THE ISOMORPHISM PROBLEM FOR CYCLIC BLOCKS AND THEIR SOURCE ALGEBRAS

Markus Linckelmann June 1994

Typeset by AMS-TEX

1 Introduction

In the present paper we determine the structure of blocks of finite groups having cyclic defect groups in terms of their source algebras. Our results confirm our conjecture in [13], namely that the source algebras of a block having a cyclic defect group P are essentially determined by an endo-permutation module for P and the Brauer tree of the block. This implies in particular, that Puig’s finiteness conjecture on source algebras of blocks for a given defect group holds for blocks with cyclic defect groups.

The methods we use imply actually a stronger result: blocks with cyclic defect groups satisfy a blocktheoretic analogon to the isomorphism problem for finite groups, namely that, roughly speaking, a suitable normalisation of the group of invertible elements of the block algebra contains a unique conjugacy class of finite subgroups isomorphic to the defect groups. This result has first been proved by L. Scott in [24] for principal blocks satisfying a T.I.-condition (or, for principal blocks in general, using the classification of finite simple groups via a theorem of H. Blau); as Scott points already out, the main difficulty for a generalisation of his results to non principal blocks consists in finding a suitable ”normalisation” of the block algebra. The solution for this problem is a technique, due to L. Puig (see e.g. [20], section 5), which uses endo-permutation modules to modify algebras (and which allows one to remove any reference to the classification of finite simple groups).

In order to describe briefly the concepts mentioned above, we fix some notation. Throughout the paper, p is a prime and O a complete discrete valuation ring having a residue field k of characteristic p.

A block b of a finite group G is a primitive idempotent of the center Z(OG) of the group algebra OG of G over O , and the algebra OGb is called the block algebra of b ; this is an indecomposable symmetric O−algebra which is O−free of finite rank as an O−module. A fundamental invariant of b is a certain conjugacy class of p−subgroups of G, called defect groups of b ; a defect group P of b is characterised by the property that it is minimal such that b can be written as a relative trace b = ∑_{x} x^{−1}ax for some

P −stable element a in OG , where in the sum x runs over a right transversal of P in G. This notion goes back to R. Brauer (see e.g. [3]).

The source algebras of b , a concept due to L. Puig [18] extending Green’s notion of sources of modules, are the algebras iOGi, where i is a primitive idempotent in the subalgebra (OGb)^{P} of P −stable elements which cannot be written as a relative trace for any proper subgroup of P , always considered as interior P −algebras; that is, endowed with the group homomorphism from P to the group of invertible elements in iOGi mapping u ∈ P to ui. All source algebras of b are isomorphic as O−algebras, but may differ by some automorphism of P considered as interior P −algebras (see e.g. [16], section 5 for more details); they are Morita equivalent to the block algebra OGb and contain all invariants currently associated with the block b (such as the generalised decomposition numbers, for instance).

Endo-permutation modules have been introduced by E. C. Dade [6]; they form a certain class of modules for p−groups, which play an important rôle in the determination of source algebras (e.g. of nilpotent blocks [20] and their extensions [11]). More precisely, an OP −module N, where P is a finite p−group, is an endo-permutation module if N^{∗} ⊗_{O} N is a permutation module, or, equivalently, if the algebra S = End_{O}(N),

considered as interior P −algebra, has a P −stable O−basis.

The Brauer tree of a block with cyclic defect groups goes back to work of Brauer [2] and Dade [5]; this is a tree with some additional information which describes in a combinatorical way the Morita equivalence class of the block algebra kG\overline{b} over k (where \overline{b} is the image of b in kG), thus equivalent to the Gabriel quiver of kG\overline{b} (this is for instance explained in Gabriel-Riedtmann [9]).

The study of blocks of finite groups in general and of blocks with cyclic defect groups in particular was initiated by R. Brauer [2]; since then many authors have published on that subject. See for instance W. Feit [8],ch.VII , for a historical account and an extensive bibliography, as well as J. L. Alperin [1] for an introduction to the subject.

We proceed as follows. After stating our main results in section 2, we recall in section 3 Broué’s notion of a stable equivalence of Morita type and its properties as far as we need them here. Section 4 is devoted to the ”local case” of a semi-direct product P ⋊ E of a cyclic p−group P by a p′−subgroup E of Aut(P). The ingredients for the step from the “local” to the “global case” are given in section 5; we make here a systematic use of stable equivalences of Morita type, which considerably simplifies parts of the proofs given in a preliminary version [15] of this paper. In section 6 finally, we conclude the proofs of our main results.

Our notation is standard. We refer to [16], section 2 to 7, for an introduction to some of Puig’s concepts and techniques we require here; our conventions are as in [17]. An O−algebra A is always associative, unitary and free of finite rank as an O−module. We say that A is split if its semisimple quotient A/J(A) is a direct sum of matrix algebras over k. An A−module is always a finitely generated unitary left module, not necessarily O−free, unless stated otherwise. If A is endowed with a group homomorphism from a finite group H to the group of invertible elements in A, that is, A is an interior H-algebra in the terminology of [18], we denote by Aut(A) the group of automorphisms of A as an O−algebra, by Aut_{H}(A) the subgroup of all automorphisms of A as interior H−algebra (i.e. which fix the image of H in A elementwise) and by Aut_{0}(A) the subgroup of Aut(A) stabilising the isomorphism classes of all simple A−modules. We denote by Ãut_{H}(A), Ãut_{0}(A) the corresponding images in the quotient Ãut(A) of Aut(A) by all inner automorphisms of A.

If σ is the structural homomorphism from H to the group of invertible elements in A defining the interior H−algebra structure on A, we say loosely, that a group automorphism φ of H extends to an algebra automorphism α of A, or that φ is induced by α, if we have σφ = ασ.

If α is an automorphism of A and U an A-module, we denote _{α}U the A−module obtained from U by restriction through α ; we use similar notations for right modules and bimodules. We denote by Ω_{A} (or Ω_{G} if A = OG) the usual Heller operator, assuming implicitly a choice of projective covers. We often use without comment the well-known fact that a Morita equivalence on the module category of an O−algebra A is induced by an automorphism if and only if it preserves the dimension of any simple module.

We allow the case O = k unless explicitly stated otherwise.

2 Statements of the main results

Let G be a finite group, b a block of G having a non trivial cyclic defect group P and I a set of representatives of the conjugacy classes of primitive idempotents in OGb ; thus {OGi}_{i∈I} is a set of representatives of the isomorphism classes of projective indecomposable OGb−modules. Denote Z the unique subgroup of order p of P.

In this section, we assume that k is a splitting field for G and its subgroups and that O has characteristic zero.

In order to make our statements, we recall some well-known properties of blocks with cyclic defect groups.

2.1. There are unique permutations ρ, σ on I and, for any i ∈ I unique submodules U_{i} , V_{i} of OGi such that there are exact sequences of OGb−modules

0 ⟶ U_{σ(i)} ⟶ OGσ(i) ⟶ V_{i} ⟶ 0,
0 ⟶ V_{ρ(i)} ⟶ OGρ(i) ⟶ U_{i} ⟶ 0,

and Z acts trivially on a source of U_{i} , for any i ∈ I .

The last condition in the above statement guarantees the uniqueness of ρ, σ, U_{i}, V_{i} (since without that condition we could exchange simultaneously ρ, σ and the U_{i}, V_{i}, where i ∈ I) .

Actually, it is possible to be much more precise about the structure of the modules U_{i}, V_{i} (see e.g. [12]); we need here from [12], section 15

2.2. The modules U_{i}, i ∈ I, are indecomposable, have vertex P, and up to isomorphism a unique common source N; the module N is then an indecomposable endopermutation OP −module having determinant one, on which Z acts trivially; moreover, Ω_{P} (N) is a source of V_{i}, i ∈ I.

Observe that since Z acts trivially on N, certainly Z does not act trivially on Ω_{P} (N) ; this is why we can use this condition for the uniqueness statement in 2.1. above.

2.3. The rational number m = (|P |−1)/|I| is an integer, and if m ≠ 1 there is exactly one orbit v of either ρ or σ on I such that, for all i ∈ v, the rank of End_{OG}(U_{i}) (if v is a ρ−orbit) or End_{OG}(V_{i}) (if v is a σ−orbit) equals m; otherwise End_{OG}(U_{i}) and End_{OG}(V_{i}) have rank one.

Usually, m is called the exceptional multiplicity and v the exceptional orbit of the block. In order to simplify the formalism, in the sequel we set v = I if m = 1.

2.4. The graph whose set of vertices is the disjoint union of the sets of ρ−orbits and σ−orbits on I and whose set of edges is the set I such that the edge i ∈ I links the ρ−orbit and σ−orbit containing i is a tree; in particular, ρσ is a transitive cycle on I and the intersection of a ρ−orbit and a σ−orbit is either empty or contains a unique element of I.

2.5. We call Brauer tree of the block b the triple (I, {ρ, σ}, v). By 2.4 above, we can visualise the Brauer tree as a tree (often referred to as the ‘Brauer tree’ in the literature; see also remark 2.8 below) together with a circular order at each set of edges having a common vertex defined by ρ and σ (which amounts to realising the tree in the plane), together with possibly an exceptional vertex corresponding to the exceptional orbit. Note that any set of two permutations on I defines a graph as in 2.4, but not necessarily a tree.

We call Brauer tree automorphism group of b the subgroup F of the symmetric group S_{I} on I stabilising {ρ, σ} and v; actually, as we prove in [17],4.5, all elements of F centralise then ρ and σ; in particular, F is a subgroup of the cyclic group of order |I| generated by ρσ in S_{I}. More generally, there is an obvious notion of isomorphism

(I, {ρ, σ}, v) ≅ (I′, {ρ′, σ′}, v′)

of Brauer trees (with the obvious notation): this is a bijective map τ : I → I′ sending v onto v′ and satisfying τ{ρ, σ}τ^{−1} = {ρ′, σ′}.

Similarly, distinguishing between ρ and σ gives an evident notion of isomorphism between the quadruples

(I, ρ, σ, v) ≅ (I′, ρ′, σ′, v′);

this is a bijective map τ : I → I′ sending v to v′ and satisfying τρτ^{−1} = ρ′ and τστ^{−1} = σ′ (thus, in particular, τ is an isomorphism of the corresponding Brauer trees).

Theorem 2.6. With the notation above, there is up to isomorphism a unique interior P −algebra A whose unit element 1_{A} is primitive in A^{P} and local in A, such that, setting S = End_{O}(N), any source algebra of OGb is isomorphic to e(S ⊗_{O} A)e as

interior P −algebra, where e belongs to the unique local point of P on S ⊗_{O} A. The

algebra A has the following properties:

(i) As interior P −algebra, A contains O(P ⋊ E) such that induction and restriction induce a stable equivalence of Morita type between A and O(P ⋊ E).

(ii) The isomorphism class of A as interior P −algebra is determined by the isomorphism class of the quadruple (I, ρ, σ, v).

(iii) Any group automorphism of P extends to an algebra automorphism of A and, if E ≠ 1, any algebra automorphism of A maps the image of P in A to some conjugate.

(iv) We have Ãut(A) ≅ Ãut_{P} (A) × Ãut_{0}(A) and Ãut_{P} (A) ≅ F; if E ≠ 1 then Ãut_{0}(A) ≅ Aut(P)/E and if E = 1 then Ãut_{0}(A) ≅ Aut(OP).

If we compare two blocks having a common cyclic defect group, then we obtain the following.

Theorem 2.7. Let b, b′ be blocks of the finite groups G, G′, respectively, having a common non trivial cyclic defect group P. Assume that k is a splitting field for all subgroups of G and G′.

Let A, A′ be the interior P −algebras in 2.6 and N, N′ be the endo-permutation OP −modules in 2.2 for b, b′, respectively.

Denote (I, {ρ, σ}, v), (I′, {ρ′, σ′}, v′) the Brauer trees of b, b′, respectively, as in 2.5.

(i) The algebras A, A′ are isomorphic as interior P −algebras if and only if they are isomorphic as O−algebras, and this in turn holds if and only if (I, ρ, σ, v) ≅ (I′, ρ′, σ′, v′). Moreover, in that case, if E ≠ 1, any O−algebra isomorphism from A to A′ maps the image of P in A to a conjugate of the image of P in A′.

(ii) The block algebras OGb and OG′b′ are Morita equivalent if and only if the Brauer trees of b and b′ are isomorphic. Moreover, in that case, we have either A ≅ A′ or A ≅ Ω_{P} (A′) as interior P −algebras.

(iii) The blocks b, b′ have isomorphic source algebras if and only if N ≅ N′ as OP −modules and (I, ρ, σ, v) ≅ (I′, ρ′, σ′, v′).

Remark 2.8 In view of 2.7 (iii) one might want to call Brauer tree the quadruple (I, ρ, σ, v) rather than (I, {ρ, σ}, v), which amounts to including, besides the Morita equivalence class, some additional “local” information about the considered block. We stick to our definition for “historical” reasons, that is, to be conformal with [1], [8], [9], for instance. Brauer’s original definition of this tree, given in [2] for the case |P | = p, uses only the ordinary decomposition numbers of b, hence yields a tree without the additional information on the cyclic order on each set of edges emanating from a common vertex, and therefore does not determine the Morita equivalence class (there are examples of blocks with cyclic defect groups such that the underlying abstract trees of the Brauer trees are isomorphic, but not the Brauer trees themselves, because there is no tree isomorphism respecting the circular order at each vertex).

3 Quoted results on stable equivalences of Morita type

Definition 3.1. (Broué [4]) Let A, B be O−algebras, M a B − A−bimodule and N an A − B−bimodule. We say that M, N induce a stable equivalence of Morita type between A and B if M, N are projective as left and right modules, and if N ⊗_{B} M ≅ A ⊕ X as A−A−bimodules, where X is a projective A−A−bimodule, and

M ⊗_{A} N ≅ B ⊕ Y as B − B−bimodules, where Y is a projective B − B−bimodule.

We consider the following special case of a stable equivalence of Morita type:

Definition 3.2. Let A, B be O−algebras and φ : B ⟶ A an O−algebra homomorphism. We say that φ induces a stable equivalence of Morita type between A and B if the bimodules _{φ}A and A_{φ} induce a stable equivalence of Morita type between A and B; that is, if (A_{φ}) ⊗_{B} (_{φ}A) ≅ A ⊕ X for some projective A − A−bimodule X and

φA_{φ} ≅ B ⊕ Y for some projective B − B−bimodule Y. If moreover, in that case, B is a subalgebra of A and φ the inclusion, we say that induction and restriction induce a stable equivalence of Morita type between A and B.

Remark 3.3. Observe that if an O−algebra homomorphism φ : B ⟶ A induces a stable equivalence of Morita type, necessarily φ is injective, since _{φ}A has a direct summand isomorphic to B as left B−module.

We recall from [17] the following properties of stable equivalences of Morita type, as far as we need them (we sketch proofs for the convenience of the reader):

Theorem 3.4. Let A, B be O−algebras such that k ⊗_{O} A, k ⊗_{O} B are self-injective

indecomposable non simple k−algebras whose semi-simple quotients are separable, M a B − A−bimodule and N an A − B−bimodule which induce a stable equivalence of Morita type between A and B.

(i) Up to isomorphism, M and N have unique indecomposable non-projective direct summands as bimodules.

(ii) If M is indecomposable as B − A−bimodule, for any simple A−module S the B−module M ⊗_{A} S is indecomposable and non projective as k ⊗_{O} B−module.

(iii)If for any simple A−module S the B−module M ⊗_{A}S is simple then the functor

M ⊗_{A} − is a Morita equivalence Mod(A) ≅ Mod(B).

Proof. (i) (Rouquier [23]) Let M′, M′′ be subbimodules of M such that M = M′⊕M′′. Then (N ⊗_{B} M′) ⊕ (N ⊗_{B} M′′) = N ⊗_{B} M ≅ A ⊕ X for some projective A − A−bimodule

X. Since A is indecomposable as A − A−bimodule we may assume that A is a direct summand of N ⊗_{B} M′. Then N ⊗_{B} M′′ is projective as A − A−bimodule. Tensoring by

M ⊗_{A} − implies that M′′ is projective as B − A−bimodule.

(ii) We may assume O = k. Note that soc(B⊗_{O}A^{0}) = soc(B)⊗soc(A^{0}), as A, B have

separable semi-simple quotients. If M is indecomposable, M has no projective direct summand and so soc(B)Msoc(A) = 0. Thus M ⊗_{A}soc(A) ≅ Msoc(A) is annihilated by

soc(B), and therefore has no non zero projective direct summand as left B−module, as B is self-injective. Since A is self-injective, any simple A−module occurs as direct summand of soc(A) and it follows that M ⊗_{A} S has no non zero projective direct

summand, thus is indecomposable non projective, as the functor M ⊗_{A} − induces a

stable equivalence.

(iii) By (i) we may assume that N is indecomposable. If M ⊗_{A} S is simple, where

S is a simple A−module, then by (ii) the A−module N ⊗_{B} M ⊗_{A} S is indecomposable,

thus isomorphic to S, as M and N induce mutually inverse stable equivalences. Since N ⊗_{B} M ≅ A ⊕ X for some projective A − A−bimodule X it follows that (A ⊗_{A} S) ⊕

(X ⊗_{A} S) ≅ S, thus X = 0. Therefore M, N induce mutually inverse equivalences

between Mod(A) and Mod(B).

Proposition 3.5. Let A, B be O−algebras such that k⊗_{O}A, k⊗_{O}B are indecomposable

non simple self-injective k−algebras, M a B−A−bimodule and N an A−B−bimodule such that M, N induce a stable equivalence of Morita type between A and B. Suppose that M, N are indecomposable. Then for any integer n, the B − A−bimodule Ω_{B⊗A^{0}}^{n}(M) is, up to isomorphism, the unique indecomposable non projective direct summand of both Ω_{B⊗B^{0}}^{n}(B) ⊗_{B} M and M ⊗_{A} Ω_{A⊗A^{0}}^{n}(A); moreover, the bimodules

Ω_{B⊗A^{0}}^{n}(M), Ω_{A⊗B^{0}}^{−n}(N) again induce a stable equivalence of Morita type between A and B.

Proof. This follows from [17],2.8 and [17],2.9.

4 The local case

We prove now the necessary results in the special case, where the group is the semidirect product P ⋊ E of a non trivial cyclic p−group P by a non trivial p′−subgroup E of Aut(P). It is well-known that P ⋊ E is a Frobenius group, it has a unique block, and every simple k(P ⋊ E)−module lifts, up to isomorphism, to a unique O−free O(P ⋊ E)−module of rank one on which then P acts trivially (here we use that E is non trivial). The Brauer tree in that case is a star with |E| edges and, if |E| ≠ |P | − 1, with the exceptional vertex in the center. With the notation of section 2, this means that the permutation ρ is the identity and σ is a transitive cycle; the modules U_{i} are the lifts of the simple k(P ⋊ E)−modules; their isomorphism classes are transitively permuted by the even powers of the Heller operator Ω_{O(P ⋊E)}, and the Brauer tree automorphism group is cyclic of order |E|.

It is also important to recall, that k(P ⋊ E) is a symmetric uniserial k−algebra as P is cyclic, and therefore its stable Auslander-Reiten quiver is a finite tube of period |E|. This implies in particular, that an automorphism of O(P ⋊ E) which stabilises the isomorphism classes of all simple modules stabilises actually the isomorphism classes of all k(P ⋊ E)−modules.

We set B = O(P ⋊ E) and keep this notation throughout this section.

Lemma 4.1. (i) As an O((P ⋊ E) × (P ⋊ E))−module, B has vertex {(u, u)}_{u∈P}, trivial source, and is periodic of period |E|.

(ii) The set {Ω_{B⊗B^{0}}^{2n}(B)}_{0≤n<|E|} is a complete set of representatives of the isomorphism classes of indecomposable O((P ⋊ E) × (P ⋊ E))−modules having vertex {(u, u)}_{u∈P} and trivial source.

Proof. Clearly b has vertex and source as stated; the period of B is easily obtained from the fact that the trivial O(P ⋊ E)−module has period |E|. Since Ω_{P}^{2} (O) ≅ O, all even powers of Heller translates of B have still same vertex and source, and as it is easy to see that there are at most |E| isomorphism classes of indecomposable modules with this vertex and source, the lemma follows easily.

The following lemma, which is crucial for our purpose, is inspired from Scott’s work in [24]; it follows actually from Scott’s results, using Roggenkamp’s generalisation [22] of Weiss’ work [25], which shows that the hypothesis on O to be unramified is not necessary.

Lemma 4.2. Suppose that O has characteristic zero. Any algebra automorphism of B maps the image of P in B to some conjugate.

Proof. Since E is non trivial, P acts trivially on every O−free B−module of rank one, and OE, considered as B−module via the natural homomorphism from B onto OE, is the sum of a system of representatives of the isomorphism classes of O−free B−modules of rank one. Thus if α ∈ Aut(B) we have _{α}(OE) ≅ OE, or equivalently, (_{α}B) ⊗_{OP} O ≅ OE. By Roggenkamp’s generalisation [22] of Weiss’

arguments in [25] it follows that _{α}B is a P × P −permutation module. Thus, as an O(P × (P ⋊ E))−module, _{α}B has a trivial source and therefore vertex Q = {(φ(u), u)}_{u∈P} for some automorphism φ of P, since _{α}B is projective as left and right module and |P | is the highest p−power dividing the O−rank of B. Therefore we have _{α}B ≅ _{φ}B as O(P × (P ⋊ E))−modules. As any such isomorphism is in particular an isomorphism of right B−modules, it comes from left multiplication by some invertible element a of B, and the condition, that this is also an isomorphism of left OP −modules means that aα(u) = φ(u)a for all u ∈ P, which proves the lemma.

Proposition 4.3. Suppose that O has characteristic zero. We have

Ãut(B) ≅ Ãut_{P} (B) × Ãut_{0}(B);

moreover, Ãut_{P} (B) is cyclic of order |E| and the map extending any automorphism of P trivially to P ⋊ E induces an isomorphism Aut(P)/E ≅ Ãut_{0}(B).

Proof. Let φ ∈ Aut(B); in view of 4.2 we may assume that φ stabilises the image of P in B, thus induces an automorphism ψ on P. Extend ψ trivially to P ⋊ E, still called ψ ; clearly then ψ ∈ Aut_{0}(B) and φψ^{−1} ∈ Aut_{P} (B). This shows that Ãut(B) = Ãut_{P} (B)Ãut_{0}(B). If φ ∈ Aut_{P} (B) then, as an O((P ⋊ E) × (P ⋊ E))−module, _{φ}B has vertex {(u, u)}_{u∈P} and trivial source, thus is isomorphic to Ω_{B⊗B^{0}}^{2n}(B) for a unique integer n, 0 ≤ n < |E| by 4.1.(ii). Therefore Ãut_{P} (B) is cyclic of order |E|; if moreover φ ∈ Aut_{0}(B) then necessarily n = 0, thus _{φ}B ≅ B as bimodules, which means that φ is an inner automorphism. This shows that Ãut_{P} (B) ∩ Ãut_{0}(B) = 1 and that the group homomorphism Aut(P) ⟶ Ãut_{0}(B) obtained by extending trivially to P ⋊ E any automorphism of P is surjective; its kernel is E as by [19],3.5 any automorphism of P induced by an inner automorphism of O(P ⋊ E) lies in E.

Remark 4.4. Lemma 4.1 and the first two statements of Proposition 4.3 are true even if E is trivial. Lemma 4.2 and the last statement of Proposition 4.3 are false in general if E is trivial. Both 4.2 and 4.3 remain true under some more general hypotheses: it suffices to take P and E non trivial abelian such that P ⋊ E is a Frobenius group and, for 4.3, such that E lies in the center of Aut(P), in order to be able to extend trivially any automorphism of P to an automorphism of P ⋊ E.

5 Algebras stably equivalent to O(P ⋊ E)

In this section is P as before a non trivial cyclic p−group and E a non trivial p′−subgroup of Aut(P); we investigate here symmetric algebras A for which there is a stable equivalence of Morita type between A and O(P ⋊ E). Note that in this situation, the algebra structure of k ⊗_{O} A is well-known: since P is cyclic, the algebra

k(P ⋊ E) is uniserial and thus k ⊗_{O}A is a Brauer tree algebra by the results of Gabriel-

Riedtmann [9]. Here we show that the Morita equivalence class of A is determined by that of k ⊗_{O} A, determine the automorphism group of A and obtain as a consequence,

that if the stable equivalence comes from induction and restriction, this determines the interior P −algebra structure of A , up to isomorphism.

Proposition 5.1. Let A be a symmetric indecomposable O−algebra such that there is a stable equivalence of Morita type between A and O(P ⋊ E). Then A is split, A has period 2|E| as an A⊗_{O}A^{0}−module and any two indecomposable A⊗_{O}A^{0}−modules

inducing a stable equivalence of Morita type on A are isomorphic to Ω_{A⊗A^{0}}^{n}(M) , Ω_{A⊗A^{0}}^{−n}(N) for a unique integer n, 0 ≤ n < 2|E|, and, up to isomorphism, unique A⊗_{O}A^{0}−modules M, N inducing a Morita equivalence on A which stabilises the iso-

morphism class of any k ⊗_{O} A−module.

Proof. Since k(P ⋊ E) is uniserial split, the simple quotient of the endomorphism algebra of any indecomposable k(P ⋊ E)−module is isomorphic to k (cf. [12],2.6), thus the same is true for any indecomposable A−module and hence A is split, too.

The statement on the period of A follows from 4.1 (i) and 3.5. Any stable equivalence on \overline{Mod}(k ⊗_{O} A) induces a graph automorphism of the stable Auslander-Reiten

quiver of k ⊗_{O} A, which is a finite tube of period |E|, as k ⊗_{O} A is stably equivalent

to the uniserial algebra k(P ⋊ E). Now any such graph automorphism is induced by some power of the Heller operator Ω_{A}; indeed, this is elementary to see for k(P ⋊ E) instead of A, and therefore true for A by 3.5. Thus if U is an A⊗_{O}A^{0}−module inducing

a stable equivalence on \overline{Mod}(k ⊗_{O} A) there is a unique integer n, 0 ≤ n < 2|E|, such

that Ω_{A⊗A^{0}}^{−n}(U) induces the identity on the stable Auslander-Reiten quiver, thus a Morita equivalence on A by 3.4.(iii). The result follows easily.

Proposition 5.2. Let A, A′ be symmetric indecomposable O−algebras such that there are stable equivalences of Morita type between both A, A′ and O(P ⋊ E). If k ⊗_{O} A

and k ⊗_{O} A′ are Morita equivalent then A and A′ are Morita equivalent.

Proof. The hypotheses imply that there is an A′ − A−bimodule M and an A − A′−bimodule N inducing a stable equivalence of Morita type between A and A′. If k ⊗_{O} A and k ⊗_{O} A′ are Morita equivalent, it follows from 5.1 that there is an integer n

such that k ⊗_{O} (Ω_{A′⊗A}^{−n}(M)) induces a Morita equivalence Mod(k ⊗_{O} A) ≅ Mod(k ⊗_{O} A′).

But then Ω_{A′⊗A}^{−n}(M) induces a Morita equivalence Mod(A) ≅ Mod(A′).

Remark 5.3. Since the Morita equivalence class of k ⊗_{O} A is determined by its

Brauer tree, the above Proposition says that the Morita equivalence class of A is still determined by its Brauer tree. This result follows also from [14],3.10, where we describe the basic algebras in terms of generators and relations, but the present proof of 5.2 using stable equivalences of Morita type is considerably shorter.

Proposition 5.4. Let A be an indecomposable symmetric O−algebra and φ, ψ : O(P ⋊ E) ⟶ A two algebra homomorphisms, each inducing a stable equivalence of Morita type between A and O(P ⋊ E). Then Im(φ) and Im(ψ) are conjugate in A; in particular, if O has characteristic zero, φ(P) and ψ(P) are conjugate in A.

Proof. Set B = O(P ⋊ E). The hypotheses imply that the B⊗_{O}B^{0}−modules _{φ}A_{ψ}

and _{ψ}A_{φ} induce a stable equivalence of Morita type on B. Thus, by 5.1, _{φ}A_{ψ} has, up to isomorphism, a unique non projective indecomposable direct summand of the form Ω_{B⊗B^{0}}^{n}(M) for some integer n and some B⊗_{O}B^{0}−module M inducing an equiva-

lence which preserves the isomorphism classes of all k ⊗_{O} B−modules. Since the stable

equivalence induced by _{φ}A_{ψ} comes from restriction and induction through φ, ψ, it preserves modulo |P | the O−ranks of any O−free B−module; thus n is necessarily even. Since any even power of Ω_{B} permutes the isomorphism classes of simple k ⊗_{O} B−modules it follows from 3.4.(iii) that Ω_{B⊗B^{0}}^{n}(M) induces a Morita equiva-

lence on Mod(B), thus Ω_{B⊗B^{0}}^{n}(M) ≅ _{α}B for some automorphism α of B. Since αB is, up to isomorphism, the unique non projective indecomposable direct summand of _{φ}A_{ψ}, tensoring by − ⊗_{B} (_{ψ}A) yields that (_{α}B) ⊗_{B} (_{ψ}A) ≅ _{α}(_{ψ}A) ≅ _{ψα}A is,

up to isomorphism, the unique non projective indecomposable direct summand of (_{φ}A_{ψ}) ⊗_{B} (_{ψ}A) ≅ _{φ}((A_{ψ}) ⊗_{B} (_{ψ}A)) ≅ (_{φ}A) ⊕ (_{φ}X) as B⊗_{O}A^{0}−module, where X is

a projective A⊗_{O}A^{0}−module. It follows that _{φ}A ≅ _{ψα}A as B⊗_{O}A^{0}−modules. Any

such isomorphism is in particular an isomorphism of right A−modules, thus induced by left multiplication with some invertible element a in A, and the condition, that this is also an isomorphism of left B−modules implies that aφ(u) = ψ(α(u))a for all u ∈ O(P ⋊ E). The last statement follows now immediately from 4.2.

Corollary 5.5. Let A be an indecomposable symmetric O−algebra containing O(P ⋊ E) as a subalgebra, such that restriction and induction induce a stable equivalence of Morita type between A and O(P ⋊ E). Every α̃ ∈ Ãut(A) has a representative α stabilizing the image of O(P ⋊ E) in A. If O has characteristic zero, we may choose α such that it stabilises simultaneously the images of P and O(P ⋊ E) in A.

Proof. If α ∈ Aut(A) then both the inclusion O(P ⋊ E) ⊂ A and the restriction of α to O(P ⋊ E) induce a stable equivalence of Morita type between A and O(P ⋊ E). Thus α(O(P ⋊ E)) and O(P ⋊ E) are conjugate in A by 5.4. The result follows easily using 4.2.

The main result of this section is the following Theorem 5.6. Let A be an indecomposable symmetric O−algebra containing O(P ⋊ E) as subalgebra, such that restriction and induction induce a stable equivalence of Morita type between A and O(P ⋊ E). The map sending any α ∈ Aut(A) which stabilises O(P ⋊ E) to its restriction α|_{O(P ⋊E)} on O(P ⋊ E) induces an injective group homomorphism

Φ : Ãut(A) ⟶ Ãut(O(P ⋊ E))

mapping Ãut_{0}(A) onto Ãut_{0}(O(P ⋊ E)).

If moreover O has characteristic zero and every automorphism of P induced by an inner automorphism of A lies in E then Φ maps Ãut_{P} (A) into Ãut_{P} (O(P ⋊ E)) and we have

Ãut(A) ≅ Ãut_{P} (A) × Ãut_{0}(A).

Proof. Set B = O(P ⋊ E). Let α ∈ Aut(A) such that α stabilises B and let β be the restriction of α to B. By our hypotheses, A = B ⊕ Y as B⊗_{O}B^{0}−modules, where Y is

a projective B⊗_{O}B^{0}−module. Thus _{α}A ≅ (_{β}B)⊕(_{β}Y) as B⊗_{O}B^{0}−modules. Therefore,

if _{α}A ≅ A as A⊗_{O}A^{0}−modules then _{β}B ≅ B as B⊗_{O}B^{0}−modules, or equivalently, if

α is an inner automorphism so is β. Since by 5.5 every element of Ãut(A) has a representative stabilising B, this shows that the map α̃ ⟶ β̃ is a well-defined group homomorphism Φ : Ãut(A) ⟶ Ãut(B).

In order to show that Φ is injective, we observe that if β is an inner automorphism,

then_{α}A ≅ A as B⊗_{O}A^{0}−modules. Thus A⊗_{B}(_{α}A) ≅ A⊗_{B}A ≅ A⊕X as A⊗_{O}A^{0}−modules

for some projective A⊗_{O}A^{0}−module X. But we have also A ⊗_{B} (_{α}A) ≅ A ⊗_{B} (A_{α^{−1}})

≅ (A ⊗_{B} A)_{α^{−1}} ≅ (A_{α^{−1}}) ⊕ (X_{α^{−1}}) thus A ≅ A_{α^{−1}}, and therefore α is an inner

automorphism. This shows that Φ is injective.

Next, let β ∈ Aut_{0}(B). Then A ⊗_{B} (_{β}A) and (A_{β}) ⊗_{B} A induce a stable equivalence

of Morita type on A stabilising the isomorphism class of any k ⊗_{O} A−module, since

β stabilises the isomorphism class of any k ⊗_{O} B−module as k ⊗_{O} B is uniserial. By

3.4, A ⊗_{B} (_{β}A) has, up to isomorphism, a unique indecomposable non projective direct

summand, which is then necessarily of the form _{α}A for some α ∈ Aut_{0}(A). By 5.5 again, we may choose α such that α stabilises B. Setting γ = β|_{B} and restricting the bimodules A ⊗_{B} (_{β}A), _{α}A to B⊗_{O}B^{0} it follows that _{β}B ≅ _{γ}B as B⊗_{O}B^{0}−modules, thus

we may assume that γ = β. This means that β ∈ Aut_{0}(B) extends to α ∈ Aut_{0}(A), or equivalently, Φ maps Ãut_{0}(A) onto Ãut_{0}(B).

Clearly, if α̃ ∈ Ãut(A) such that Φ(α̃) ∈ Ãut_{P} (B) then α̃ has a representative α whose restriction to P is the identity, thus α̃ ∈ Ãut_{P} (A).

Conversely, if α̃ ∈ Ãut_{P} (A), then α̃ has a representative α whose restriction to P is the identity and which by 5.5 satisfies α(B) = B^{a} and α(P) = P^{a} for some invertible element a in A; since α(P) = P, this means that P = P^{a}. Therefore, if we assume additionally that every automorphism of P induced by an inner automorphism of A lies in E, thus is induced by an inner automorphism of B, this means that we may choose the element a in such a way that a induces the identity on P, too. But this in turn means that we may choose a representative α which stabilises B and induces the identity on P; that is, α|_{B} ∈ Aut_{P} (B). This shows that Φ maps Ãut_{P} (A) into Ãut_{P} (B), and now the last formula follows easily from the corresponding formula for B in 4.3.

Remark 5.7. The automorphisms of P induced by an inner automorphism of O(P ⋊ E) are precisely the automorphisms in E; this is a very particular case of Puig’s fusion theorem [19],3.1 on fusions in block algebras. The condition in 5.6 that any automorphism of P induced by an inner automorphism of A lies in E means therefore, very roughly speaking, that the “subalgebra O(P ⋊ E) of A controls A−fusions on P ”.

We note a few consequences of 5.5 and 5.6.

Corollary 5.8. Suppose that O has characteristic zero. Let A be an indecomposable symmetric O−algebra containing O(P ⋊ E) as a subalgebra such that induction and restriction induce a stable equivalence of Morita type between A and O(P ⋊ E).

(i) Any algebra automorphism of A maps the image of P in A to some conjugate. (ii) Any group automorphism of P extends to an algebra automorphism of A.

If we suppose moreover that any automorphism of P induced by an inner automorphism of A lies in E, we have:

(iii) An element α̃ of Ãut(A) lies in Ãut_{P} (A) if and only if _{α}A ≅ Ω_{A⊗A^{0}}^{2n}(A) for some integer n, where α is any representative of α̃ in Aut(A).

Proof. Statement (i) is an immediate consequence of 5.5.

By the last statement of 4.3 we can extend any automorphism φ of P to an automorphism, still called φ, in Aut_{0}(O(P ⋊ E)). As the group homomorphism Φ in 5.6 maps Ãut_{0}(A) onto Ãut_{0}(O(P ⋊ E)) we can extend φ to an automorphism of A.This prooves (ii).

For (iii) we use the fact that by 5.6, α̃ lies in Ãut_{P} (A) if and only if α̃ has a representative α stabilising O(P ⋊ E) such that the restriction β of α to O(P ⋊ E) lies in Aut_{P} (O(P ⋊ E)). By 4.1.(ii) this is equivalent to _{β}B ≅ Ω_{B⊗B^{0}}^{2n}(B) for some integer n. Since _{α}A is up to isomorphism the unique indecomposable non projective direct summand of the A⊗_{O}A^{0}−bimodules (_{α}A) ⊗_{B} A ≅ (A_{α^{−1}}) ⊗_{B} A ≅ (A_{β^{−1}}) ⊗_{B} A ≅

A ⊗_{B} (_{β}A) ≅ A ⊗_{B} ((_{β}B) ⊗_{B} A) ≅ A ⊗_{B} (Ω_{B⊗B^{0}}^{2n}(B) ⊗_{B} A), it follows easily from 3.5 that

this is the case if and only if _{α}A ≅ Ω_{A⊗A^{0}}^{2n}(A) for the same integer n.

Corollary 5.9. Suppose that O has characteristic zero. Let A, A′ be symmetric indecomposable O−algebras containing both O(P ⋊ E) as subalgebra, such that induction and restriction induce stable equivalences of Morita type between both A, A′ and O(P ⋊ E). Then A, A′ are isomorphic as interior P −algebras if and only if A, A′ are isomorphic as O−algebras.

Proof. Suppose there is an O−algebra isomorphism α : A ⟶ A′. Then it follows from 5.4 that α maps the image of P in A to a conjugate of the image of P in A′.

Thus we may assume that α maps the image of any element u of P in A to the image of φ(u) in A′, where φ is an automorphism of P. by 5.8(ii), φ extends to an algebra automorphism of A′, still called φ. This means that φ^{−1}α : A ⟶ A′ is an isomorphism of interior P −algebras. The converse is trivial.

Remark 5.10. In [17],4.7 we proved, that the automorphism group of the basic algebras of A is isomorphic to F ×Ãut_{0}(A), where F is the Brauer tree automorphism group of the Brauer tree of A. Thus 5.6 implies that Ãut_{P} (A) is isomorphic to a subgroup of F; in particular, Ãut_{P} (A) is cyclic of order dividing |F | and thus dividing |E|. In the next section we prove that if A is a source algebra of a block having P as a defect group, then actually we have the isomorphism Ãut_{P} (A) ≅ F; the reason is that using our results in [13], we show that any Morita equivalence on A preserves in that case the dimensions of all simple modules, thus is induced by an algebra automorphism of A.

6 From the local case to the general case

In this section b is a block of a finite group G having a non trivial cyclic defect group P; we denote by H the normaliser in G of the unique subgroup Z of P of order p and by c the block of H corresponding to b (i.e. such that Br_{P} (c) = Br_{P} (b); see[12],8.16).

There are two facts of fundamental importance for the study of cyclic blocks: first, OGb and OHc are stably equivalent through induction and restriction (this is a classical result; see e.g. Green [10]) and second, OHc and O(P ⋊ E) are Morita equivalent. In other words, the “general case” and the “local case” are linked by a stable equivalence. As one expects, there is actually a stable equivalence of Morita type:

Proposition 6.1. With the notation above, the OGb − OHc−bimodule bOGc and the OHc − OGb−bimodule cOGb induce a stable equivalence of Morita type between OGb and OHc.

The proof of 6.1 requires the following technical result (see [12],11.4); we freely use the notation for the Brauer homomorphism and the relative trace as described, for instance, in [16] section 3, as well as some standard properties of idempotents (see e.g. [16] section 2).

Lemma 6.2. For any non trivial subgroup Q of P we have Br_{Q}(b) = Br_{Q}(c); in particular, c − bc ∈ (OG)_{1}^{H}.

Proof. Let Q be a non trivial subgroup of P and a ∈ (OG)^{P}. We have T r_{P}^{G}(a) = ∑_{x}T r_{P^{x}∩Q}^{Q}(a^{x}) , where x runs over a set of representatives of the double cosets P \G/Q

in G. As P^{x}∩Q = 1 if x ∈ G−H, applying Br_{Q} yields Br_{Q}(T r_{Q}^{G}(a)) = Br_{Q}(T r_{P}^{H}(a)), thus Br_{Q}((OG)_{P}^{G}) = Br_{Q}((OG)_{P}^{H}) = Br_{Q}((OH)_{P}^{H}), where the last equality holds since C_{G}(P) ⊂ H. Consequently, Br_{Q}(b) and Br_{Q}(c) are primitive idempotents in the ideal Br_{Q}((OG)_{P}^{G}) of the commutative subalgebra Br_{Q}((OG)^{G}) of kC_{G}(Q) and therefore are either equal or orthogonal. Since Br_{P} (Br_{Q}(b)) = Br_{P} (b) = Br_{P} (c) = Br_{P} (Br_{Q}(c)) they cannot be orthogonal. As c − bc ∈ (OG)_{P}^{H} and c − bc ∈ ker(Br_{Q}) for any non trivial subgroup Q of P, the last statement follows easily.

Proof of 6.1. Clearly bOGc and cOGb are projective as left and right modules.

We show first, that the OHc − OHc−bimodule cOGb ⊗_{OGb}bOGc ≅ cOGbc has OHc

as unique non projective indecomposable direct summand, up to isomorphism. Since OH is a direct summand of OG as OH − OH−bimodule, clearly OHc is a direct summand of cOGc as OHc − OHc−bimodule.

If x ∈ G−H we have P ∩P^{x} = 1; thus it follows easily that the P ×P −permutation module O[HxH] is projective as P ×P −module and therefore cO[HxH]c is projective as OHc − OHc−bimodule, since it is relatively P × P −projective and projective as P × P −module. Consequently, cOGc has OHc as unique non projective indecomposable direct summand, up to isomorphism. Now cOGc = cOGbc ⊕ cOG(c − bc), and we only have to observe that OHc is not a direct summand of cOG(c − bc). This however follows easily, using Higman’s criterion, as c − bc ∈ (OG)_{1}^{H} by 6.2, and therefore cOG(c − cb) is projective as OHc − OHc−bimodule.

It remains to show that the OGb − OGb−bimodule bOGc ⊗_{OHc}cOGb has OGb as

unique indecomposable non projective direct summand, up to isomorphism. Since Br_{P} (b) = Br_{P} (c), in a primitive decomposition of cb in (OG)^{P} occurs an idempotent belonging to a local point of P on OGb. This implies, by [18],3.5, that multiplication by c induces a Morita equivalence between OGb and cOGbc. It suffices thus to show that cOGbc ⊗_{OHc}cOGbc has cOGbc as unique non projective indecomposable direct

summand, up to isomorphism, as cOGbc − cOGbc−bimodule. Now multiplication induces a surjective bimodule homomorphism cOGbc ⊗_{OHc}cOGbc ⟶ cOGbc, which

has a section as OHc − cOGbc−bimodule homomorphism, namely the map sending a ∈ cOGbc to bc ⊗ a. Thus it has a section as a cOGbc − cOGbc−bimodule homomorphism, since any cOGbc − cOGbc−bimodule is relatively OHc − cOGbc−projective; in particular, cOGbc is a direct summand of cOGbc ⊗_{OHc}cOGbc. There is no other

non projective direct summand, since even as OHc − OHc−bimodule, OHc is, up to isomorphism, the unique non projective direct summand of cOGbc, thus of

cOGbc ⊗_{OHc}cOGbc, too.

Remark 6.3 The proof of proposition 6.1 uses only the T.I.-property; that is, P ∩ P^{x} = 1 for any x ∈ G − H.

Observe that the bimodules bOGc and cOGb need not be indecomposable, since the idempotent cb need not be primitive in (OG)^{H}.

Proposition 6.4. Let f be a primitive idempotent in (OGb)_{P}^{H} such that Br_{P} (f) = Br_{P} (b).

(i) Multiplication by f induces a Morita equivalence between OGb and fOGf. (ii) The OGb − OHc−bimodule bOGf and the OHc − OGb−bimodule fOGb are

indecomposable and induce a stable equivalence of Morita type between OGb and OHc.

(iii) Multiplication by f on OHc is an injective homomorphism OHc ⟶ fOGf of interior H−algebras which induces a stable equivalence of Morita type between OHc and fOGf.

Proof. Since Br_{P} (f) = Br_{P} (b), in a primitive decomposition of f in (OGb)^{P} occurs an idempotent belonging to a local point of P on OGb; thus (i) follows from [18],3.5. Statement (ii) follows easily from 6.1 and the observation that bOGf and fOGb are, up to isomorphism, the unique indecomposable non projective direct summands of bOGc and cOGb, since f is primitive in (OGb)^{H} and not contained in (OGb)_{1}^{H}. Statement (iii) is an immediate consequence of (i) and (ii).

The idempotent f in 6.4 need not be unique, but it is unique up to conjugacy, since it is easy to see that f belongs to the unique point β of H on OGb satisfying an inclusion P_{γ} ⊂ H_{β} ⊂ G_{{b}} for some local point γ of P on OGb.

The next step is the reduction of 6.4 to the level of the source algebras of OHc.

Proposition 6.5. Let f be a primitive idempotent in (OGb)_{P}^{H} such that Br_{P} (f) = Br_{P} (b); let i and j be primitive idempotents in (OGb)^{P} and (OHc)^{P}, respectively, such that Br_{P} (i) = Br_{P} (j) ≠ O.

(i) Multiplication by f is an injective homomorphism jOHj ⟶ jfOGjf of interior P −algebras, which induces a stable equivalence of Morita type between jOHj and jfOGjf.

(ii) There is, up to isomorphism, a unique indecomposable endo-permutation OP −module N having P as vertex and determinant one, such that, setting S = End_{O}(N), we have an isomorphism of interior P −algebras

jOHj ≅ S ⊗_{O} O(P ⋊ E);

moreover, Z acts trivially on N.

Proof. Multiplication by j induces Morita equivalences both between OHc and jOHj, as well as fOGf and jfOGjf, thus (i) follows from 6.4 (iii). Statement (ii) is a reformulation of the results in [12],10.4,13.1,15.1 on the source algebras of OHc.

Observe that jfOGjf is “almost” a source algebra of OGb, since Br_{P} (i) = Br_{P} (j) = Br_{P} (jf) is a primitive idempotent in kC_{G}(P); in other words, we may choose i in a primitive decomposition of jf in (OGb)^{P}, and then jf = i + i′ for some idempotent i′ in ker(Br_{P}), whence there is a unique local point of P on jfOGjf, it has multiplicity one and contains i.

Using 6.5 we can be more precise:

Proposition 6.6. With the notation of 6.5, there is up to isomorphism a unique interior P −algebra A containing O(P ⋊ E) as unitary interior P −subalgebra, such that jfOGjf ≅ S ⊗_{O} A as interior P −algebras. The algebra A has then the following

properties:

(i) Induction and restriction induce a stable equivalence of Morita type between A and O(P ⋊ E).

(ii) The unit element 1_{A} is primitive in A^{P} and local in A.

(iii) The source algebra iOGi of b is isomorphic to

e(S ⊗_{O} A)e,

where e belongs to the unique local point of P on S ⊗_{O} A; in particular, OGb and A

are Morita equivalent.

Proof. Combine 6.5(i), (ii) to get an injective homomorphism

S ⊗_{O} O(P ⋊ E) ≅ jOHj ⟶ jfOGjf

which induces a stable equivalence of Morita type. In particular, jfOGjf contains a unitary subalgebra isomorphic to the matrix algebra S. Thus by [18] 2.1, jfOGjf ≅ S ⊗_{O} A, where A is the centralisor in jfOGjf of the image of S in jfOGjf. Clearly

A contains then the image of O(P ⋊ E) in such a way, that induction and restriction still induce a stable equivalence of Morita type. This proves (i). By 6.4(ii), bOGf is indecomposable as OGb−OHc−bimodule, thus fOGf is indecomposable as fOGf − OHc−bimodule, thus jfOGjf is indecomposable as jfOGjf −jOHj−bimodule and therefore A is indecomposable as A − O(P ⋊ E)−bimodule. This means that 1_{A} is primitive in A^{P ⋊E}. Then, by [16] 7.10, P ⋊ E acts transitively on the set of points of P on A. Since Br_{P} (jf) = Br_{P} (i) is primitive in kC_{G}(P), clearly P has a unique local point on A, which then is necessarily the unique point of P on A, and as its multiplicity is one, it follows that 1_{A} is primitive in A^{P}. This implies (ii). The last statement follows from (i), as we may choose the idempotent i in jfOGjf.

Recall the notation (I, {ρ, σ}, v) from section 2 for the Brauer tree of OGb.

Proposition 6.7. Suppose that O has characteristic zero. With the notation of 6.6, the isomorphism class of A as interior P −algebra is determined by the isomorphism class of the quadruple (I, ρ, σ, v).

Proof. By 5.9, we have to show , that the isomorphism class of A as O−algebra is determined by (I, ρ, σ, v). It is a well-known fact (see e.g. [9]) that the Morita equivalence class of k ⊗_{O} A is determined by the Brauer tree (I, {ρ, σ}, v); thus, by

5.2, this is still true for the Morita equivalence class of A. Now the dimensions of the simple A−modules are determined by the formulae [13],3.13, which show that they are determined by (I, ρ, σ, v). Whence the isomorphism class of A as O−algebra is determined by (I, ρ, σ, v).

We need the following technical facts on fusions, which are a particular case of [20],5.3; we give here a short ad hoc proof:

Lemma 6.8. With the notation of 6.6, (i) for any φ ∈ Aut(P) we have Res_{φ}(S) ≅ S as interior P −algebras; (ii) any automorphism of P induced by an inner automorphism of A lies in E.

Proof. Statement (i) follows from the easy fact that since P is cyclic, the isomorphism class of the endo-permutation OP −module N is invariant under any automorphism of P. If φ ∈ Aut(P) is induced by an inner automorphism of A, then, using (i), φ is still induced by an inner automorphism of S ⊗_{O} A, since any automorphism of S is an

inner automorphism by the Skolem-Noether Theorem.Clearly an inner automorphism of S ⊗_{O}A which stabilises the image of P in S ⊗_{O}A stabilises also the unique local point

of P on S ⊗_{O} A, thus φ is already induced by an inner automorphism of the source

algebra e(S ⊗_{O}A)e of OGb, where e belongs to the unique local point of P on S ⊗_{O}A. By

Puig’s theorem [19] 3.1 (or its corollary [19],3.5) on fusions in block source algebras, necessarily φ belongs to E, which proves (ii).

Proposition 6.9. With the notation of 6.6, we have Ãut_{P} (A) ≅ F, where F is the Brauer tree automorphism group of (I, {ρ, σ}, v).

Proof. Clearly an algebra automorphism α of A induces a Brauer tree automorphism τ and then α lies in Aut_{0}(A) if and only if τ = Id_{I}. Conversely, if τ is any Brauer tree automorphism, then τ centralises ρ and σ by [17] 4.5, thus τ = (ρσ)^{n} = (σρ)^{n} for some integer n, as ρσ is a transitive cycle on I. Note that by 2.1 we have Ω_{G}^{2}(U_{i}) ≅ U_{σρ(i)} and Ω_{G}^{2}(V_{i}) ≅ V_{ρσ(i)} for all i ∈ I. Therefore Ω_{G}^{2n}(U_{i}) ≅ U_{τ(i)} and Ω_{G}^{2n}(V_{i}) ≅ V_{τ(i)} for all i ∈ I. A combinatorical argument, using the composition series of the uniserial modules k ⊗_{O} U_{i} and k ⊗_{O} V_{i} shows that Ω_{G}^{2n}(S_{i}) ≅ S_{τ(i)} for all i ∈ I, where

S_{i} is the simple quotient of OGi. As A and OGb are Morita equivalent it follows that the functor Ω_{A⊗A^{0}}^{2n}(A) ⊗_{A} − permutes the set of isomorphism classes of simple

A−modules (cf. 3.4(ii)) and is therefore a Morita equivalence (cf. 3.4(iii)). This Morita equivalence preserves the dimensions of simple modules, as the dimensions of the simple A−modules corresponding to S_{i}, S_{τ(i)} (through any Morita equivalence) coincide by the formulae [13],3.13, as τ commutes to ρ and σ. Thus Ω_{A⊗A^{0}}^{2n}(A) ≅ _{α}A for some α ∈ Aut(A). By 5.8(iii) and 6.8(ii) we have α̃ ∈ Ãut_{P} (A), and now 6.9 follows easily.

The proofs of 2.6 and 2.7 are now easy consequences of the preceding results.We use the notation introduced in section 2. We may assume that E ≠ 1, since if E = 1 the block b is nilpotent, and in that case, Theorem 2.6 and 2.7 follow trivially from the structure of the source algebras of nilpotent blocks as determined by Puig in [20].

Proof of Theorem 2.6. The first statement of 2.6 follows directly from 6.6, except that we have to show that the isomorphism class of the source algebra iOGi in 6.6 does not depend on the choice of i. By [16],5.6, it suffices to show that Res_{φ}(iOGi) ≅ iOGi as interior P −algebras for any φ ∈ Aut(P). Now this is clear, since Res_{φ}(A) ≅ A as any automorphism of P extends to an automorphism of A (cf. 5.8(ii)) and Res_{φ}(S) ≅ S (cf. 6.7(i)). Statement (i), (ii) and (iii) follow then from 6.6(i), 6.7 and 5.8, respectively; finally, (iv) is a consequence of 5.6 and 6.8.

.

Proof of Theorem 2.7. Clearly, A, A′ are isomorphic as interior P −algebras if and only they are isomorphic as O−algebras, by 5.9. Using 6.7 it is easy to see, that this holds if and only if (I, ρ, σ, v) ≅ (I′, ρ′, σ′, v′), thus if and only if there is an isomorphism of τ of the corresponding Brauer trees satisfying τρτ^{−1}. In that case, it follows from 5.4 that actually any O−algebra isomorphism A ≅ A′ maps the image of P in A to some conjugate of the image of P in A′. This proves (i).

Clearly OGb, OG′b′ are Morita equivalent if and only A, A′ are Morita equivalent (6.6 (iii)), and by 5.2, 5.3 this holds if and only if there is an isomorphism of the Brauer trees τ : (I, {ρ, σ}, v) ⟶ (I′{ρ′, σ′}, v′).

In that case, if τρτ^{−1} = ρ′ we have A ≅ A′ by 6.7, and if τρτ^{−1} = σ′ it is not difficult to see that A ≅ Ω_{P} (A′), since applying the Heller operator Ω_{P} to the interior P −algebra A′ as defined in [16],section 7, yields an interior P −algebra which is still Morita equivalent to A′ and containing O(P ⋊ E) as a subalgebra such that restriction and induction induce a stable equivalence of Morita type and where the rôles of ρ′ and σ′ are exchanged. Thus (ii) follows.

If OGb, OG′b′ have isomorphic source algebras, they have the same matrices of decomposition numbers, and as N, N′ are determined, up to isomorphism, by the signs of the non ordinary entries of the decomposition matrices of OGb, OG′b′, respectively, it follows that N ≅ N′. But then clearly A ≅ A′ as interior P −algebras and now (iii) follows from (i).

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Markus Linckelmann CNRS, Université Paris 7 UFR Mathématiques 2, place Jussieu 75251 Paris Cedex 05

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