On stable equivalences and blocks with one simple module

Radha Kessar, Markus Linckelmann

Abstract Using a stable equivalence due to Rouquier, we show that Alperin’s weight conjecture holds for any p-block of a finite group with defect 2 whose Brauer correspondent has a unique isomorphism class of simple modules.

1 Introduction

Throughout this paper we denote by O a complete discrete valuation ring with residue field k = O/J(O) of prime characteristic p and quotient field K of characteristic zero. Given a finite group G, a block algebra B of OG is an indecomposable direct factor of OG as O-algebra; we denote by ℓ(B) the number of isomorphism classes of simple k ⊗_{O} B-modules.

Theorem 1.1. Let G be a finite group and B a block algebra of OG having a defect group of order at most p^{2}. Denote by C the Brauer correspondent of B and suppose that K, k are splitting fields for B, C. If ℓ(C) = 1 then ℓ(B) = 1, the inertial quotient of B is abelian, the decomposition matrices of B and C are equal and there is a p-permutation equivalence between B and C inducing an isotypy between B and C all of whose signs are positive.

Broué’s Abelian Defect Conjecture predicts more precisely that B and C are derived equivalent. If true, a result of Roggenkamp and Zimmermann would imply that B and C are actually Morita equivalent. This is known to hold if the defect groups of B are cyclic or Klein four because in that case the hypothesis of having a unique isomorphism class of simple modules implies that B and C are nilpotent, hence Morita equivalent to OP. In order to prove Theorem 1.1 we may therefore assume that p is odd, that a defect group P of B is elementary abelian of rank 2 and that the inertial quotient of B is non trivial. We will see that this forces the inertial quotient to be abelian, and hence the Brauer correspondent C is a quantum complete intersection; see [3], [4], [13]. The main ingredient to prove Theorem 1.1 is Rouquier’s stable equivalence between B and its Brauer correspondent, obtained from “gluing” together various derived equivalences at local levels. Since stable equivalences between block algebras preserve the character group L^{0}(B) of generalised characters which vanish on p-regular elements, isometry arguments turn out to work particularly well for blocks with one simple module, because in that case the rank of L^{0}(B) is equal to |Irr_{K}(B)| − 1 and hence this subgroup will contain enough information to reconstruct the number of irreducible characters of any block stably equivalent to B.

Remark 1.2. By work of Kiyota in [14], if p = 3 then Alperin’s weight conjecture holds for all blocks with an elementary abelian defect group of order 9 except possibly when the inertial quotient is cyclic of order 8 or quaternion of order 8 and the block is non principal (the case of non principal blocks with semi-dihedral inertial quotient of order 16 is attributed to A. Watanabe in [14, §0, Table 1]). In particular, with the notation of 1.1, Kiyota’s results imply that if p = 3 then ℓ(B) = 1 if and only if ℓ(C) = 1.

2 Notation and quoted results

We review in this section some classic background material, adapted to symmetric algebras, mostly for the purpose of introducing our notation. Let A be an O-algebra which is finitely generated free as O-module. We denote by mod(A) the category of finitely generated left A-modules, by D^{b}(A) the bounded derived category of mod(A) and by \underline{mod}(A) the relatively O-stable category of mod(A) (identifying to zero, in mod(A), all relatively O-projective modules). We denote by R_{K}(A) and R_{k}(A) the Grothendieck groups of finitely generated K ⊗_{O} A-modules and k ⊗_{O} A-modules, respectively. For any finitely generated module U over K ⊗_{O} A or k ⊗_{O} A we denote by [U] the image of U in R_{K}(A) or R_{k}(A), respectively. The group R_{K}(A) is free of finite rank, having as basis the set of images, denoted Irr_{K}(A), of simple K ⊗_{O} A-modules in R_{K}(A). We set k(A) = |Irr_{K}(A)|, the number of isomorphism classes of simple K ⊗_{O} A-modules. Similarly, the group R_{k}(A) is free of finite rank, having as basis the set of images, denoted Irr_{k}(A), of simple k ⊗_{O} A-modules in R_{k}(A). We set ℓ(A) = |Irr_{k}(A)|, the number of isomorphism classes of simple k ⊗_{O} A-modules. We denote by Pr_{O}(A) the subgroup of R_{K}(A) generated by the images of modules of the form K ⊗_{O} U, where U is a finitely generated projective A-module. Denote by I a set of representative of the conjugacy classes of primitive idempotents in A; then {Ai | i ∈ I} is a set of representatives of the isomorphism classes of projective indecomposable A-modules and {Ai/J(A)i | i ∈ I} is a set of isomorphism classes of the simple k ⊗_{O} A-modules. For i ∈ I denote by Φ_{i} the image of K ⊗_{O} Ai in R_{K}(A) and by φ_{i} the image of Ai/J(A)i in R_{k}(A). The set

IPr_{O}(A) = {Φ_{i} | i ∈ I}

generates Pr_{O}(A) and we have Irr_{k}(A) = {φ_{i} | i ∈ I}. Similarly,we denote by Pr_{k}(A) the subgroup of R_{k}(A) generated by the images of modules of the form k ⊗_{O} U, where U is a finitely generated projective A-module; equivalently, Pr_{k}(A) is generated by the set {[k ⊗_{O} Ai] | i ∈ I}. Assume in addition K ⊗_{O} A that is split semi-simple. We define a bilinear form

< , >_{A} : R_{K}(A) × R_{K}(A) ⟶ ℤ

on R_{K}(A) by setting < [U], [V] >_{A}= dim_{K}(Hom_{K⊗_{O}A}(U, V)) for any two finitely generated K ⊗_{O} A-modules U, V and extending this to R_{K}(A) in the obvious way. Since K⊗_{O}A is split semi-simple, this form is symmetric, the set Irr_{K}(A) is an orthonormal basis of R_{K}(A), every finitely-generated K ⊗_{O} A-module is a finite direct sum of simple K ⊗_{O} A-modules and any simple K ⊗_{O} A-module X is isomorphic to a direct summand of K ⊗_{O} A. Intersecting a direct summand X of K ⊗_{O} A with the image 1 ⊗ A of A in K ⊗_{O} A yields an O-free A-module Y satisfying K ⊗_{O} Y ≅ X. Thus, for any finitely generated K ⊗_{O} A-module X there is an O-free A-module Y satisfying K ⊗_{O} Y ≅ X. Moreover, if Y′ is another O-free A-module satisfying K ⊗_{O} Y′ ≅ X then k ⊗ Y and k ⊗ Y′ have identical composition factors (with multiplicities) because the multiplicity of a simple k ⊗_{O} A-module S in a composition series of k ⊗_{O} Y is equal to

dim_{k}(Hom_{A}(Ai, k ⊗_{O} Y)) = rank_{O}(Hom_{A}(Ai, Y)) = dim_{K}(Hom_{K⊗_{O}A}(K ⊗_{O} Ai, X)) = dim_{K}(iX)

where i is a primitive idempotent in A satisfying Ai/J(A)i ≅ S. We denote by

d_{A} : R_{K}(A) ⟶ R_{k}(A)

the decomposition map sending [X] to [k ⊗_{O} Y], where X is a finitely-generated K ⊗_{O} A-module and Y a finitely generated O-free A-module Y satisfying K ⊗_{O} Y ≅ X. By the above remarks, for χ ∈ Irr_{K}(A) and i ∈ I there are unique integers d_{i}^{χ} such that

d_{A}(χ) = ∑_{i∈I} d_{i}^{χ} φ_{i}

and we have d_{i}^{χ} = dim_{K}(iX), where X is a simple K ⊗_{O} A-module such that χ = [X]. Again since K ⊗_{O} A is split, d_{i}^{χ} is also the multiplicity of X in a decomposition of K ⊗_{O} Ai, hence

Φ_{i} = ∑_{χ∈Irr_{K}(A)} d_{i}^{χ}χ

The matrix D = (d_{i}^{χ}), with χ ∈ Irr_{K}(A) and i ∈ I, is the decomposition matrix of A. The matrix C = (d_{S_{j}}^{Φ_{i}}), with i, j ∈ I, is the Cartan matrix, denoted C, of A. The above considerations imply the well-known fact C = D^{t} · D. We denote by L^{0}(A) the subgroup consisting of all Y ∈ R_{K}(A) such that < X, Y >_{A}= 0 for all X ∈ Pr(A).

Lemma 2.1. Let A be an O-algebra which is finitely generated free as O-module such that K ⊗_{O} A is split semi-simple. We have L^{0}(A) = ker(d_{A}). In addition, if the Cartan matrix C of k ⊗_{O} A is non singular then Pr_{O}(A) ∩ L^{0}(A) = {0}, the decomposition map induces an isomorphism Pr_{O}(A) ≅ Pr_{k}(A), and R_{k}(A)/Pr_{k}(A) is a finite abelian group of order | det(C)|.

Proof. Any element η in R_{K}(A) can be written in the form [K ⊗_{O} Y_{1}] − [K ⊗_{O} Y_{2}] for some finitely generated O-free A-modules Y_{1} and Y_{2}. Since Pr_{O}(A) is generated by the images Φ_{i} of the modules K ⊗_{O} Ai, with i running over a set of representative I of the conjugacy classes of primitive idempotents, we get that η ∈ L^{0}(A) if and only if < Φ_{i}, η >_{A}= 0 for all i ∈ I. This is equivalent to rank_{O}(iY_{1}) = rank_{O}(iY_{2}), hence to dim_{k}(ik ⊗_{O} Y_{1}) = dim_{k}(ik ⊗_{O} Y_{2}) for all i ∈ I. This, in turn, is just a reformulation of η ∈ ker(d_{A}). If the Cartan matrix of k ⊗_{O} A is non singular then P r_{O}(A) ∩ ker(d_{A}) = {0}, thus d_{A} induces an isomorphism Pr_{O}(A) ≅ Pr_{k}(A). The last statement is an elementary fact.

A fundamental result of Brauer states that for group algebras, and hence block algebras, the decomposition map is surjective. One of the numerous applications of this fact is Brauer’s reciprocity, and this can be formulated for symmetric algebras. An O-algebra A is called symmetric if A is isomorphic, as A-A-bimodule, to its O-dual A^{∗} = Hom_{O}(A, O). Let A be a symmetric O-algebra such that K ⊗_{O} A is split semi-simple and such that k ⊗_{O} A is split. Suppose that the Cartan matrix of A is non singular and that the decompostion map d_{A} : R_{K}(A) → R_{k}(A) is surjective. The scalar product < , >_{A} restricts to a bilinear form, still denoted

< , >_{A}: Pr_{O}(A) × R_{K}(A) ⟶ ℤ

This bilinear form vanishes on Pr_{O}(A) × L^{0}(A), hence induces a bilinear form

< , >′_{A}: Pr_{O}(A) × R_{k}(A) ⟶ ℤ

Note that Pr_{O}(A) ≅ Pr_{k}(A) since the Cartan matrix of A is assumed to be non singular, so < , >′_{A} can also be viewed as bilinear form from Pr_{k}(A) × R_{k}(A) to ℤ. If need arises we use the same notation < , >′_{A} for the ℚ-bilinear form from ℚ ⊗_{ℤ} Pr_{O}(A) × ℚ ⊗_{ℤ} R_{k}(A) to ℚ obtained via extension of coefficients. The following is Brauer’s reciprocity:

Proposition 2.2. Let A be a symmetric O-algebra such that K ⊗_{O} A is split semi-simple and such that k ⊗_{O} A is split. Suppose that the Cartan matrix of A is non singular, that the decomposition map d_{A} : R_{K}(A) → R_{k}(A) is surjective, and that k ⊗_{O} A has no projective simple module. Let I be a system of representatives of the conjugacy classes of primitive idempotents. For any i ∈ I denote by Φ_{i} the image in Pr_{O}(A) of K ⊗_{O} Ai and by φ_{i} the image of Ai/J(A)i in R_{k}(A).

(i) We have < Φ_{i}, φ_{j} >′_{A}= δ_{i,j} for any i, j ∈ I. (ii) We have L^{0}(A)^{⊥} = Pr_{O}(A).

Proof. Since d_{A} is surjective, for any i ∈ I and any χ ∈ Irr_{K}(A) there are integers m_{i}^{χ} satisfying

φ_{i} = ∑_{χ∈Irr_{K}(A)} m_{i}^{χ} · d_{A}(χ) = ∑_{j∈I} ∑_{χ∈Irr_{K}(A)} m_{i}^{χ}d_{j}^{χ}φ_{j}

which implies that ∑_{χ∈Irr_{K}(A)} m_{i}^{χ}d_{j}^{χ} = δ_{i,j} for all i, j ∈ I. We also have Φ_{i} = ∑_{χ∈Irr_{K}(A)} d_{i}^{χ}X and hence

< Φ_{i}, φ_{j} >′_{A}=< Φ_{i}, ∑_{χ∈Irr_{K}(A)} m_{j}^{χ}χ >_{A}= ∑_{χ∈Irr_{K}(A)} d_{i}^{χ}m_{j}^{χ} = δ_{i,j}

proving (i). The hypotheses imply that extending coefficients yields

ℚ ⊗_{ℤ} R_{K}(A) = (ℚ ⊗_{ℤ} L^{0}(A)) ⊕ (ℚ ⊗_{ℤ} Pr_{O}(A))

Let θ ∈ L^{0}(A)^{⊥}. Then, in ℚ ⊗_{ℤ} R_{K}(A), we have 1_{K} ⊗ θ = ∑_{i∈I} q_{i} ⊗ Φ_{i} for some q_{i} ∈ ℚ. For any i ∈ I we have

q_{i} =< θ, φ_{i} >′_{A}= ∑_{χ∈Irr_{K}(A)} m_{i}^{χ} < θ, χ >

which is an integer as θ ∈ R_{K}(A). This shows (ii).

If A is a block algebra of a group algebra OG for some finite group G such that A has a non trivial defect group, then by results of Brauer, A satisfies the hypotheses of the above Proposition. In addition, for any χ ∈ Irr_{K}(A) there is λ ∈ L^{0}(A) such that < λ, χ >_{A}≠ 0. Indeed, if not then χ ∈ L^{0}(A)^{⊥} = Pr_{O}(A), but then χ would correspond to an irreducible character which vanishes on all p-singular elements, hence which belongs to a block of defect zero. The following observation is useful for explicit calculations of determinants of Cartan matrices:

Proposition 2.3. Let G be a finite group and B a block algebra of OG and denote by C_{B} the Cartan matrix of k ⊗_{O} B. Let {ψ_{s} | 1 ≤ s ≤ r} be a basis of the abelian group L^{0}(B) and denote by F the matrix F = (< ψ_{s}, ψ_{t} >)_{1≤s,t≤r}. We have det(F) = det(C_{B}).

Proof. Since the decomposition map d_{B} : R_{K}(B) → R_{k}(B) is surjective, with kernel L^{0}(B) and since Pr_{O}(B) ≅ Pr_{k}(B), the map d_{A} induces an isomorphism of abelian groups

R_{K}(B)/(L^{0}(B) ⊕ Pr_{O}(B)) ≅ R_{k}(B)/Pr_{k}(B)

which is a finite abelian group of order det(C_{B}). Denote by {Φ_{i} | 1 ≤ i ≤ ℓ(B)} the canonical basis of Pr_{O}(B) of the images of the projective indecomposable B-modules in some order. The union {ψ_{s} | 1 ≤ s ≤ r} ∪ {Φ_{i} | 1 ≤ i ≤ ℓ(B)} is a basis of the subgroup L^{0}(B) ⊕ Pr_{O}(B) of R_{K}(B). Writing this basis in terms of the canonical basis Irr_{K}(B) of R_{K}(B) yields a square matrix of the form (E|D), where D is the decomposition matrix of B. The absolute value of the determinant of this matrix is the order of the group R_{K}(B)/(L^{0}(B) ⊕ Pr_{O}(B)), hence equal to det(C_{B}). Any column of E is perpendicular to any column of D, because L^{0}(B) and Pr_{O}(B) are orthogonal subgroups. Thus the product (E|D)^{t} · (E|D) is a block diagonal matrix whose blocks are the matrices E^{t} ·E = F and D^{t} ·D = C_{B}. This implies that det(C_{B})^{2} = det(E|D)^{2} = det(F) det(C_{B}), whence the result.

The following observation is a useful tool to compute L^{0}(B) for block algebras B with normal defect:

Lemma 2.4. Let P be a finite p-group, E a p′-subgroup of Aut(P) and α ∈ H^{2}(E; O^{×}). Set A = O_{α}(P ⋊ E). Label the set Irr_{K}(A) = {χ_{i} | 1 ≤ i ≤ k(A)} in such a way that the subset Irr_{K}(A|P) = {χ_{i} | 1 ≤ i ≤ ℓ(A)} consists of all irreducible characters with P in their kernel. For ℓ(A) + 1 ≤ j ≤ k(A) and 1 ≤ i ≤ ℓ(A) denote by a_{i,j} the unique integers such that Res_{E}(χ_{j}) = ∑_{i=1}^{ℓ(A)} a_{i,j}Res_{E}(χ_{i}), where Res_{E}(χ_{i}) is the restriction to O_{α}E of χ_{i}. Then the set

A = {χ_{j} − ∑_{i=1}^{ℓ(A)} a_{i,j}χ_{i} | ℓ(A) + 1 ≤ j ≤ k(A)}

is a basis of the free abelian group L^{0}(A).

Proof. Since E is a p′-group, every χ_{i} lifts a simple k_{α}E-module S_{i}, for 1 ≤ i ≤ ℓ(A), and hence the numbers a_{i,j} are the decomposition numbers corresponding to the characters χ_{j} and simple module S_{i}. Thus the set of characters of the projective indecomposable A-modules is

IPr_{O}(A) = {χ_{i} + ∑_{j=ℓ(A)+1}^{k(A)} a_{i,j}χ_{j} | 1 ≤ i ≤ ℓ(A)}

A trivial verification shows that the elements in A are perpendicular to this set, hence belong to L^{0}(A). Since each element in A has one irreducible character appearing with multiplicity 1, the set A is a basis of L^{0}(A).

3 On stable equivalences and isometries

It is a well-known fact that a stable equivalence between two block algebras induces an isometry on the groups of generalised characters which are orthogonal to projective characters. Extending, if possible, this isometry to an isometry between the character groups is one of the standard strategies in character theory. We review this briefly and prove that an isometry obtained in this way is always perfect in the sense of Broué [6]. Let A, B be O-algebras which are finitely generated free as O-modules such that K ⊗_{O} A, K ⊗_{O} B are split semisimple. Then any B-A-bimodule M which is finitely generated projective as right A-module induces group homomorphisms

Φ_{M} : R_{K}(A) ⟶ R_{K}(B)
φ_{M} : R_{k}(A) ⟶ R_{k}(B)

satisfying Φ_{M}([K ⊗_{O} U]) = [K ⊗_{O} M ⊗_{A} U] and φ_{M}([k ⊗_{O} U]) = [k ⊗_{O} M ⊗_{A} U] for any finitely generated O-free A-module U. Moreover, we have d_{B} ∘Φ_{M} = φ_{M} ∘d_{A}. If both A, B are symmetric and M is an B-A-bimodule which is finitely generated projective as left B-module and as right A-module then the functors M ⊗_{A} − and M^{∗} ⊗_{B} − between the categories mod(A) and mod(B) of finitely generated modules over A and B, respectively, are both left and right adjoint to each other. In that situation we say that M induces a stable equivalence of Morita type between A and B if M ⊗_{A} M^{∗} ≅ B ⊕ W for some projective B-B-bimodule W and M^{∗} ⊗_{B} M ≅ A ⊕ V for some projective A-A-bimodule V. The functors M ⊗_{A} − and M^{∗} ⊗_{B} − induce inverse equivalences between the relatively O-stable categories \underline{mod}(A) and \underline{mod}(B).

Proposition 3.1. Let A, B be symmetric O-algebras such that K ⊗_{O} A and K ⊗_{O} B are split semi-simple and such that the Cartan matrices of k ⊗_{O} A and k ⊗_{O} B are non singular. Let M be a B-A-bimodule which is finitely generated as left B-module and as right A-module. Suppose that M induces a stable equivalence of Morita type between A and B.

(i) Φ_{M} and Φ_{M^{∗}} induce inverse isomorphisms R_{K}(A)/Pr_{O}(A) ≅ R_{K}(B)/Pr_{O}(B). (ii) Φ_{M} and Φ_{M^{∗}} induce inverse isomorphisms R_{k}(A)/Pr_{k}(A) ≅ R_{k}(B)/Pr_{k}(B); in particular, | det(B)| = | det(A)|. (iii) Φ_{M} and Φ_{M^{∗}} induce inverse isometries L^{0}(A) ≅ L^{0}(B).

Proof. Since the Cartan matrix of k ⊗_{O} A is non singular we have Pr(A) ∩ L^{0}(A) = {0}, by Lemma 2.1; similarly for B. Note that Φ_{M} sends Pr_{O}(A) to Pr_{O}(B) and L^{0}(A) = ker(d_{A}) to L^{0}(B) = ker(d_{B}), thus induces a group homomorphisms R_{K}(A)/Pr_{O}(A) → R_{K}(B)/Pr_{O}(B) and R_{k}(A)/Pr_{k}(A) → R_{k}(B)/Pr_{k}(B). The map Φ_{M^{∗}} ∘ Φ_{M} is the map induced by tensoring with the bimodule M^{∗} ⊗_{B} M ≅ A ⊕ W, where W is a projective A-A-bimodule. Thus Φ_{W} maps R_{K}(A) to Pr_{O}(A), and hence for any χ ∈ R_{K}(A) we have Φ_{M^{∗}}(Φ_{M}(χ)) = χ + ζ for some ζ ∈ Pr_{O}(A). Since Φ_{M^{∗}} ∘ Φ_{M} maps L^{0}(A) to itself and since Pr(A) ∩ L^{0}(A) = {0} this implies that if χ ∈ L^{0}(A) then ζ = 0. Thus Φ_{M} and Φ_{M^{∗}} induce inverse isomorphisms R_{K}(A)/Pr_{O}(A) ≅ R_{K}(B)/Pr_{O}(B), R_{k}(A)/Pr_{k}(A) ≅ R_{k}(B)/Pr_{k}(B) and L^{0}(A) ≅ L^{0}(B). Since the functors M ⊗_{A} − and M^{∗} ⊗_{B} − are adjoint we get in particular for χ, χ′ ∈ L^{0}(A) that

< Φ_{M}(χ), Φ_{M}(χ′) >_{B}=< χ, Φ_{M^{∗}}(Φ_{M}(χ′)) >_{A}=< χ, χ′ >_{A}

which shows that the isomorphisms between L^{0}(A) and L^{0}(B) are isometries.

The surjectivity of the decomposition map is invariant under stable equivalences of Morita type:

Proposition 3.2. Let A, B be symmetric O-algebras such that K ⊗_{O} A, K ⊗_{O} B are split semisimple and such that k ⊗_{O} A, k ⊗_{O} B are split. Suppose there is a stable equivalence of Morita type between A and B. Then the decomposition map d_{A} : R_{K}(A) → R_{k}(A) is surjective if and only if the decomposition map d_{B} : R_{K}(B) → R_{k}(B) is surjective.

Proof. Let M be a B-A-bimodule which is finitely generated projective as left B-module and as right A-module and which induces a stable equivalence of Morita type betweem A and B. Suppose that d_{A} is surjective. Then the map d̄_{A} : R_{K}(A)/Pr_{O}(A) → R_{k}(A)/Pr_{k}(A) induced by d_{A} is surejective. Thus, using the fact that d_{A} and d_{B} commute with the maps Φ_{M} and φ_{M} as described above, the map d̄_{B} : R_{K}(B)/Pr_{O}(B) → R_{k}(B)/Pr_{k}(B) induced by d_{B} is surjective. Since Im(d_{B}) contains Pr_{k}(B) = d_{B}(Pr_{O}(B)) it follows that d_{B} is surjective.

Let A, B be block algebras of OG, OH, respectively, where G, H are finite groups. Assume that K, k are splitting fields for G and H. We show next that if an isometry L^{0}(A) ≅ L^{0}(B) induced by a stable equivalence of Morita type extends to an isometry R_{K}(A) ≅ R_{K}(B), then this is a perfect isometry. We use the following notation. If M is a B-A-bimodule which is finitely generated projective as left and right module, then in particular, M is O-free of finite rank and hence determines an element χ_{M} in R_{K}(B⊗_{O} A^{0}). We can also regard χ_{M} as the character of M as O(H ×G)-module, and then χ_{M} is perfect by a result of Broué in [6]; that is, for any (y, x) ∈ H ×G the character value χ_{M}(y, x) is divisible, in O, by the orders of C_{H}(y) and C_{G}(x), and χ_{M}(x, y) = 0 if exactly one of x, y is p-regular. Any simple K ⊗_{O} A-module X and simple K ⊗_{O} B-module Y determines a simple K ⊗_{O} (B ⊗_{O} A^{0})-module Y ⊗_{K} X^{∗}, where X^{∗} = Hom_{K}(X, K), and if χ is the image of X in R_{K}(A) and η the image of Y in R_{K}(B) we denote by η · χ^{∗} the image in R_{K}(B ⊗_{O} A^{0}) of Y ⊗_{K} X^{∗}. If Φ : R_{K}(A) ≅ R_{K}(B) is an isometry, then for any χ ∈ Irr_{K}(A) we have Φ(χ) = δ_{χ}η_{χ} for some δ_{χ} ∈ {±1} and some η_{χ} ∈ Irr_{K}(B). Using the above notation we can consider Φ as an element in R_{K}(B ⊗_{O} A^{0}), denoted χ_{Φ}, by setting

χ_{Φ} = ∑_{χ∈Irr_{K}(A)} Φ(χ) · χ^{∗}

Following Broué [6], the isometry Φ is called perfect if χ_{Φ} is perfect when viewed as character of H × G. One of the main features of a perfect isometry between the two blocks A and B is that it induces an isomorphism of the centers Z(A) ≅ Z(B); see [6] for more details.

Proposition 3.3. Let G, H be finite groups, A a block algebra of OG and B a block algebra of OH. Suppose that K, k are splitting fields for G and H. Let M be a B-A-bimodule which is finitely generated projective as left B-module and as right A-module such that M induces a stable equivalence of Morita type between A and B. Assume that the isometry L^{0}(A) ≅ L^{0}(B) induced by Φ_{M} extends to an isometry Φ : R_{K}(A) ≅ R_{K}(B). Then χ_{Φ} − χ_{M} ∈ Pr_{O}(A ⊗_{O} B^{0}); in particular, Φ is a perfect isometry.

Proof. For x ∈ G denote by c(x) the conjugacy class of x in G and by τ_{x} the restriction to A of the O-linear map OG → O sending x′ ∈ c(x) to 1 and every other group element to 0. If x is p-singular then τ_{x} ∈ K ⊗_{ℤ} L^{0}(A), and if x runs over the p-singular elements of G then τ_{x} runs over a spanning set of the K-space K ⊗_{ℤ} L^{0}(A) of K-valued central functions on K ⊗_{O} A which are orthogonal to Pr_{O}(A). Extend Φ, Φ_{M} in the obvious way to K ⊗_{ℤ} R_{K}(A). Since Φ and Φ_{M} coincide on L^{0}(A) we have Φ_{M}(τ_{x}) = Φ(τ_{x}). Since Φ_{M} is induced by tensoring with M we also have

Φ_{M}(τ_{x})(y) = \frac{1}{|G|} ∑_{s∈G} χ_{M}(y, s)τ_{x}(s) = \frac{|c(x)|}{|G|}χ_{M}(y, x)

and a similar reasoning yields

Φ(τ_{x})(y) = \frac{|c(x)|}{|G|}χ_{Φ}(y, x)

This shows that χ_{Φ}(y, x) = χ_{M}(y, x) for any p-singular x ∈ G and any y ∈ H. Exchanging the roles of A, B shows that also χ_{Φ}(y, x) = χ_{M}(y, x) for any p-singular y ∈ H and any x ∈ G. Thus χ_{Φ} − χ_{M} vanishes outside the p-regular elements of H × G, hence belongs to Pr_{O}(A ⊗_{O} B^{0}). Since χ_{M} is perfect by the assumptions on M this implies that χ_{Φ} is perfect.

In what follows, A, B are symmetric O-algebras such that K ⊗_{O} A, K ⊗_{O} B are split semisimple. Given a B-A-bimodule M which is finitely generated projective as right A-module, the group homomorphism Φ_{M} : R_{K}(A) → R_{K}(B) induced by the functor M ⊗_{A} − depends only on the image [K⊗_{O}M] of M in R_{K}(B⊗_{O}A^{op}) because we have an obvious isomorphism K⊗_{O}(M ⊗_{A}U) ≅ (K ⊗_{O} M) ⊗_{(K⊗_{O}A)} (K ⊗_{O} U) for any finitely generated O-free A-module U. Rouquier’s stable equivalence for blocks of defect 2 is in fact induced by a complex rather than a bimodule; we describe briefly how these are linked (what follows are well-known formalities, included for the convenience of the reader). A bounded complex X of B-A-bimodules is said to induce a stable equivalence, if the components of X are finitely generated projective as left B-modules, right A-modules, and if there are isomorphisms of complexes of bimodules X ⊗_{A} X^{∗} ≅ B ⊕ Z and X^{∗} ⊗_{B} X ≅ A ⊕ Y with Y and Z homotopy equivalent to bounded complexes of projective B-B-bimodules and A-A-bimodules, respectively. By a result of Rickard in [20], there is a canonical functor D^{b}(B ⊗_{O} A^{op}) → \underline{mod}(B ⊗_{O} A^{op}). The following well-known lemma shows that if M is the image in \underline{mod}(B ⊗_{O} A^{op}) of a complex X inducing a stable equivalence, then M induces a stable equivalence of Morita type between A and B.

Lemma 3.4. Let A, B be symmetric O-algebras and X a bounded complex of B-A-bimodules inducing a stable equivalence. Then the image M of X in \underline{mod}(B⊗_{O}A^{op}) induces a stable equivalence of Morita type.

Proof. A formal verification shows that the image of X^{∗} in \underline{mod}(A ⊗_{O} B^{op}) is isomorphic to M^{∗} and that the image of X ⊗_{A} X^{∗} in \underline{mod}(B ⊗_{O} B^{op}) is isomorphic to M ⊗_{A} M^{∗}. By the assumptions on X, the image of X ⊗_{A}X^{∗} in \underline{mod}(B⊗_{O} B^{op}) is isomorphic to B. Similary, the image of X^{∗}⊗_{B} X in \underline{mod}(A ⊗_{O} A^{op}) is isomorphic to A, whence the result.

The purpose of the next proposition is to show that if X is a splendid complex inducing a stable equivalence between block source algebras A, B and if the isometry L^{0}(A) ≅ L^{0}(B) induced by the image M of X in \underline{mod}(B⊗_{O} A^{op}) extends to an isometry R_{K}(A) ≅ R_{K}(B), then X can be modified by a projective B-A-bimodule in such a way that the resulting image in the Grothendieck group of B-A-bimodules is a p-permutation equivalence - and hence Φ is in fact part of an isotypy. In order to state this properly, we use the following notation. For U a bounded complex of finitely generated O-free A-modules set [K ⊗_{O} U] = ∑_{i∈ℤ}(−1)^{i}[K ⊗_{O} U_{i}] ∈ R_{K}(A), where U_{i} is the component of U in degree i for i ∈ ℤ. If U, U′ are quasi-isomorphic bounded complexes of finitely generated O-free A-modules then K ⊗_{O} U, K ⊗_{O} U′ are homotopy equivalent complexes of K ⊗_{O} A-modules (because both are split and quasi-isomorphic as K⊗_{O}A is split semi-simple and the functor K⊗_{O}− is exact); in particular, [K⊗_{O}U] = [K⊗_{O}U′]. For X a bounded complex of B-A-bimodules which are finitely generated projective as right A-modules, the functor X ⊗_{A} − induces a group homomorphism Φ_{X} : R_{K}(A) → R_{K}(B) by setting Φ_{X}([K ⊗_{O} U]) = [K ⊗_{O} (X ⊗_{A} U)]; one checks easily that Φ_{X} = ∑_{i∈ℤ} (−1)^{i} Φ_{X_{i}}, where X_{i} is the component of X in degree i. By the above remarks the map Φ_{X} depends only on the image [K ⊗_{O} X] of X in R_{K}(B ⊗_{O} A^{op}), and hence if X → X′ is a quasi-isomorphism of bounded complexes of B-A-bimodules which are finitely generated projective as right A-modules then Φ_{X} = Φ_{X′}. Furthermore, if Y is a bounded complex of A-B-bimodules which are finitely generated projective as right B-modules then Φ_{Y} ∘ Φ_{X} = Φ_{Y ⊗_{B}X}. If A, B are block source algebras with a common defect group P, a complex of B-A-modules X is called splendid if its components are finite direct sums of summands of the B-A-bimodules B ⊗_{OQ} A, with Q running over the subgroups of P. A p-permutation equivalence between A and B is essentially a splendid generalised B-A-bimodule inducing an isomorphism R_{K}(A) ≅ R_{K}(B); this concept is due to Boltje and Xu [5]. We refer to [16] for more details regarding the terminology involving splendid complexes and p-permutation equivalences between source algebras.

Proposition 3.5. Let A, B be block source algebras with a common defect group such that K ⊗_{O} A, K ⊗_{O} B are split semi-simple and such that k ⊗_{O} A, k ⊗_{O} B are split. Let X be a bounded complex of B-A-bimodules inducing a stable equivalence, and let M be a B-A-bimodule such that M is isomorphic, in \underline{mod}(B ⊗_{O} A^{op}), to the canonical image of X under Rickard’s functor D^{b}(B ⊗_{O} A^{op}) → \underline{mod}(B ⊗_{O} A^{op}). If the isometry L^{0}(A) ≅ L^{0}(B) induced by M extends to an isometry Φ : R_{K}(A) ≅ R_{K}(B) then there is a split bounded complex of projective B-A-bimodules W such that Φ = Φ_{X⊕W} and Φ^{−1} = Φ_{(X⊕W)^{∗}}. If moreover X is splendid then the image of X ⊕ W in the Grothendieck group of B-A-bimodules is a p-permutation equivalence between A and B.

Proof. By construction of M, the characters [K ⊗_{O} X] and [K ⊗_{O} M] differ by the character of a generalised projective B ⊗_{O} A-module, and the character [K ⊗_{O} M] differs from the character determined by Φ by the character of a generalised projective B ⊗_{O} A-module. Thus there is a split bounded complex of projective B-A-bimodules W such that [K ⊗_{O} (X ⊕ W)] is the character of Φ, hence Φ = Φ_{X⊕W}. Since tensoring by the dual of X ⊕ W is left and right adjoint to the functor given by tensoring with X ⊕ W, the statement Φ^{−1} = Φ_{(X⊕W)^{∗}} follows immediately. Thus the composition of the maps Φ_{X⊕W} and Φ_{(X⊕W)^{∗}} is the identity on R_{K}(B), hence

[K ⊗_{O} ((X ⊕ W) ⊗_{A} (X ⊕ W)^{∗})] = [K ⊗_{O} B]

Since also

(X ⊕ W) ⊗_{A} (X ⊕ W)^{∗} ≃ B ⊕ Z′

for some bounded complex Z′ of projective B-B-bimodules this forces that [K ⊗_{O} Z′] = 0. But if the character of a generalised projective B-B-bimodule is zero, then actually the generalised bimodule is zero itself because the Cartan matrix of B is non-singular. Thus the image of Z′ in the Grothendieck group of B-B-bimodules (with respect to split exact sequences) is zero. Equivalently, the images of (X ⊕ W) ⊗_{A} (X ⊕ W)^{∗} and B in the Grothendieck group of B-B-bimodules are equal. Since X is splendid and W consists of projective B-B-bimodules, the complex X ⊕ W is still splendid, and hence its image in the Grothendieck group of B-A-bimodules is a p-permutation equivalence.

4 Stable equivalences and one simple module

Theorem 4.1. Let A, B be symmetric O-algebras such that K ⊗_{O} A, K ⊗_{O} B are split semisimple and such that k ⊗_{O} A, k ⊗_{O} B are split and have non singular Cartan matrices. Suppose that ℓ(A) = 1 and that for any η ∈ Irr_{K}(B) there is λ ∈ L^{0}(B) such that < η, λ >_{B}≠ 0. Denote by d_{1}, d_{2}, .., d_{r} the different dimensions of the simple K ⊗_{O} A-modules and by m_{i} the number of isomorphism classes of simple K ⊗_{O} A-modules of dimension d_{i}, where 1 ≤ i ≤ r. Assume that d_{i} divides d_{i+1} and that m_{i} > 2\frac{d_{i+1}}{d_{i}} for 1 ≤ i ≤ r − 1. Suppose that there is a stable equivalence of Morita type between A and B. Then there is an isomorphism R_{K}(A) ≅ R_{K}(B) which maps Irr_{K}(A) onto Irr_{K}(B) and induces an isomorphism Pr(A) ≅ Pr(B). In particular, ℓ(B) = 1 and the decomposition matrices of A and B are equal.

Proof. Since k ⊗_{O} A has a unique isomorphism class of simple modules, the rank of Pr_{O}(A) is one, and hence the rank of L^{0}(A) is |Irr_{K}(A)| − 1. Since the stable equivalence of Morita type between A and B induces an isometry L^{0}(B) ≅ L^{0}(A) we get |Irr_{K}(A)| ≤ |Irr_{K}(B)|. Note that |Irr_{K}(A)| = ∑_{i=1}^{r} m_{i}. Since k ⊗_{O} A is also split it is a matrix algebra over its basic algebra. We therefore may assume that A is basic. Similarly, we may assume that B is basic. Then A, as left A-module, is projective indecomposable, and we have

[K ⊗_{O} A] = ∑_{χ∈Irr_{K}(A)} d^{χ} · χ

where d^{χ} = dim_{K}(X) if χ = [X] for some simple K ⊗_{O} A-module X. For 1 ≤ i ≤ r denote by Λ_{i} the subset of Irr_{K}(A) of isomorphism classes of simple K ⊗_{O} A-modules of dimension d_{i}. In particular, |Λ_{i}| = m_{i}. By regrouping simple modules of the same degree we may rewrite this in the form

[K ⊗_{O} A] = ∑_{i=1}^{r} d_{i}( ∑_{χ∈Λ_{i}} χ)

Thus the elements χ − χ′, with χ, χ′ ∈ Λ_{i}, where 1 ≤ i ≤ r, and χ − \frac{d_{i+1}}{d_{i}} χ′ with χ ∈ Λ_{i+1}, χ′ ∈ Λ_{i}, where 1 ≤ i ≤ r − 1 are all in L^{0}(A). More precisely, if we choose an element χ_{i} ∈ Λ_{i} and set Λ′_{i} = Λ_{i} − {χ_{i}} then the following set is a basis of the free abelian group L^{0}(A):

B = ∪_{i=1}^{r} \{χ_{i} − χ′_{i} | χ′_{i} ∈ Λ′_{i}\} ∪ \{χ_{i+1} − \frac{d_{i+1}}{d_{i}} χ_{i} | 1 ≤ i ≤ r − 1\}

Indeed, B is clearly linearly independent and has ∑_{i=1}^{r} (m_{i} − 1) + r − 1 = |Irr_{K}(A)| − 1 elements. One checks that if a ℤ-linear combination of elements in B is divisible, in R_{K}(A), by a positive integer q then all coefficients of that ℤ-linear combination are divisible by q, which shows that B is indeed a basis of L^{0}(A). Note that |Λ_{i}| > 2\frac{d_{i+1}}{d_{i}} ≥ 4 for 1 ≤ i ≤ r − 1. Denote by M a B-A-bimodule which is finitely generated projective as left B-module and as right A-module such that M and M^{∗} induce a stable equivalence of Morita type between A and B. In particular, by Proposition 3.1, the map Φ_{M} restricts to an isometry L^{0}(A) ≅ L^{0}(B). For any i such that 1 ≤ i ≤ r, the elements χ − χ′, with χ, χ′ ∈ Λ_{i}, have norm 2, and thus, are mapped by Φ_{M} to elements of norm 2, hence an element of the form η − η′ for some η, η′ ∈ Irr_{K}(B). Thus if m_{i} = 2 then Λ_{i} determines a unique subset Δ_{i} = {η, η′} having exactly two elements, but this does not determine a canonical bijection between Λ_{i} and Δ_{i}. If m_{i} ≥ 3 then Λ_{i} contains a third element χ′′, and then < χ − χ′, χ − χ” >= 1, and hence χ − χ′′ is mapped to either η − η′′ or η′′ − η′ for some η ∈ Irr_{K}(B) different from both η, η′. Thus, if m_{i} ≥ 3 then there is a subset Δ_{i} of Irr_{K}(B) and a sign δ_{i} ∈ {±} such that Φ_{M} induces a uniquely determined bijection χ ↦ η_{χ} between Λ_{i} and Δ_{i} with the property

Φ_{M}(χ − χ′) = δ_{i}(η_{χ} − η_{χ′})

for all χ, χ′ ∈ Λ_{i}. This case applies to all indices i such that 1 ≤ i ≤ r − 1 because of the assumption m_{i} > 2\frac{d_{i+1}}{d_{i}}. We denote by η_{i} the image, in Δ_{i}, of χ_{i} under this bijection. If m_{r} = 1 we set Δ_{r} = Irr_{K}(B) − ∪_{i+1}^{r−1}Δ_{i}. Since |Irr_{K}(A)| ≤ |Irr_{K}(B)|, the set Δ_{r} is non empty. After possibly replacing M by Ω_{B⊗_{O}A^{0}}(M) we may assume that δ_{1} = 1. We will show inductively that the signs δ_{i} are all 1. Let i be an integer such that δ_{i} = 1 and such that 1 ≤ i ≤ r − 1. Consider the element

μ_{i} = χ_{i+1} − \frac{d_{i+1}}{d_{i}} χ_{i}

in L^{0}(A). We have < μ_{i}, χ_{i} − χ′ >_{A}= −\frac{d_{i+1}}{d_{i}} for all χ′ ∈ Λ′_{i}. Thus

Φ_{M}(μ_{i}) = (a − \frac{d_{i+1}}{d_{i}})η_{i} + a ∑_{η∈Δ′_{i}} η + Ψ_{i}

for some integer a and some element Ψ in R_{K}(B) not involving any of the elements in Δ_{i}. If m_{i+1} ≥ 2 then Ψ_{i} ≠ 0 because for any χ′ ∈ Λ′_{i+1} we have < μ_{i}, χ_{i+1} − χ′ >= 1 and hence Ψ_{i} must involve one of the two characters occurring in Φ_{i}(χ_{i+1} − χ′). Taking norms on both sides in the above equality yields

1 + \frac{d_{i+1}^{2}}{d_{i}^{2}} ≥ (a − \frac{d_{i+1}}{d_{i}})^{2} + (m_{i} − 1)a^{2} = m_{i}a^{2} − 2a\frac{d_{i+1}}{d_{i}} + \frac{d_{i+1}^{2}}{d_{i}^{2}}

which is equivalent to

1 ≥ m_{i}a^{2} − 2a\frac{d_{i+1}}{d_{i}}

Since m_{i} > 2\frac{d_{i+1}}{d_{i}} this implies

1 ≥ m_{i}a^{2} − 2a\frac{d_{i+1}}{d_{i}} ≥ m_{i}(a^{2} − a) ≥ 0

and all numbers involved in these inequalities are integers, hence equal to 1 or 0. Thus a is either 1 or 0. Note that if Ψ_{i} ≠ 0 (which happens in particular for 1 ≤ i ≤ r − 2) then the left most inequalities are proper inequalities which forces a = 0, hence

Φ_{M}(μ_{i}) = −\frac{d_{i+1}}{d_{i}} η_{i} + Ψ_{i}

Comparing norms again forces that the norm of Ψ_{i} is 1. If m_{i+1} ≥ 3 then for any χ′, χ′′ ∈ Λ′_{i+1} we have

< Φ(μ_{i}), δ_{i+1}(η_{i+1} − η_{χ′}) >=< μ_{i}, χ_{i+1} − χ′ >_{A}= 1
< Φ(μ_{i}), δ_{i+1}(η_{χ′} − η_{χ′′}) >_{B}=< μ_{i}, χ′ − χ′′ >_{A}= 0

which shows that Φ_{M}(μ_{i}) involves η_{i+1}. Since μ_{i} cannot come from a module, the signs force

Φ_{M}(μ_{i}) = η_{i+1} − \frac{d_{i+1}}{d_{i}} η_{i}

hence δ_{i+1} = 1. The only thing that remains to see is that we can exclude the pathological case a = 1 and Ψ_{i} = 0. This can occur only for i = r − 1. What happens in that case is that then m_{r} = 1. By the assumptions, every element of Irr_{K}(B) occurs in at least one element of L^{0}(B). Moreover, |Irr_{K}(B)| ≥ |Irr_{K}(A)|, so all elements η ∈ Δ_{r} occur in Φ_{M}(μ_{r−1}), contradicting Ψ_{r−1} = 0. The bijections Λ_{i} ≅ Δ_{i} constructed above yield therefore an isometry Φ : R_{K}(A) ≅ R_{K}(B) mapping Irr_{K}(A) to Irr_{K}(B). This implies that Irr_{k}(B) has a unique element because |Irr_{K}(B)| − |Irr_{k}(B)| is equal to the rank of L^{0}(B), hence to the rank of L^{0}(A) which in turn is equal to |Irr_{K}(A)|−|Irr_{k}(A)| = |Irr_{K}(A)|−1. Since Φ extends the isometry L^{0}(A) ≅ L^{0}(B) induced by Φ_{M} it must send Pr_{O}(A) to Pr_{O}(B). For the same reason the inverse of Φ sends Pr_{O}(B) to Pr_{O}(A). Thus Φ induces an isomorphism Pr_{O}(A) ≅ Pr_{O}(B), and both groups are isomorphic to ℤ. In particular, Φ maps the image in R_{K}(A) of a projective indecomposable A-module to the image in R_{K}(B) of a projective B-module. The statement on the decomposition matrices follows.

5 Proof of Theorem 1.1

Definition 5.1 (cf. [12]). A finite group H is said to be of central type if H has an ordinary irreducible character χ satisfying χ(1)^{2} = |H : Z(H)|.

Proposition 5.2. Let P be an elementary abelian p-group of order p^{2}, and let E ≤ Aut(P) be a p′-group. Suppose that there exists a central extension

1 → Z → Ẽ → E → 1

with Z a cyclic p′-group such that there is a K-valued linear character of Z which is covered by a unique irreducible character of Ẽ. Then E is abelian.

Proof. The hypothesis implies that Z = Z(Ẽ) and that Ẽ is of central type. We first show that E has abelian Sylow 2-subgroups. Let Ũ be a Sylow 2-subgroup of Ẽ, and let U := ŨZ/Z ≅ Ũ/Ũ ∩Z, a Sylow 2-subgroup of E. By [10, Lemma 2.2] or [8, Theorem 2], Ũ is also of central type and Ũ ∩ Z = Z(Ũ). Now E and hence U is a subgroup of GL_{2}(p). Let R be a Sylow 2-subgroup of GL_{2}(p) containing U. First suppose that p ≡ 3(mod 4). Then R has a cyclic subgroup R_{0} of index 2. So, U_{0} := R_{0} ∩ U is a cyclic subgroup of index at most 2 in U. On the other hand, by [10, Corollary 3.2], either U = 1 or U_{0} is not self-centralising in U. Since U_{0} has index at most 2 in U, it follows that U is abelian.

Let us suppose now that p ≡ 1(mod 4). In this case, R ≅ (R_{1} × R_{2}) ⋊ ⟨t⟩ where R_{1} = ⟨g⟩ and R_{2} = ⟨h⟩ are cyclic groups of equal order, t has order 2 and tgt^{−1} = g, tht^{−1} = gh^{−1}. In particular, R_{1} = Z(R). Set R_{0} = R_{1} × R_{2} and for 0 ≤ i ≤ 2 set U_{i} = U ∩ R_{i}. For any subgroup V of R, denote by Ṽ the inverse image of V in Ũ and denote by X_{V} the set of ordinary irreducible characters of Ṽ covering θ. Note that if V_{1} ≤ V_{2} are subgroups of U, with V_{1} normal in V_{2}, then X_{V_{2}} is the set of irreducible characters of Ṽ_{2} covering some character in X_{V_{1}}.

If U_{0} = U, then U is abelian and we are done. We assume therefore that U_{0} has index 2 in U. If U_{0} is cyclic, then by the same argument as above, U is abelian. Thus, we may assume that U_{0} is of rank 2, say U_{0} is a direct product of a cyclic group of order 2^{α} and a cyclic group of order 2^{β} with α ≥ β. We claim that β = α − 1. Indeed, let V be a cyclic subgroup of U_{0} of order 2^{α}. Since U/U_{0} has order 2, X_{U_{0}} has two elements which are transitively permuted by Ũ (otherwise there would be at least two characters of Ũ which cover characters of X_{U} and hence cover θ). Thus there are at most two orbits in the action of Ũ_{0} on X_{V}. Since V is cyclic of order 2^{α}, |X_{V} | = 2^{α} and since |U_{0} : V | = 2^{β} each orbit of Ũ_{0} on X_{V} has at most 2^{β} elements. Thus, α ≥ β ≥ α − 1. If β = α, then |Ũ/Z(Ũ)| = |U| = 2^{2α+1} is not a square, a contradiction to Ũ being of central type. Thus, β = α − 1, proving the claim.

We note here also that if V is a cyclic subgroup of U_{0} of order 2^{α} which is normal in U, then the fact that there is only one orbit of Ũ on X_{V} along with the fact that |U : V | = 2^{α} = |X_{V} | means that Ũ acts faithfully, freely and transitively on X_{V}. In particular, CŨ(Ṽ) = Ṽ.

Next, we claim that U_{1} = U ∩ R_{1} has order 2^{α}. For this, first we note that since U_{0}/U_{1} is a subgroup of R_{0}/R_{1}, U_{0}/U_{1} is cyclic. So, U_{1} has order at least 2^{α−1}. Suppose if possible that U_{1} has order 2^{α−1}. Then U_{1} has a complement in U_{0} (any ⟨v⟩ such the coset of v generates U_{0}/U_{1} is a complement to U_{1} in U_{0}). Write U_{1} = ⟨g^{a}⟩ and let W = ⟨g^{i}h^{j}⟩ be a complement to U_{1} in U_{0}. The order of W being 2^{α}, it follows that the order of h^{j} is greater than or equal to the order of g^{i} for otherwise, the unique involution of ⟨g⟩ (which is also the unique involution of U_{1}) would be a power of g^{i}h^{j}. So, denoting by n_{2} the 2-part of a natural number n, we have that i_{2} ≥ j_{2}. In particular, (j + 2i)_{2} = j_{2}. Let s be an element of U − U_{0}. Since sg^{i}h^{j}s^{−1} = g^{j+2i}g^{−i}h^{−j} ∈ U_{0} we get that g^{j+2i} ∈ U_{1}, that is, (j + 2i)_{2} ≥ a_{2}. So, i_{2} ≥ j_{2} = (j + 2i)_{2} ≥ a_{2} which means that the order of g^{i}h^{j} is less than or equal to the order of g^{a}. This proves the claim.

So, U_{1} is a cyclic normal subgroup of U of order 2^{α} contained in U_{0}. Hence, by the above remarks, CŨ(Ũ_{1}) = Ũ_{1}. Write Ũ ∩ Z = ⟨z⟩ and let ũ ∈ Ũ_{0} be a generator of Ũ_{1}/Ũ ∩ Z. Since U_{1} ≤ Z(U), the rule ṽ → [ũ, ṽ] defines a homomorphism from Ũ → Ũ ∩Z, with kernel CŨ(Ũ_{1}) = Ũ_{1}. Since Z is cyclic, this means in particular that Ũ/Ũ_{1} and hence U/U_{1} is cyclic. Let v ∈ U be such that vU_{1} generates U/U_{1}. Then, v ∉ U_{0}, so v = g^{i}h^{j}t for some i and j. Then v^{2} = g^{2i}h^{j}th^{j}t = g^{2i}h^{j}g^{j}h^{−j} = g^{2i+j} ∈ R_{1} ∩ U = U_{1}. This means that α = 1 , hence |U| has order 4 and is abelian.

Suppose, if possible that E is not abelian. Since E is a p′-group, the faithful two-dimensional representation of E on the field of p-elements represented by the inclusion E ≤ Aut(P) is absolutely irreducible and lifts to a faithful 2-dimensional irreducible representation of E over ℂ. By the classification of the non-abelian finite subgroups of GL_{2}(ℂ) (see for instance [9, Theorem 26.1]), either E has a normal abelian subgroup of index 2 or E/Z(E) is isomorphic to one of A_{4}, S_{4} or A_{5} and Z(E) consists of scalar matrices (considering E as subgroup of GL_{2}(ℂ)). In particular, Z(E) is cyclic. But A_{4}, S_{4} and A_{5} all have non-abelian Sylow 2-subgroups, whereas by the above E has abelian Sylow 2-subgroups. Thus, E has a normal abelian subgroup, say N of index 2. So the Sylow l-subgroups of E for odd primes l are abelian, normal in E and centralise each other. Furthermore, we have shown above that the Sylow 2-subgroups of E are abelian. Thus it suffices to show that any 2-element of E centralises the Sylow l-subgroup of E for any odd prime l.

Let l be an odd prime, Q the Sylow l-subgroup of E and u a 2-element of E. Since Q is abelian and normal in E, Q = C_{Q}(u) × [⟨u⟩, Q]. Let Q̃ be the Sylow l-subgroup of Ẽ, so that Q := Q̃Z/Z and let ũ be a 2-element in Ũ lifting u. By [10, Lemma 2.2], Q̃ is of central type with Z(Q̃) = Z ∩ Q̃. So Q is non-cyclic, hence of l-rank 2. By conjugating Q into the subgroup of diagonal matrices in GL_{2}(ℂ), we see that Q/Z(GL_{2}(ℂ)) ∩ Q is cyclic. Thus, C_{Q}(u) is non-trivial. Also, we may assume that [⟨u⟩, Q] is non-trivial as otherwise u centralises Q and we are done. Thus, both C_{Q}(u) and [⟨u⟩, Q] are non-trivial cyclic groups. Let Q̃_{1} be the inverse image of C_{Q}(u) in Q̃ and let Q̃_{2} be the inverse image of [⟨u⟩, Q] in Q̃. Both Q̃_{1} and Q̃_{2} are abelian groups; along with Z ∩ Q̃, they generate Q̃ and Z(Q̃) = Z ∩ Q̃. Hence CQ̃_{2}(Q̃_{1}) ≤ Z ∩ Q̃. Now , ũ stablises the normal series 1 ≤ Z ∩ Q̃ ≤ Q̃_{1}, Q̃_{1} is an l-group and u is an l′ -element. Hence, ũ centralises Q̃_{1}. Also, ũ centralises Z ∩ Q̃ = CQ̃_{2}(Q̃_{1}). Thus by the A × B-lemma (applied with A = ⟨u⟩, B = Q̃_{1} and P = Q̃_{2}), it follows that ũ centralises Q̃_{2} and hence Q̃.

Proposition 5.3. Let P be an elementary abelian p-group of order p^{2}, and let E ≤ Aut(P) be an abelian p′-group. Suppose that there exists a central extension

1 → Z → Ẽ → E → 1

with Z a cyclic p′-group such that there is a K-valued linear character θ of Z which is covered by a unique irreducible character of Ẽ. Then |E| = l^{2} for some natural number l dividing (p − 1). Let e_{θ} ∈ OZ be the central idempotent corresponding to θ and set A := O(P ⋊ Ẽ)e_{θ}, m := \frac{p−1}{l}. The degree of an element of Irr_{K}(A) is either l or l^{2} and Irr_{K}(A) has 2p − 1 elements of degree l and m^{2} elements of degree l^{2}.

Proof. By [8, Lemma2], E is a direct product of two isomorphic groups. In particular, |E| = l^{2} for some l. If E acts irreducibly on P, (where P is viewed as a vector space over the field of p elements), then E = C_{E}(E) is cyclic, a contradiction. Thus, P is a direct product P_{1} × P_{2} of two 1-dimensional E-invariant spaces. Hence E is conjugate to a subgroup of the group H of diagonal matrices in GL_{2}(p), l divides (p − 1) and E is a direct product of cyclic groups of order l. But H has a unique subgroup isomorphic to the direct product of two cyclic groups of order l, hence E is conjugate to that unique subgroup. In other words, E = E_{1} × E_{2}, where E_{1} acts faithfully and regularly on P_{1} and centralises P_{2} and E_{2} acts faithfully and regularly on P_{2} and centralises P_{1}.

Set N = P ⋊ Ẽ and consider an irrreducible K-valued character ξ = ξ_{1} × ξ_{2} of P, where ξ_{i} is an irreducible character of P_{i}, i = 1, 2. If ξ_{1} and ξ_{2} are both trivial, then there is exactly one character of N covering ξ × θ and that has dimension l. Supose that ξ_{1} is trivial and ξ_{2} is non-trivial. The inertial subgroup of ξ × θ is the inverse image Ẽ_{1} of E_{1} in Ẽ, Ẽ/Ẽ_{1} is cyclic of order l and |Ẽ : Ẽ_{1}| = l. Thus, there are l irreducible characters of Ẽ covering ξ × θ, each of dimension l. Also, the Ẽ-orbit of ξ has l-elements. Thus, there are 2(p − 1) irreducible characters covering characters of the form ξ_{1} × ξ_{2} × θ where exactly one of ξ_{1} and ξ_{2} is trivial, and they all have dimension l. Finally, if neither ξ_{1} nor ξ_{2} is trivial, then the inertial subgroup of ξ in Ẽ is Z, hence there is exactly one character of N covering ξ × θ and it has dimension l^{2}. The Ẽ-orbit of ξ has size l^{2}. Thus, there are m^{2} characters of N covering characters of the form ξ_{1} × ξ_{2} × θ where none of ξ_{1} and ξ_{2} is trivial, and each of these has dimension l^{2}.

Proof of Theorem 1.1. Since the theorem is well known when P is cyclic, we may assume that P is elementary abelian of order p^{2}. By Rouquier’s work [21, 6.3] (see also [16, Theorem A.2]) there is a stable equivalence of Morita type between B and C given by a splendid complex (in the slightly more restrictive sense of the notion “splendid” as used in [16]). By the structure theory of blocks with normal defect groups [15], [17, 14.6] and Proposition 5.2 the Brauer correpondent C of B has as source algebra a twisted group algebra of the form A = O_{α}(P ⋊E) for some abelian p′-subgroup E of Aut(P) and some α ∈ H^{2}(E; O^{×}). Then for a suitable central extension

1 → Z → Ẽ → E → 1

of E by a cyclic p′-group Z, determined by α, the algebra A is isomorphic to a block algebra O(P ⋊ Ẽ)e_{θ} for some block e_{θ} of P ⋊ Ẽ. By Proposition 5.3, A and a source algebra of B satisfy the hypotheses of Theorem 4.1, with a stable equivalence which is induced by a splendid complex. Thus, by 3.5, this stable equivalence induces a p-permutation equivalence. Since the blocks B, C have the same local structure it follows from [16, Theorem 1.4] that this p-permutation equivalence induces an isotypy.

6 Block cohomlogy need not be an Ext-algebra

One of the technical difficulties with non principal p-blocks is that they do not have, in general, a canonical module which would play the role of the trivial module. If there were such a module, one reasonable expectation would be that its Ext-algebra should be isomorphic to the block cohomology because this is what happens in the principal block case. In the context of a block algebra B which has, up to isomorphism, a unique simple module S, it may be tempting to think that S would play that role. However, even the smallest known and well understood example in [3], [4] of a non nilpotent block with one isomorphism class of simple modules has the property that its block cohomology is not isomorphic to the Ext-algebra of a module over that block, and this is what we are going to describe in this section. Let k be an algebraically field of odd prime characteristic p and let P = C_{p} × C_{p} be an elementary abelian group of order p^{2}. Set G = P ⋊ Q_{8}, where Q_{8} is a quaterion group of order 8 with Z(Q_{8}) acting trivially on P and with Q_{8}/Z(Q_{8}) ≅ C_{2} × C_{2} acting in such a way that each factor C_{2} inverts a generator of the corresponding factor C_{p} of P. Then C_{G}(P) = P × Z(Q_{8}). Thus kG has two blocks, namely the principal block b_{0} = \frac{1}{2}(1 + z) and the block b_{1} = \frac{1}{2}(1 − z), where z is the generator of Z(Q_{8}). The block algebra B_{1} = kGb_{1} has, up to isomorphism, a unique simple module S. We have dim_{k}(S) = 2; more precisely, Res_{Q_{8}}^{G}(S) is the unique 2-dimensional simple kQ_{8}-module. Both blocks have the same local structure, namely the fusion system of the group G, and hence the block cohomology of B_{1} is H^{∗}(B_{1}) ≅ H^{∗}(B_{0}) = H^{∗}(G, k).

Proposition 6.1. With the notation above, there is no finitely generated B_{1}-module V such that Ext_{B_{1}}^{∗}(V, V) ≅ H^{∗}(B_{1}).

Proof. Suppose that there is a finitely generated B_{1} module V such that Ext_{B_{1}}(V, V) ≅ H^{∗}(B_{1}). Then in particular Ext_{B_{1}}^{0}(V, V) ≅ End_{B_{1}}(V) ≅ H^{0}(B_{1}) ≅ k. But since all composition factors of V are isomorphic to S this implies V ≅ S. The generator z of Z(Q_{8}) acts as −1 on S, hence also on its dual S^{∗}, and thus as identity on S ⊗_{k} S^{∗}. This shows that S ⊗_{k} S^{∗} is a module over the principal block algebra B_{0}. we have B_{0} ≅ k(P ⋊ (C_{2} × C_{2})), and hence B_{0} has four pairwise non isomorphic simple modules all of which have dimension one. Since P acts as identity on S it follows that S ⊗_{k} S^{∗} is a semi-simple four-dimensional B_{0}-module. By a result of Benson and Carlson [2, Theorem 2.1] or [1, Theorem 3.1.9], the trivial B_{0}-module occurs exactly with multiplicity 1 in S ⊗_{k} S^{∗}. The remaining three summands are non-trivial one-dimensional modules. If one regards S and S ⊗_{k} S^{∗} as kQ_{8}-modules, then S is stable under any automorphism of Q_{8}, hence so is S ⊗_{k} S^{∗}. The one-dimensional non trivial kQ_{8}-modules are permuted transitively by an automorphism of order three if Q_{8}. Thus, as B_{0}-module, S ⊗_{k} S^{∗} = k ⊕ T_{1} ⊕ T_{2} ⊕ T_{3} is a direct sum of four pairwise non-isomorphic simple B_{0}-modules of dimension one, with k the trivial module. We have

Ext_{B_{1}}^{∗}(S, S) ≅ Ext_{kG}^{∗}(k, S ⊗ S^{∗}) ≅ H^{∗}(G; k) ⊕ Ext_{kG}^{∗}(k, T_{1} ⊕ T_{2} ⊕ T_{3})

Since the Ext-quiver of B_{0} is connected, the summand Ext_{kG}^{∗}(k, T_{1} ⊕ T_{2} ⊕ T_{3}) is not zero. But then Ext_{B_{1}}^{∗}(S, S) cannot be isomorphic to H^{∗}(B_{1}).

Acknowledgement. Most of this work was done at MSRI, Berkeley, during the semester on representation theory, Spring 2008 - and the authors would like to thank the institute for providing a stimulating research environment. Moreover, the authors would like to thank Shigeo Koshitani for valuable discussions.

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