datas:

• If P is cyclic, then we choose q ≡ |P | − 1(mod p|P |) and Q of order q^{3}.

• If P is quaternion, then we build two finite 2-nilpotent groups by choosing two distinct prime numbers q: first q ≡ |P | − 1(mod 2|P |), and then q ≡ \frac{|P |}{2} + 1(mod |P |). In each case, Q has order q^{3}.

• If P is semi-dihedral, then we choose q ≡ \frac{|P |}{2} − 1(mod |P |) and Q of order q^{5}.

These choices of congruences for the prime q will be motivated in the next section, where we are going to build simple modules for these p-nilpotent groups.

3 Construction of simple kG-modules

Let us first recall some basic facts about the simple kQ-modules, where Q is an extra-special qgroup of order q^{1+2m} and exponent q for a prime number q, distinct from p = char(k), and an integer m ≥ 1. Write x_{1}, …, x_{m}, y_{1}, …, y_{m} the generators of Q satisfying the relations x_{i}^{q} = y_{i}^{q} = 1, ^{y_{i}}x_{i} = x_{i}[y_{i}, x_{i}], [y_{i}, x_{i}] = [y_{j}, x_{j}] and [x_{i}, x_{j}] = [x_{i}, y_{j}] = [y_{i}, y_{j}] = 1, ∀ 1 ≤ i ≠ j ≤ m. Let also z = [y_{1}, x_{1}] be the generator of the center Q′ of Q (which coincides with the subgroup generated by the commutators) and let A be the elementary abelian normal subgroup of Q of rank m + 1, generated by x_{1}, …, x_{m} and z. We know, by Proposition 7.7 in [Hu], that Q has q − 1 simple kQ-modules of dimension q^{m} and q^{2m} simple kQ-modules of dimension 1. Let us describe them. Let ω ∈ k be a primitive q-throot of unity and consider the degree 1 representation given by:

\begin{matrix}ψ : A & ⟶ k^{∗} \\ x_{i} & ⟼ 1, 1 ≤ i ≤ m \\ z & ⟼ ω. \end{matrix}

We have ^{g}ψ ≠ ψ, ∀g ∉ A. In terms of modules, if we write k_{ψ} the kA-module of dimension 1 defined by a · λ = ψ(a) · λ, ∀a ∈ A, ∀λ ∈ k, then this is equivalent to saying that k_{ψ} ≇ ^{g}k_{ψ}. It follows by Clifford theory (see Chapter 11 in [CR]), that the induced kQ-module L = k_{ψ}↑_{A}^{Q} is simple, of dimension q^{m}. As k is supposed to be algebraically closed, we have q − 1 distinct primitive q-th roots of unity in k and so we get q −1 non isomorphic simple kQ-modules of dimension q^{m}. On the other hand, we get the q^{2m} simple kQ-modules of dimension 1 by considering the representations

\begin{matrix}ρ : Q & ⟶ k^{∗} \\ x_{i} & ⟼ ω_{i}, 1 ≤ i ≤ m \\ y_{i} & ⟼ θ_{i}, 1 ≤ i ≤ m \\ z & ⟼ 1, \end{matrix}

where ω_{i} and θ_{i}, 1 ≤ i ≤ m are q-th roots of unity in k.

Before going further, recall that, as kQ is semi-simple, the decomposition map induces a bijection between simple kQ- and KQ-modules (see [Se]). In other words, we can lift in characteristic 0 any simple kQ-module L into a unique simple KQ-module L_{K} (up to isomorphism). It follows that if M_{K} is a simple KQ-module which lifts another simple kQ-module M, then L ≅ M, if and only if L_{K} ≅ M_{K}, if and only if the scalar product (L_{K}, M_{K})_{Q} = 1 (otherwise (L_{K}, M_{K})_{Q} = 0). In our situation, M is a simple kQ-module of dimension 1, if and only if M↓_{Q′}^{Q} ≅ k, where k denotes the trivial kQ-module. From the above description of the simple kQ-modules, we can also notice that any kQ-module M of dimension q^{m} is simple, if and only if M↓_{Q′}^{Q} ≅ (k_{ω})^{q^{m}}, where k_{ω} is the kQ-module of dimension 1 defined by z · λ = ωλ, ∀ λ ∈ k, for a primitive q-th root of unity ω (otherwise M↓_{Q′}^{Q} ≅ k^{q^{m}}). So we have proved:

Lemma 3.1 Let L and M be two kQ-modules of dimension q^{m}, lifted to the KQ-modules L_{K} and M_{K}, respectively. Assume that L is simple. Then, L and M are isomorphic if and only if (L_{K}↓_{Q′}^{Q}, M_{K}↓_{Q′}^{Q})_{Q′} = q^{2m}.

The next question that we are going to discuss betreats extension of modules. Indeed, we have constructed simple kQ-modules, but we are interested in simple kG-modules. In order to get some simple kG-modules, we are going to prove that these kQ-modules can be extended to kG, as they will still be simple. More precisely, we are going to state some analogous results to the well known ones we have in characteristic 0, for a splitting field of G. We even get some ”better” result, as we can prove that the extension is unique, up to isomorphism (recall that in characteristic 0, one must introduce an argument on the determinant of a representation to get the unicity of the extension).

Lemma 3.2 Let G = Q ⋊ P a finite p-nilpotent group, where Q is a p′-group and P is a p-Sylow subgroup of G. Let L be a simple kQ-module, P -invariant, i.e. ^{u}L ≅ L for all u ∈ P. Then there exists a unique simple kG-module L̃, up to isomorphism, such that L̃↓_{P}^{G}≅ L.

Proof . Let us first fix some notation. If H is a finite group, we denote by S_{K}(H), respectively S_{k}(H), the sets of isomorphism classes of simple KH-, respectively kH-modules, R_{K}(H), respectively R_{k}(H) the Grothendieck groups of H and d_{H} : R_{K}(H) ⟶ R_{k}(H) the decomposition map. As we have already recalled, p | |̸Q| implies that kQ is semi-simple and we have a bijection d : S_{K}(Q) ⟶ S_{k}(Q), induced by d_{Q}. Let L_{K} ∈ d^{−1}([L]), i.e., up to isomorhism, L_{K} is the unique KQ-module (obviously simple) that lifts L in characteristic 0. We claim that L_{K} is P -invariant. Indeed, as ^{u}L ≅ L, we have ^{u}L_{K} ∈ d^{−1}([^{u}L]) = d^{−1}([L]) = [L_{K}] and so ^{u}L_{K} ≅ L_{K}, ∀ u ∈ P. Moreover |G/Q| and |Q| are coprime and so L_{K} extends to G in a (always simple) KG-module L̃_{K} (see Theorem 22.3 in [Hu]). Let L_{O} an OG-module such that L̃_{K} ≅ K ⊗_{O} L_{O} and let L̃ = L_{O}/PL_{O}, where P denotes the unique maximal ideal of O. It follows that L̃ is a simple kG-module and satisfies L̃↓_{Q}^{G}≅ L_{O}↓_{Q}^{G} /P(L_{O}↓_{Q}^{G}) and K ⊗_{O} L_{O}↓_{Q}^{G}≅ L̃_{K}↓_{Q}^{G}≅ L_{K}. So L̃↓_{Q}^{G}∈ [L], i.e. L̃ extends L.

To prove the uniqueness of L̃, let us use for a while the point of view of representations, instead of modules and write ρ : Q ⟶ Aut_{k}(V) and ρ̃ : G ⟶ Aut_{k}(V) the two representations corresponding to L and L̃, respectively, where V is a k-vector space isomorphic to L and L̃. We have ρ̃↓_{Q}^{G}= ρ. Assume there is another extension ρ̂ : G ⟶ Aut_{k}(V) of ρ (i.e. satisfying also ρ̂↓_{Q}^{G}= ρ). Consider the map

\begin{matrix}s : G & ⟶ Aut_{kQ}(V) \\ g & ⟼ ρ̂(g)ρ̃(g)^{−1}, ∀ g ∈ G. \end{matrix}

It can be easily checked that s is well defined, as s(g) ∈ Aut_{k}(V), and s(g)(h·v) = h·s(g)(v), ∀ h ∈ Q, ∀ v ∈ V and ∀ g ∈ G. Schur’s Lemma implies that Aut_{kQ}(V) ≅ k^{∗}. It follows that ∀ g, h ∈ G we have s(gh) = ρ̂(g)ρ̂(h)ρ̃(h)^{−1}ρ̃(g)^{−1} = ρ̂(g)s(h)ρ̃(g)^{−1} = ρ̂(g)ρ̃(g)^{−1}s(h) = s(g)s(h), i.e. s is a group homomorphism. Moreover, if g ∈ Q, then s(g) = ρ̂(g)ρ̃(g)^{−1} = ρ(g)ρ(g)^{−1} = 1, i.e. Q ≤ Ker(s). So s induces a group homomorphism

\begin{matrix}s̄: G/Q & ⟶ k^{∗} \\ ḡ & ⟼ s(g)(v), ∀ g ∈ ḡ ∈ G/Q, \end{matrix}

for any fixed non-zero vector v ∈ V. But G/Q ≅ P is a finite p-group and k^{∗} has no primitive p-th root of unity. It follows that s̄ is the identity morphism and so 1 = s̄(ḡ) = s(g) = ρ̂(g)ρ̃(g)^{−1}, ∀ g ∈ ḡ ∈ G/Q. Consequently, we have ρ̂ = ρ̃, which proves the uniqueness of the extension ρ̃, and so, in terms of kG-modules, the extension L̃ of L is unique (up to isomorphism).

□

Let us turn back to the constructed simple kQ-module L. To prove the existence of a unique extension L̃ of L, it suffices to verify that L is P -invariant. Let u ∈ P and consider the kQ-modules L and ^{u}L of dimension q^{m}, with m = 1, if P is cyclic or quaternion, and m = 2, if P is semi-dihedral. Recall that ^{u}L is a kQ-module for the action of Q given by g ·u⊗l = u⊗(^{u^{−1}}g ·l), ∀ g ∈ Q, ∀ l ∈ L. Lemma 3.1 tells us that L ≅ ^{u}L if and only if (^{u}L_{K}↓_{Q′}^{Q}, L_{K}↓_{Q′}^{Q})_{Q′} = q^{2m}. But z acts on L by the scalar matrix ω Id, and so does it also on ^{u}L, as u and z commute. Thus we have

(^{u}L_{K}↓_{Q′}^{Q}, L_{K}↓_{Q′}^{Q})_{Q′} = q^{−1} ∑_{i=0}^{q−1} Tr(z^{i}, ^{u}L_{K}) Tr(z^{−i}, L_{K}) = q^{−1} ∑_{i=0}^{q−1}(q^{m}ω^{i})(q^{m}ω^{−i}) = q^{2m}.

We conclude then that L is P -invariant and extends to a unique (up to isomorphism) simple kG-module L̃ of dimension q^{m}. Moreover, as dim(L̃) and p are coprime, L̃ has vertex P.

4 Source of L̃

In this section we are going to determine, up to isomorphism, the sources of the simple modules L̃ we constructed in the previous section. We know, by Theorem 30.5 in [Th], that these sources are indecomposable endo-permutation modules. Moreover, Theorem 8.39 and Remark 8.41 in [Pu1] imply that these sources belong to a subset of the set of torsion endo-permutation modules. But, as this latter result is not published, we will not use it in our argumentation. We use the previous notation and set X | Y, if X and Y are two modules such that X is (isomorphic to) a direct summand of Y. We want to find an indecomposable endo-permutation module S, with vertex P, satisfying L̃ | S↑_{P}^{G} and S | L↓_{P}^{G}= (L↓_{N}^{G})↓_{P}^{N} (by transitivity of restriction), where N = N_{G}(P). It follows, by uniqueness of the cap of an endo-permutation module, that the module S we are looking for must be a direct summand of the Green correspondent M of L, i.e. M is the unique indecomposable direct summand with vertex P of L̃↓_{N}^{G} such that L̃ | M↑_{N}^{G}. Let us write

(∗∗)  L̃↓_{N}^{G} | M↑_{N}^{G}↓_{N}^{G} ≅ M ⊕ ( ⨁_{g∈[G/N], g≠1} \underbrace{^{g}M↓_{^{g}N∩N}^{^{g}N}↑_{^{g}N∩N}^{N}}_{(I)} ).

The next lemma shows that M↓_{P}^{N} is an indecomposable kP -module, and so M↓_{P}^{N} is the source S of L̃.

Lemma 4.1 Let N = A × P be a finite group where A is an abelian p′-group and P is a p-group, and let V be an indecomposable kN-module. Then, V ↓_{P}^{N} is indecomposable.

Proof . As A is of order prime to p, the group algebra kA is semi-simple. Write Irr_{Br}(A) the set of irreducible Brauer characters of A (which is in bijection with the set of irreducible K-characters of A). We have V ↓_{A}^{N} ≅ ⨁_{χ∈Irr_{Br}(A)} V_{χ}, where V_{χ} denotes the isotypic component of V ↓_{A}^{N} corresponding to the irreducible Brauer character χ. The kA-module V_{χ} ≅ ⨁_{dim V_{χ}} k_{χ} is invariant under the action of P. Indeed, let a ∈ A, u ∈ P and v ∈ V_{χ}. As V is a kN-module and a and u commute, we have a · (u · v) = (au) · v = (ua) · v = u · (a · v) = u · χ(a)v = χ(a)(u · v), and so u · v ∈ V_{χ}. This implies that V_{χ} is a direct summand of V as kN-module. But V is indecomposable and so V ≅ V_{χ}, for a χ ∈ Irr_{Br}(A). Let us consider the indecomposable kN-modules V_{A} = k_{χ} and V_{P} = k_{χ^{−1}} ⊗ V, on which N acts as follow. Let g ∈ N, say g = au for an a ∈ A and an u ∈ P. We have g·λ = χ(a)λ, ∀ λ ∈ V_{A} and g·(λ⊗v) = (g·λ)⊗(g·v) = χ^{−1}(a)λ⊗χ(a)(u·v) = λ⊗(u·v), ∀ λ⊗v ∈ V_{P}. So, P acts trivially on V_{A}, because u·λ = λ, ∀ u ∈ P, ∀ λ ∈ k, and similarly, A acts trivially on V_{P}. Moreover, from the isomorphism of kN-modules k ≅ (k_{χ} ⊗ k_{χ^{−1}}), we deduce that V ≅ V_{A} ⊗ V_{P} as kN-modules. Thus, V ↓_{P}^{N} ≅ V_{A}↓_{P}^{N} ⊗V_{P}↓_{P}^{N} ≅ k⊗V_{P}↓_{P}^{N} ≅ V_{P}↓_{P}^{N}. But V_{P} is indecomposable and A acts trivially on V_{P}. So the restriction V_{P}↓_{P}^{N} is indecomposable. Consequently, V ↓_{P}^{N} is indecomposable.

□

4.1 Cyclic case

Assume that P is cyclic. In the finite p-nilpotent group G we constructed, P is a trivial intersection subgroup (see Lemma 2.2). So the direct summands in (I) have trivial vertex, i.e. they are projective kG-modules and their dimension is divisible by |P |. We then get from (∗∗) the congruence dim L̃ ≡ −1 (mod |P |). The choice of q and Lemma 4.1 imply that dim S ≡ −1 (mod |P |). As kP is uniserial and S is indecomposable, this forces S to be isomorphic to Ω_{P}^{1} (k).

4.2 Quaternion case

Assume that P is quaternion. In both finite p-nilpotent groups having P as 2-Sylow subgroup, P is a trivial intersection subgroup. So, as in the cyclic case, the direct summand in (I) have trivial vertex, i.e. they are projective and their dimension is divisible by |P |. If q ≡ |P | − 1 (mod 2|P |), then the relation (∗∗) implies dim S ≡ −1(mod |P |). If q ≡ \frac{|P |}{2} + 1 (mod |P |), then (∗∗) implies dim S ≡ \frac{|P |}{2} + 1 (mod |P |). Let us prove that in both cases S is endo-trivial. Set P = <u, v | u^{2^{n−1}} = 1, u^{2^{n−2}} = v^{2}, ^{v}u = u^{−1}> (so |P | = 2^{n}) and w = u^{2^{n−2}} the unique element of P of order 2, i.e. the generator of the center Z of P. Lemma 2.9 in [CaTh] tells us that it suffices to prove that S↓_{Z}^{P} is endo-trivial and so we just need to know how w acts on L̃. As the calculations are the same for both congruences of q, we don’t need to distinguish the cases. Call a and b the two integers such that 0 ≤ a, b < q, ^{w}x = x^{−1}z^{a} and ^{w}y = y^{−1}z^{b}, and consider the k-basis {y^{i} ⊗ 1 | 0 ≤ i < q} of L̃. Set w · y^{i} ⊗ 1 = ω^{(i−l)b}(y^{−i−a} ⊗ 1), ∀ 0 ≤ i < q, where l is the unique integer with 0 ≤ l < q and 2l ≡ −a(mod q), and extend this action to Q ⋊ <w>, by setting (gw) · y^{i} ⊗ 1 = g · (w · y^{i} ⊗ 1), ∀ g ∈ Q, ∀ 0 ≤ i < q. This gives L̃ the (unique, by Lemma 3.2) structure of Q ⋊ <w>-module, as it satisfies:

1 · y^{i} ⊗ 1 = w^{2} · y^{i} ⊗ 1 = w · ω^{(i−l)b}(y^{−i−a} ⊗ 1) = y^{i} ⊗ 1,
(wx) · y^{i} ⊗ 1 = (x^{−1}z^{a}w) · y^{i} ⊗ 1 = ω^{(i−l)b−i}(y^{−i−a} ⊗ 1) = w · (x · y^{i} ⊗ 1) and
(wy) · y^{i} ⊗ 1 = (y^{−1}z^{b}w) · y^{i} ⊗ 1 = ω^{(i+1−l)b}(y^{−i−1−a} ⊗ 1) = w · (y · y^{i} ⊗ 1).

The matrix of the action of w is

(w, L̃) = (\begin{matrix}& & ω^{lb} \\ & ⋯ & & & 0 \\ ω^{−lb} \\ & & & & & ω^{−(l+1)b} \\ & 0 & & & ⋯ \\ & & & ω^{(l+1)b} \end{matrix}) ,

with a unique non-zero element, equal to 1, on the diagonal, in the (l, l) coordinate, i.e. an eigenvector for the eigenvalue 1. We also notice that the other columns of the matrix pair to form \frac{q−1}{2} Z-invariant subspaces W_{i} = k<y^{i} ⊗ 1, y^{−i−a} ⊗ 1>, for all i strictly between l and \frac{q−1+2l}{2}, and on which Z acts freely. As all these subspaces intersect trivially, their sum is direct and form then a kZ-submodule of L̃↓_{Z}^{G}. Counting the dimension, we deduce that L̃↓_{Z}^{G} decomposes as a direct sum k ⊕ (kZ)^{\frac{q−1+2l}{2}} and consequently we have S↓_{Z}^{P} ≅ k ⊕ S′, for a free kZ-module S′, by the Krull-Schmidt Theorem. Thus S is endo-trivial.

Let us recall that there are eight indecomposable endo-trivial kP -modules with vertex P (up to isomorphism), whose dimensions determine their orders and allow us to recover generators of T (P) (see Section 6 in [CaTh]). Indeed, up to isomorphism, two have dimension |P | − 1 and order 4: Ω_{P}^{1} (k) and its dual Ω_{P}^{3} (k); one has dimension |P | + 1 and order 2: Ω_{P}^{2} (k) (which is self-dual); two have dimension \frac{|P |}{2} + 1 and order 2: an ”exceptionnal” indecomposable endo-trivial kP -module V with vertex P and its dual; two have dimension \frac{|P |}{2} − 1 and order 2: Ω_{P}^{1} (V) and its dual; and the trivial kP -module, of dimension 1. So, comparing S with the elements in this list we draw the following conclusions. For q ≡ |P | − 1 (mod 2|P |), we have dim S = |P | − 1 and S is isomorphic either to Ω_{P}^{1} (k), or to Ω_{P}^{3} (k), i.e. [S] has order 4 in D(P). For q ≡ \frac{|P |}{2} + 1 (mod |P |), we have dim S = \frac{|P |}{2} + 1 and S is isomorphic either to V, or to its dual and [S] has order 2 in D(P). In particular, we notice that the classes of the two modules realized as sources generate all T (P).

4.3 Semi-dihedral case

Assume that P is semi-dihedral, say P = <u, v | u^{2^{n−1}} = v^{2} = 1, ^{v}u = u^{2^{n−2}−1}> of order 2^{n}, and use the same notation as in section 2. Recall that P is not a trivial intersection subgroup of G, and that the direct summands of (I) have trivial vertex or vertex cyclic of order 2, conjugate to <v>. So, their dimension is divisible by 2^{n−1} and the relation (∗∗) implies that dim S ≡ 1(mod \frac{|P |}{2}). We want to prove that S is isomorphic to the endo-trivial kP -module Ω_{P}^{1} (Ω_{P/<v>}^{1}(k)) of dimension \frac{|P |}{2}+1. By injectivity of restriction and results of section 7 in [CaTh], we must show that Ω_{R}^{2}(k) | S↓_{R}^{P} and that S↓_{E}^{P} is an endo-trivial module belonging to the class of the trivial module, where R is the maximal quaternion subgroup of P and E is an elementary abelian subgroup of P of order 4.

Let us start with S↓_{R}^{P}. For i = 1 and 2, consider the simple kQ_{i}-modules L_{i} of dimension q satisfying L_{i}↓_{Q′}^{Q_{i}} ≅ (k_{ω})^{q}, for the same primitive root ω appearing in the simple kQ-module L. We assert that the k-vector space X = L_{1} ⊗ L_{2} is a kQ-module for the action given as follows: ∀ g ∈ G, ∃ g_{i} ∈ Q_{i}, i = 1, 2 such that g = g_{1}g_{2}. We set then g ·l_{1} ⊗l_{2} = g_{1} ·l_{1} ⊗g_{2} ·l_{2}, ∀ l_{1} ⊗l_{2} ∈ X. This action is well defined, as g_{1}g_{2} = h_{1}h_{2}, if and only if there exists 0 ≤ a < q such that g_{1} = h_{1}z^{a} and g_{2} = h_{2}z^{−a}, and then we have g_{1}g_{2}·l_{1}⊗l_{2} = h_{1}z^{a}·l_{1}⊗h_{2}z^{−a}·l_{2} = ω^{a}ω^{−a}(h_{1}·l_{1}⊗h_{2}·l_{2}) = h_{1}h_{2}·l_{1}⊗l_{2}. Moreover, we have 1 · l_{1} ⊗ l_{2} = l_{1} ⊗ l_{2} and ((g_{1}g_{2})(h_{1}h_{2})) · l_{1} ⊗ l_{2} = (g_{1}g_{2})((h_{1}h_{2}) · l_{1} ⊗ l_{2}), ∀ g_{i}, h_{i} ∈ Q_{i}, ∀ i = 1, 2 and ∀ l_{1} ⊗ l_{2} ∈ X.

Lemma 4.2 X is isomorphic to L and consequently X is a simple kQ-module which extends to the kG-module L̃.

Proof . By Lemma 3.1, X ≅ L, if and only if (X_{K}↓_{Q′}^{Q}, L_{K}↓_{Q′}^{Q})_{Q′} = q^{4}, as X and L have dimension q^{2} and L is simple. We have z^{i} · (l_{1} ⊗ l_{2}) = (z^{i} · l_{1}) ⊗ l_{2} = ω^{i}(l_{1} ⊗ l_{2}), ∀ l_{1} ⊗ l_{2} ∈ X and it follows then that (X_{K}↓_{Q′}^{Q}, L_{K}↓_{Q′}^{Q})_{Q′} = (L_{K}↓_{Q′}^{Q}, L_{K}↓_{Q′}^{Q})_{Q′} = q^{4}.

□

Let H = Q ⋊ R and H_{i} = Q_{i} ⋊ R, i = 1 and 2. By construction, H_{1} and H_{2} are isomorphic to the 2-nilpotent group constructed for the quaternion case, with q ≡ |R| − 1(mod 2|R|).

Lemma 4.3 Let L̃_{i} be the extension of the simple kQ_{i}-module L_{i} to the 2-nilpotent group H_{i}, for i = 1 and 2. Then L̃_{1} ⊗ L̃_{2} is the unique simple kH-module (up to isomorphism) that extends X, for the action of H given by g_{1}g_{2}s · l_{1} ⊗ l_{2} = g_{1}s · l_{1} ⊗ g_{2}s · l_{2}, ∀ l_{1} ⊗ l_{2} ∈ L̃_{1} ⊗ L̃_{2}, ∀ g_{i} ∈ Q_{i}, i = 1, 2 and ∀ s ∈ R. Moreover, L̃_{1} ⊗ L̃_{2} ≅ L̃↓_{H}^{G}.

Proof . By definition of the external tensor product, L̃_{1} ⊗ L̃_{2} is a kQ-module for the action ”component by component” of Q (see definition 10.15 in [CR]) and it is also a kR-module for the diagonal action of R. So we just need to check that both actions are compatible, i.e. if s ∈ R, g_{i} ∈ Q_{i}, i = 1, 2 and l_{1} ⊗ l_{2} ∈ L̃_{1} ⊗ L̃_{2}, then (sg_{1}g_{2}) · l_{1} ⊗ l_{2} = s · (g_{1}g_{2} · l_{1} ⊗ l_{2}). We have:

(sg_{1}g_{2}) · l_{1} ⊗ l_{2} = (^{s}g_{1} ^{s}g_{2}s) · l_{1} ⊗ l_{2} = ^{s}g_{1}s · l_{1} ⊗ ^{s}g_{2}s · l_{2} = sg_{1} · l_{1} ⊗ sg_{2} · l_{2} = s · (g_{1} · l_{1} ⊗ g_{2} · l_{2}).

It follows that L̃ ↓_{H}^{G} and L̃_{1} ⊗ L̃_{2} are extensions of X to H. So they must be isomorphic, by Lemma 3.2.

□

By transitivity of restriction it follows that L̃↓_{R}^{G}≅ (L̃_{1} ⊗ L̃_{2})↓_{R}^{H} ≅ (L̃_{1}↓_{R}^{H_{1}}) ⊗ (L̃_{2}↓_{R}^{H_{2}}). By the previoussection, we know that L̃_{i}↓_{R}^{H_{i}} has an indecomposable direct summand L_{o}, with vertex R, isomorphic to Ω_{R}^{1}(k) or to Ω_{R}^{3}(k), for i = 1 and 2. Thus Ω_{R}^{2}(k) | L̃↓_{R}^{G}.

Consider L̃↓_{E}^{G}, with E = <w, v> and B = {Y_{ij} = (y_{1}^{i} ⊗ 1) ⊗ (y_{2}^{j} ⊗ 1) | 0 ≤ i, j < q}, as a k-basis of L̃. Let us write again a, b and l for the same integers as in the previous subsection, that is 0 ≤ a, b, l < q, ^{w}x = x^{−1}z^{a}, ^{w}y = y^{−1}z^{b} and 2l ≡ −a (mod q). Set v · Y_{ij} = Y_{ji}, w · Y_{ij} = ω^{(i+j+a)b}Y_{−i−a,−j−a} and extend k-linearly this action to L̃. It is well defined, as we have Y_{ij} = 1 · Y_{ij} = v^{2} · Y_{ij} = w^{2} · Y_{ij} and v · w · Y_{ij} = ω^{(i+j+a)b}Y_{−j−a,−i−a} = w · v · Y_{ij}, ∀ Y_{ij} ∈ B. Moreover, if we define gs · Y_{ij} = g · (s · Y_{ij}), ∀ g ∈ Q, ∀ s ∈ E and ∀ Y_{ij} ∈ B, then we get a well defined k[Q ⋊ E]-module, as it can be easily checked that the actions of Q and E are compatible. So, from unicity of extensions, we deduce that we have described the (unique) action of E on L̃.

Lemma 4.4 L̃↓_{E}^{G} is isomorphic to k ⊕ (k[E/<v>])^{\frac{q−1}{2}} ⊕ (k[E/<vw>])^{\frac{q−1}{2}} ⊕ (kE)^{( \frac{q−1}{2} )^{2}} as KEmodule.

Proof . Consider the following k-subspaces of L̃ ([α] denotes the integer part of the real number α):

k<Y_{ll}>, W_{i} = k<Y_{ii}, ω^{(2i+a)b}Y_{l−i,l−i}>, ∀ [\frac{l + 1}{2}] ≤ i ≠ l ≤ [\frac{q + l}{2} ],

W′_{i} = k<Y_{l+i,l−i}, Y_{l−i,l+i}>, ∀ 0 < i ≤ \frac{q − 1}{2}, and , ∀ 0 ≤ i, j < q, avec δ_{ij} = δ_{i+j,−a} = 0,

W_{ij} = k<Y_{ij}, Y_{ji}, ω^{(i+j+a)b}Y_{−i−a,−j−a}, ω^{(i+j+a)b}Y_{−j−a,−i−a}>.

There are \frac{q−1}{2} distinct subspaces W_{i} and W′_{i}, and (\frac{q−1}{2})^{2} distinct subspaces W_{ij}. They are all permutation kE-submodules of L̃. Indeed, k<Y_{ll}> is isomorphic to the trivial kE-module, W_{i} is isomorphic to k<Y_{ii}>↑_{<v>}^{E}≅ k[E/<v>] as kE-module, because v acts trivially on W_{i}, while w acts

freely, and that is true for any integer i such that [\frac{l+1}{2}] ≤ i ≠ l ≤ [\frac{q+l}{2}]. The same argument

shows that W′_{i} is isomorphic to k[E/<vw>] as kE-module, ∀ 0 < i ≤ \frac{q−1}{2}. The subspaces W_{ij} are free kE-modules of rank 1, as they are isomorphic to k<Y_{ij}>↑_{1}^{E}. Moreover, all these modules intersect trivially, as the vectors Y_{ij} are linearly independent (over k) and no one appears in two different subspaces k<Y_{ll}>, W_{i}, W′_{i} or W_{ij}. It follows that the kE-submodule they generate is a direct sum and it has dimension 1 + (\frac{q−1}{2}) · 2 + (\frac{q−1}{2}) · 2 + (\frac{q−1}{2})^{2} · 4 = q^{2} = dim L̃. So we have

L̃↓_{E}^{G} ≅ k ⊕ (k[E/<v>])^{\frac{q−1}{2}} ⊕ (k[E/<vw>])^{\frac{q−1}{2}} ⊕ (kE)^{( \frac{q−1}{2} )^{2}}.

□

In particular, we deduce from this that L̃↓_{Z}^{G}≅ k ⊕ (kZ)^{( \frac{q^{2}−1}{2} )}, as kZ-modules, and so S↓_{Z}^{G}≅ k ⊕ F, where Z = <w> is the center of P and F a free kZ-module.

Lemma 4.5 Let P be a semi-dihedral 2-group, Z be its center and M an indecomposable endopermutation kP -module with vertex P. Suppose that the restriction M ↓_{Z}^{P} is an endo-trivial kZmodule. Then M is endo-trivial.

Proof . As M is indecomposable with vertex P, Lemma 1.2 in [BoTh] implies that M is endotrivial, if and only if Def_{N_{P} (Q)/Q}^{N_{P} (Q)} Res_{N_{P} (Q)}^{P}([M]) = [k], for all non-trivial subgroups Q. If Q is a non-trivial subgroup of P, then either Q contains Z, or Q is conjugate to <v> and its normalizer is conjugate to E. In this latter case, N_{P} (Q)/Q is cyclic of order 2 and D(N_{P} (Q)/Q) = {[k]}. It follows that Def_{N_{P} (Q)/Q}^{N_{P} (Q)} Res_{N_{P} (Q)}^{P}([M]) = [k].

Suppose that Q contains Z. Then we can decompose the map Def_{N_{P} (Q)/Q}^{N_{P} (Q)} Res_{N_{P} (Q)}^{P} as the

composition Def_{N_{P} (Q)/Q}^{N_{P} (Q)/Z} Res_{N_{P} (Q)/Z}^{P/Z} Def_{P/Z}^{P}. By hypothesis, M ↓_{Z}^{P} is isomorphic to k ⊕ F, for a free kZ-module F. Let A = End_{k} M. Then A is isomorphic to k ⊕ F′ as kZ-module, for a free kZmodule F′. Thus we have, Def_{P/Z}^{P}([A]) = [A^{Z}/A_{1}^{Z}] = [k] = [End_{k} k], and so Def_{P/Z}^{P}([M]) = [k]. It

follows that Def_{N_{P} (Q)/Q}^{N_{P} (Q)/Z} Res_{N_{P} (Q)/Z}^{P/Z} Def_{P/Z}^{P}([M]) = [k] and hence M is endo-trivial. Recall that A^{Z} denotes the k-subalgebra of A formed by all fixed points of A under the action (by conjugacy) of Z, and A_{1}^{Z} is the image of the relative trace map A ⟶ A^{Z}, φ ↦ φ + ^{w}φ, ∀ φ ∈ A.

□

Let us apply these two results to S, as S is an indecomposable endo-permutation kP -module with vertex P and we have S ↓_{Z}^{P} ≅ k ⊕ F. From Lemma 4.5, we deduce that S is endo-trivial. Moreover, as S↓_{E}^{P} | L̃↓_{E}^{G}, Lemma 4.4 implies that S↓_{E}^{P}∈ [k]. So, S is an endo-trivial kP -module and its restrictions to R and E belong to the classes of Ω_{R}^{2}(k) and k respectively. By injectivity of the restriction T (P) ⟶ T (R) ⊕ T (E) (see Theorem 2.8 in [CaTh]), we conclude that S is isomorphic to the indecomposable endo-trivial kP -module Ω_{P}^{1} (Ω_{P/<v>}^{1}(k)), of dimension \frac{|P |}{2} + 1.

5 Inflation, tensor induction and tensor product

In the previous section, we explicitly realized a few examples of endo-permutation modules as sources of simple modules for finite p-nilpotent groups. Now, instead of going on with explicit examples, we are going to provide tools that will allow for realizations of many other modules.

5.1 Inflation

Let P′ be a normal subgroup of P and write P̄ the quotient P/P′. Let M be an indecomposable endo-permutation kP̄-module with vertex P, such that we can explicitly construct a finite pnilpotent group G having P̄ as p-Sylow subgroup and a simple kG-module L with source isomorphic to M. We are going to prove that we can explicitly realize InfP ; P̄ M as source (recall that this is an indecomposable endo-permutation kP -module with vertex P, see Proposition 3.17 in [Da1]). Write G = Q ⋊ P̄, where Q is a finite p′-group, and σ : P̄ ⟶ Aut(Q) the group homomorphism corresponding to the conjugation of elements of Q by elements of P in G. Consider the map σ′ : P ⟶ Aut(Q), u ↦ σ(ū), ∀ u ∈ P. It is a group homomorphism and so defines a finite p-nilpotent group G′ = Q ⋊ P. Moreover, as σ′(u) = Id_{Q}, ∀ u ∈ P′, the subgroup {1} × P′ is normal in G′ and the quotient is isomorphic to G. Let L′ = Inf_{G}^{G′} L. It is a simple kG′-module with vertex P and its source is isomorphic to InfP ; P̄ M. Indeed, we have by hypothesis M | L↓G ; P̄, and

so InfP ; P̄ M | InfP ; P̄(L↓G ; P̄) ≅ L′↓_{P}^{G′}.

5.2 Tensor induction

Let C be a subgroup of P and M an indecomposable endo-permutation kC-module with vertex C, such that we can explicitly construct a finite p-nilpotent group G and a simple kG-module L with source isomorphic to M. Write G = Q ⋊ C, where Q has order prime to p. Suppose moreover that the restriction L↓_{Q}^{G} is a simple kQ-module. In this section, we are going to realize the cap of Ten_{C}^{P} M as source of a simple module for a finite p-nilpotent group having P as p-Sylow subgroup.

Let [P/C] = {v_{0}, …, v_{r}} be a set of representatives of the left cosets of P/C and assume v_{0} = 1. For each 0 ≤ i ≤ r, we formally define a group ^{v_{i}}Q = {^{v_{i}}g | g ∈ Q}, where the composition law is given by ^{v_{i}}g^{v_{i}}h = ^{v_{i}}(gh), ∀ g, h ∈ Q. Let H = ∏_{i=0}^{r} ^{v_{i}}Q. We let P act on H as follows. Let (^{v_{i}}h_{i})_{i=0}^{r} ∈ H and u ∈ P. As left multiplication by u acts by permutation on [P/C], there exists an unique σ_{u} ∈ S_{r+1} (where S_{r+1} acts on {0, …, r}), such that uv_{i} = v_{σ_{u}(i)}c(u, i) for an

unique c(u, i) ∈ C, ∀ 0 ≤ i ≤ r. We set then (^{u}^{v_{i}}h_{i})_{i=0}^{r} = (^{v_{i}}(^{c(u,σ_{u}^{−1}(i))}h_{σ_{u}^{−1}(i)}))_{i=0}^{r}. We can then

check directly that ^{1}h = h and that ^{(uu′)}h = (^{u}^{u′}h), ∀ h ∈ H, ∀ u, u′ ∈ P (as σ_{uu′} = σ_{u}σ_{u′} and

c(uu′, i) = c(u, σ_{u′}(i))c(u′, i), ∀ 0 ≤ i ≤ r). So we have a p-nilpotent group G = H ⋊ P, in which the groups ^{v_{i}}Q, ∀ 0 ≤ i ≤ r are all conjugates. Moreover, for all 0 ≤ i ≤ r, the k[^{v_{i}}Q]-module v_{i}(L↓_{Q}^{G}) is simple and so, as k is algebraically closed, the external tensor product L̃ = ⨂_{i=0}^{r} ^{v_{i}}(L↓_{Q}^{G}) is a simple kH-module.

Set, for all u ∈ P and for all x = ⊗_{i=0}^{r}x_{i} ∈ L̃, u ∗ x = ⨂_{i=0}^{r} ^{v_{i}(}c(u, σ_{u}^{−1}(i)) · x_{σ_{u}^{−1}(i)}), where · denotes the action of G on L. Then, L̃ turns out to be a kP -module, if we extend this action k-linearly to L̃. Indeed, ∀ u, u′ ∈ P and ∀ x = ⊗_{i=0}^{r}x_{i} ∈ L̃, we have 1 ∗ x = x et u ∗ (u′ ∗ x) =

= u ∗ (⨂_{i=0}^{r}^{v_{i}(}c(u′, σ_{u′}^{−1}(i)) · x_{σ_{u′}^{−1}(i)}))= ⨂_{i=0}^{r} ^{v_{i}}(c(u, σ_{u}^{−1}(i))·(c(u′, σ_{u′}^{−1}(σ_{u}^{−1}(i))) · x_{σ_{u′}^{−1}(σ_{u}^{−1}(i))})) =

= ⨂_{i=0}^{r} ^{v_{i}}((c(u, σ_{u}^{−1}(i))c(u′, σ_{u′}^{−1}(σ_{u}^{−1}(i)))) · x_{σ_{u′}^{−1}(σ_{u}^{−1}(i))}) = ⨂_{i=0}^{r} ^{v_{i}(}c(uu′, σ_{uu′}^{−1}(i)) · x_{σ_{uu′}^{−1} (i)}) =

= (uu′) ∗ x.

Then, if we set (hu) ∗ x = h · (u ∗ x), ∀ h ∈ H, ∀ u ∈ P and ∀ x ∈ L̃, the actions of P and H are compatible, and so L̃ becomes a kG′-module. Indeed, ∀ h ∈ H, ∀ u ∈ P and ∀ x ∈ L̃, we have:

(uh) ∗ x = ⨂_{i=0}^{r} ^{v_{i}(}^{c(u,σ_{u}^{−1}(i))}h_{σ_{u}^{−1}(i)}) · (u ∗ x) = ⨂_{i=0}^{r} ^{v_{i}(}^{c(u,σ_{u}^{−1}(i))}h_{σ_{u}^{−1}(i)} · (c(u, σ_{u}^{−1}(i)) · x_{σ_{u}^{−1}(i)})) =

= ⨂_{i=0}^{r} ^{v_{i}(}c(u, σ_{u}^{−1}(i))h_{σ_{u}^{−1}(i)} · x_{σ_{u}^{−1}(i)}) = u ∗ (h · x).

Moreover, we have L̃↓_{P}^{G′} ≅ Ten_{C}^{P}(L↓_{C}^{G}), as they are isomorphic as k-vector spaces (of dimension (dim L)^{r+1}) and as the action of P on the generators coincides on both modules. As the k[^{v_{i}}C]- module ^{v_{i}}M is a direct summand of ^{v_{i}}(L↓_{C}^{G}), for all 0 ≤ i ≤ r, it follows that Ten_{C}^{P} M | Ten_{C}^{P}(L↓_{C}^{G}) as kP -modules and so the source of L̃ is isomorphic to the cap of Ten_{C}^{P} M.

Remark 5.1 Let G = Q ⋊ C be a finite p-nilpotent group, with Q having order prime to p, and C a p-group, and let L be a simple kG-module. The additional hypothesis we made above, namely that L↓_{Q}^{G} is a simple kQ-module, is not restrictive. Indeed, if S is a simple kQ-module belonging to a block e of kQ and I = {g ∈ G | ^{g}S ≅ S} is the inertial subgroup of S, then, by Clifford theory, we know that S can be extended to a simple kI-module S̃ (which belongs to the block e of kI). Moreover, the relative trace from I to G sends b on a block of kG and the k-algebras kIe and kGb are Morita equivalent. This equivalence lets S̃ correspond to the simple kG-module Ind_{I}^{G}(S̃). It follows that, if L is a simple kG-module, then its restriction to kQ is not a simple module, in general. However, L↓_{Q}^{G} is a finite direct sum ⊕_{i=0}^{r} ^{v_{i}}S of conjugate simple modules (we can assume that v_{0} = 1), having conjugate inertial subgroups I_{i}, which are strictly contained in G. We also know that L ≅ Ind_{I_{0}}^{G} S̃ and so, in this situation, instead of constructing the simple kG′-module L̃ using L↓_{Q}^{G}, we consider the simple kQ-module S. Notice, however, that in every situation we realized until now, the additional hypothesis we make here was fulfilled.

5.3 Tensor product

Let C be a family of indexes (e.g. C is a set of conjugacy classes of non-trivial cyclic subgroups of P). Suppose that for each C ∈ C there is an indecomposable endo-permutation kP -module M_{C}, such that we can explicitly construct a p-nilpotent group G_{C} = Q_{C} ⋊ P (Q_{C} of order prime to p) and a simple kG_{C}-module L_{C} with source isomorphic to M_{C}. We prove the following result.

Theorem 5.2 If L_{C}↓_{Q_{C}}^{G_{C}} is a simple kQ_{C}-module, ∀ C ∈ C, then there exists a p-nilpotent group G and a simple kG-module L̃ with source isomorphic to the cap of ⊗_{C∈C}M_{C}.

Proof . Let G = (∏_{C∈C} Q_{C}) ⋊ P, where P acts diagonally on ∏_{C∈C} Q_{C}, i.e.

(^{u}h_{C})_{C∈C} = (^{u}h_{C})_{C∈C}, ∀u ∈ P, ∀(h_{C})_{C∈C} ∈ ∏_{C∈C} Q_{C}.

This is a well defined semi-direct product, as P ∩ ( ∏_{C∈C} Q_{C}) = {1}, (^{1}h_{C})_{C∈C} = (h_{C})_{C∈C}

and (^{(uu′)}h_{C})_{C∈C} = (^{(uu′)}h_{C})_{C∈C} = ( (^{u}^{u′}h_{C}))_{C∈C} = (^{u}^{u′}(h_{C})_{C∈C}), ∀u, u′ ∈ P and ∀(h_{C})_{C∈C} ∈

∏_{C∈C} Q_{C}. The external tensor product (over k) L = ⨂_{C∈C}(L_{C} ↓_{Q_{C}}^{G_{C}}) is a simple k[∏_{C∈C} Q_{C}]- module, as k is algebraically closed. Moreover, we can extend L to a simple kG-module L̃, by letting P act diagonally: u · ⊗_{C∈C}l_{C} = ⊗_{C∈C}u · l_{C}, and setting then ((h_{C})_{C∈C}u) · ⊗_{C∈C}l_{C} = (h_{C})_{C∈C} · (u · ⊗_{C∈C}l_{C}), ∀u ∈ P, ∀ (h_{C})_{C∈C} ∈ ∏_{C∈C} Q_{C} and ∀ ⊗_{C∈C} l_{C} ∈ L̃, where · denotes the action of P on the kG_{C}-module L_{C}, ∀ C ∈ C. Next calculation shows that the actions of P and ∏_{C∈C} Q_{C} are compatible:

(u((h_{C})_{C∈C})·⊗_{C∈C}l_{C} = (^{u}h_{C})·u·⊗_{C∈C}l_{C} = ⊗_{C∈C}u·h_{C} ·l_{C} = u·⊗_{C∈C}h_{C} ·l_{C} = u·(h_{C})_{C∈C} ·⊗_{C∈C}l_{C}.

So we get a finite p-nilpotent group G, with P as p-Sylow subgroup and a simple kG-module L̃, with vertex P, because the dimension of each L_{C}, and thus the dimension of L̃, is prime to p. By hypothesis, M_{C} is a direct summand of L_{C}↓_{P}^{G_{C}} , ∀C ∈ C. It follows that ⨂_{C∈C} M_{C} is a direct summand of ⨂_{C∈C}(L_{C}↓_{P}^{G_{C}}), which is clearly isomorphic to L̃↓_{P}^{G} as kP -module. Hence, the source of the simple kG-module L̃ is isomorphic to the cap of ⊗_{C∈C}M_{C}.

□

Let us apply this to the following two examples.

Corollary 5.3 Let p = 2 and P be a quaternion 2-group. Then every indecomposable torsion endopermutation kP -module with vertex P can be explicitly realized as source of a simple kG-module for a finite 2-nilpotent group G having P as 2-Sylow subgroup.

Proof . It follows by [CaTh] that every indecomposable torsion endo-permutation module belongs to the class of an endo-trivial module. More precisely, we have T^{t}(P) ≅ ℤ/4ℤ×ℤ/2ℤ, where ℤ/4ℤ is generated by the class of a module of dimension |P |−1 and ℤ/2ℤ by the class of an ”exceptional” endo-trivial module, of dimension \frac{|P |}{2} + 1. Thus, as we noticed in section 4, the classes of the two modules we realized in section 3 generate the whole torsion subgroup D^{t}(P) of D(P) and so, by Theorem 5.2, every indecomposable torsion endo-permutation kP -module with vertex P can be explicitly realized (up to isomorphism).

□

Corollary 5.4 Let p be an odd prime and P a metacyclic p-group. Then every indecomposable torsion endo-permutation kP -module with vertex P can be explicitly realized as source of a simple kG-module for a finite p-nilpotent group G having P as p-Sylow subgroup.

Proof . By the main theorem of [Ma], the set {[Ten_{C}^{P} Inf_{C/Φ(C)}^{C} Ω_{C/Φ(C)}^{1}(k)], C ∈ C} , where C is the set of conjugacy classes of non-trivial cyclic subgroups of P, forms a basis of the 𝔽_{2}-vector space D^{t}(P) (all torsion elements have order 2). In other words, every indecomposable torsion endo-permutation kP -module with vertex P is isomorphic to a direct summand of a kP -module of the form ⊗_{C∈C′} Teninf_{C/Φ(C)}^{P} Ω_{C/Φ(C)}^{1}(k), where C′ ⊆ C. In section 3, we realized explicitly Ω_{C/Φ(C)}^{1}(k) as source, as C/Φ(C) is cyclic of order p. Thus, by subsections 5.1 and 5.2, we can do the same for the cap of each Teninf_{C/Φ(C)}^{P} Ω_{C/Φ(C)}^{1}(k), where C ∈ C, and this construction satisfies the hypothesis of Theorem 5.2. It follows that every indecomposable torsion endo-permutation kP -module with vertex P can be explicitly realized as source.

□

6 Conclusion

If we sum up the results we obtained in the previous sections, then we come to the following conclusion.

Theorem 6.1 Let p be a prime, P a finite p-group and k an algebraically closed field of characteristic p. Then the cap of any kP -module of the form ⨂_{(Q/R)∈S} Ten_{Q}^{P} Inf_{Q/R}^{Q}(M_{Q/R}), where M_{Q/R} is an indecomposable torsion endo-trivial module with vertex Q/R, and S is a set of sections of P that are cyclic, or also quaternion or semi-dihedral, in case p = 2, can be explicitly realized as source of a simple module with vertex P for some finite p-nilpotent group.

Proof . By section 5, if we know how to realize explicitly M_{Q/R}, ∀ Q/R ∈ S, then we are able to realize the cap of a tensor product ⨂_{Q/R∈S} Ten_{Q}^{P} Inf_{Q/R}^{Q}(M_{Q/R}). Thus, it is enough to check that we have an explicit realization for any indecomposable torsion endo-trivial module with vertex P, for a finite cyclic, quaternion or semi-dihedral p-group P. And, in fact, we just need to consider a set of modules whose classes generate the torsion subgroup T^{t}(P) of T (P), as tensor product corresponds to linear combination in D(P). If P is cyclic, then T^{t}(P) is generated by the class of Ω_{P}^{1} (k), that we realized as our first example. If p = 2 and P is quaternion, then the statement is proved in Corollary 5.3. Finally, if p = 2 and P is semi-dihedral, then, T^{t}(P) is cyclic of order 2, generated by the class of the module we realized explicitly in the semi-dihedral example. So the statement is proved.

□

Let us now take into account Theorem 8.39 and Remark 8.41 in [Pu1]. We can sum up them into the following statement: let p be a prime, k an algebraically closed field with characteristic p and G a finite p-solvable group having a p-Sylow subgroup P and a simple kG-module L with vertex P. Then, the source of L is the cap of a module of the shape ⨂_{Q/R∈S} Ten_{Q}^{P} Inf_{Q/R}^{Q}(M_{Q/R}), where the set S and the modules M_{Q/R} are as in Theorem 6.1. Hence, the following corollary is straightforward.

Corollary 6.2 Let p be a prime, P a finite p-group and k an algebraically closed field of characteristic p. Then any indecomposable endo-permutation kP -module M with vertex P that can be a source of a simple module with vertex P for a finite p-solvable group can be explicitly realized as source for a finite p-nilpotent group.

Finally, we would like to come back to the last remark of the introduction to draw the main consequence of our theorem, in case both following conjectures hold, namely that, if the source of a simple module is an endo-permutation module, then its class should be a torsion element in the Dade group, and that there are no other indecomposable torsion endo-permutation modules as the ones that are isomorphic to the cap of a module of the form ⨂_{Q/R∈S} Ten_{Q}^{P} Inf_{Q/R}^{Q}(M_{Q/R}), as in our main theorem. At least, if we trust Jon Carlson’s recent result, then this latter conjecture holds for p odd and so, if the first conjecture is also true, then it follows that any torsion endopermutation module is the source of a simple module and it can be explicitly realized as it for a finite p-nilpotent group (p odd).

References

[BoTh] S. Bouc, J. Thévenaz, The group of endo-permutation modules, Invent. math. 139 (2000), 275–349.

[CaTh] J. Carlson, J. Thévenaz, Torsion endo-trivial modules, Algebr. Represent. Theory 3 (2000), 303–335.

[CR] C. W. Curtis, I. Reiner, Methods of representation theory with applications to finite groups and orders I, Pure and applied mathematics, John Wiley and Sons, 1981.

[Da1] E. C. Dade, Endo-permutation modules over p-groups, I, II, Ann. Math. 107 (1978), 459– 494, 108 (1978), 317–346.

[Da2] E. C. Dade, Blocks with cyclic defect groups I, Ann. Math. 84 (1966), 20-48.

[DH] K. Doerk, T. Hawkes, Finite solvable groups, Walter de Gruyter, 1992.

[Go] D. Gorenstein, Finite groups, Harper and Row, 1968.

[Hu] B. Huppert, Character theory of finite groups, Walter de Gruyter, 1998.

[Is] I. M. Isaacs, Character theory of finite groups, Acad. Press, 1976.

[Ma] N. Mazza, The Dade group of a metacyclic p-group, J. Algebra, to appear.

[Pu1] L. Puig, Notes sur les p-algèbres de Dade, Preprint, 1988.

[Pu2] L. Puig, Affirmative answer to a question of Feit, J. Algebra 131 (1990), 513–526.

[Se] J.-P. Serre, Représentation linéaires des groupes finis, Hermann, 1978.

[Se2] J.-P. Serre, Corps locaux, Hermann, 1962.

[Th] J. Thévenaz, G-algebras and modular representation theory, Oxford, 1995.