Equivalences of Blocks of Group Algebras

Michel Broué

Ecole Normale Supérieure, Paris

Three lectures given at the International Conference on Representations of Algebras Ottawa, August 1992

Let O be a complete local noetherian ring, whose field of fractions has characteristic zero and residue field has non–zero characteristic. A block algebra over O is an indecomposable summand of the algebra of a finite group over O.

We introduce briefly (and justify through examples) several types of equivalences. Three levels of equivalences between block algebras seem to be relevant: Morita equivalence, Rickard (derived) equivalence, stable equivalence of Morita type.

We give a classification of various classical “invariants” of block algebras (such as numerical defect, decomposition matrices, defect of irreducible characters, etc.) depending on the type of equivalence we consider between block algebras.

After recalling why, when switching from the algebra point of view to the group point of view, the source algebra is a suitable replacement for the basic algebra, we try to give suitable “group theoretic” refinements of the previous equivalences.

This is an introductory survey : almost no proof is given, the comments are brief and the applications short. We emphasize the “algebra–theoretic approach”, which should be viewed as a first approximation to the methods used in block theory, as we try to explain in the last paragraph. In order to simplify the exposition, we restrict ourselves, most of the time, to the case of principal blocks.

1. Basic context and Notation

Let A be a left and right noetherian ring. We denote by _{A}mod the abelian category of finitely generated left A–modules,

and by _{A}proj the category of finitely generated projective left A–modules. We denote by R(A) the Grothendieck group of _{A}mod and by R^{pr}(A) the Grothendieck group of _{A}proj. If X is an object of _{A}mod (resp. of _{A}proj), we denote by [X] its representative in R(A) (resp. in R^{pr}(A)).

We denote by mod_{A} the abelian category of finitely generated right A–modules, and by proj_{A} the category of finitely generated projective right A–modules. For B another ring, we denote by _{A}mod_{B} the category of finitely generated (A, B)– bimodules.

1991 Mathematics Subject Classification. 20, 20G. Key words and phrases. Finite Groups, Algebras, Representations.

Typeset by AMS-TEX

Let G be a finite group, and let p be a prime number. Let K be a finite extension of the field of p–adic numbers ℚ_{p} which contains the |G|-th roots of unity. Thus the group algebra KG is a split semi–simple K–algebra. Let O be the ring of integers of K over ℤ_{p}. We denote by 𝔭 the maximal ideal of O, and we set k := O/𝔭. If JkG denotes the Jacobson radical of the group algebra kG, the algebra kG/JkG is a split semi–simple k–algebra.

By extension of scalars we get two functors

OGmod → _{KG}mod and _{OG}mod → _{kG}mod .
[figure: K; O; k = O/𝔭; ℚ_{p}; ℤ_{p}; 𝔽_{p} = ℤ_{p}/pℤ_{p}]

The Cartan–Decomposition triangle.

For the following classical facts, we refer the reader to [Se], part III.

The set Irr(KG) of representatives in R(KG) of the irreducible KG–modules is an orthonormal basis of R(KG) for the scalar product defined by

⟨[X], [X′]⟩ := dim Hom_{KG}(X, X′) .

The set Irr(kG) of representatives in R(kG) of the irreducible kG–modules is a ℤ–basis of R(kG), while the set Pim(kG) of representatives in R^{pr}(kG) of the indecomposable projective kG–modules is a ℤ–basis of R^{pr}(kG). The pairing

R^{pr}(kG) × R(kG) → ℤ

defined by

⟨[P], [X]⟩ := dim Hom_{kG}(P, X)

(P an object of _{kG}proj, X an object of _{kG}mod) defines a duality between R^{pr}(kG) and R(kG).

Let X be a finitely generated KG–module. Let X_{0} be a finitely generated O–free OG–module such that X = K ⊗_{O} X_{0}. Then the corresponding element [k ⊗ X_{0}] in R(kG) depends only on X, and this defines the decomposition map

dec^{G}: R(KG) → R(kG) .

The reduction modulo 𝔭 defines an isomorphism R^{pr}(OG)⟶^{∼}R^{pr}(kG). Identifying R^{pr}(OG) and R^{pr}(kG) through this isomorphism, the adjoint of the decomposition map is the linear map ^{t}dec^{G}: R^{pr}(OG) → R(KG) which sends the representative of a projective OG–module X onto the representative of the KG–module K ⊗_{O} X.

Finally, the Cartan map Car^{G}: R^{pr}(kG) → R(kG) is the linear map which sends the representative in R^{pr}(kG) of a projective kG–module X onto its representative in R(kG).

(T (G))  [figure: R(KG); ^{dec^{G}}; R(kG); R^{pr}(kG); _{_{t}dec_{G}}; _{Car_{G}}]

1.1. Theorem.

(1) The cokernel of Car^{G} is a finite p–group, whose exponent is the order of a Sylow p–subgroup of G.

(2) The map dec^{G} is onto, and the image of ^{t}dec^{G} is a pure submodule of R(KG).

(3) Car^{G} = dec^{G} ·^{t}dec^{G} .

2. Blocks

The decomposition of the unity element of OG into a sum of orthogonal primitive central idempotents 1 = ∑ e corresponds to the decomposition of the algebra OG into a direct sum of indecomposable two–sided ideals OG = ⨁ A (A = OGe), called the blocks of OG. For A a block of O, we set KA := K ⊗_{O} A and kA := k ⊗_{O} A.

By reduction modulo 𝔭, a primitive central idempotent remains primitive central, and consequently kG = ⨁ kA is still a decomposition into a direct sum of indecomposable two–sided ideals, called the blocks of kG.

OG = ⨁ A
↓  ↓
kG = ⨁ kA

The augmentation map OG → O factorizes through a unique block of OG called the principal block and denoted byA_{0}(OG).

2.A. The invariants of a block.

Let e be a central idempotent of OG and let A := OGe be the corresponding algebra (note that we are not assuming e necessarily primitive, so what follows applies to direct sums of blocks). The idempotent e is the unity element of the algebra A, and A is a symmetric O–algebra for the form

t: A → O , ∑_{g∈G} a(g)g ↦ a(1) .

Center and projective center. View A as an (A, A)–bimodule. The ring End_{A}(A)_{A} of its endomorphisms is the center Z(A) of A. The set of projective endomorphisms (endomorphisms which factorize through a projective (A, A)–bimodule) is an ideal of Z(A) which is denoted by Z^{pr}(A) and called the projective center of A.

2.1. Proposition. We have Z^{pr}(A) = {∑_{g∈G} gag^{−1} | (a ∈ A)} .

The set of projective endomorphisms of the (kA, kA)–bimodule kA is denoted by Z^{pr}(kA). It is equal to the image of Z^{pr}(A) through the reduction modulo 𝔭 Z(A) → Z(kA).

c-d-triangle and associated invariants. The Grothendieck groups R(KA), R(kA) and R^{pr}(kA) are summands of the Grothendieck groups R(KG), R(kG) and R^{pr}(kG), and the maps dec^{G}, ^{t}dec^{G}, Car^{G} restrict to maps which define the “c-d-triangle” of the block A,

(T (A))  [figure: R(KA); ^{dec^{A}}; R(kA); R^{pr}(kA); _{_{t}dec_{A}}; _{Car_{A}}]

which we view as endowed with its “metric structure” given by the dualities

R(KA) × R(KA) → ℤ and R^{pr}(kA) × R(kA) → ℤ .

We denote by Irr(KA) (resp. Irr(kA), Pim(kA)) the set of representatives in the corresponding Grothendieck group of the irreducible KA–modules (resp. of the irreducible kA–modules, of the projective indecomposable kA–modules), called the canonical basis of the corresponding ℤ–modules.

The matrix of Car^{A} on the canonical basis is called the Cartan matrix of A and denoted by C^{A}, while the matrix of dec^{A} on the canonical basis is called the decomposition matrix of A and denoted by D^{A}.

It is traditional to set

k(A) := |Irr(KA)| and l(A) := |Irr(kA)| = |Pim(kA)| .

The O–rank of Z(A) equals k(A), while the rank of Z^{pr}(kA) equals the number of trivial invariant factors of the map Car^{A}. We set

l^{pr}(A) := dim Z^{pr}(kA) .

The exponent of the cokernel of Car^{A} divides the order of a Sylow p–subgroup of G and so has the shape p^{d(A)}. The integer d(A) is called the defect of the block A.

Defects of irreducible KA–modules.

If X is an irreducible KA–module, we set p^{d(X)} := (\frac{|G|}{dim X} )_{p} , and we call the

integer d(X) the defect of X.

2.2. Proposition.

(1) We have d(A) = sup{d(X) | (X ∈ Irr(KA))}.

(2) Let X_{0} be an O–free A–module such that X = K ⊗_{O} X_{0}. Let End_{A}^{pr}(X_{0}) be the ideal of End_{A}(X_{0}) consisting of the projective endomorphisms of X_{0}. Then End_{A}(X_{0})/End_{A}^{pr}(X_{0}) = p^{d(A)−d(X)}O .

Remark.

Let P be a p–group. Then we have JOP = 𝔭OP + AOP where JOP denotes the Jacobson radical of OP and AOP denotes the augmentation ideal of OP. So OP is itself a block and we have l(OG) = 1. The c-d-triangle is trivial :

[figure: R(KP); dim; R(kP); R^{pr}(kP); reg; |P |]

where the map “reg” maps the generator |P | of R^{pr}(kP) onto the representative of the regular representation of KP. Notice that, on the other hand, the category _{OP} mod is far from being trivial. If P is neither cyclic nor (for p = 2) dihedral, semidihedral or generalized quaternion, then the algebra OP is wild.

2.B. Problems of block theory.

Block theory, as introduced and developed by Richard Brauer, originated mainly in the problem of the classification of finite simple groups. As a first approximation, we may say that the main problem of block theory is to compare the category _{OG}mod to the “local” categories _{ON_{G}(P)}mod, where P runs over the set of non–trivial p– subgroups of G, and N_{G}(P) denotes the normalizer of P in G.

Remark.

• Let A_{0} be the principal block of OG. The structure of _{A_{0}}mod is closely related to the structure of the group G itself — more precisely, to the structure of G/O_{p′}(G), where O_{p′}(G) denotes the largest normal subgroup of G whose order is relatively prime to p.

For example, let P be a Sylow p–subgroup of G. The following assertions are equivalent:

(i) l(A_{0}) = 1,

(ii) _{A_{0}}mod is equivalent to _{OP} mod, (iii) G is p–nilpotent, i.e., isomorphic to the semi–direct product O_{p′}(G) ⋊ P.

• The situation is more complicated for non–principal blocks. Indeed, there are blocks A of non–abelian simple groups G which satisfy one of the following equivalent properties (“defect zero”) :

(i) There is X ∈ Irr(KA) such that (dim X)_{p} equals the order of a Sylow p– subgroup of G,

(ii) _{A}mod is equivalent to _{O}mod. For example, the block defined by the Steinberg character of GL_{n}(p^{m}) has the above properties.

More generally, the “nilpotent blocks” A (see [BrPu], [Pu]) are such that _{A}mod ≃ OP mod for a certain p–group P. We give here an example of such a block in GL_{n}(ℓ^{m}) for ℓ ≠ p (see [Br1] for more details). We view GL_{n}(ℓ^{m}) as the group of fixed points of the algebraic group G := GL_{n}(\overline{𝔽}_{ℓ}) under the action of the usual Frobenius endomorphism F. Let T be a maximal torus of G and θ: T^{F} → K^{×} a character of T^{F} such that :

• the order of θ is prime to p, • θ is in general position in GL_{n}(ℓ^{m}) (i.e., an element which normalizes T and

fixes θ must centralize T). We denote by R_{T}^{G} : R(KT^{F}) → R(KG^{F}) the linear map defined by Deligne and Lusztig (see [DeLu]). There is a block A(T, θ) of OGL_{n}(ℓ^{m}) such that

Irr(KA(T, θ)) = {ε_{G}ε_{T}R_{T}^{G}(θη)}_{η}

where η runs over the set of characters of T^{F} whose order is a power of p. The category A(T,θ)mod is equivalent to _{OT_{p}^{F}} mod where T_{p}^{F} denotes the Sylow p–subgroup of T^{F}.

3. Morita equivalences between blocks

From now on, we denote by G and H two finite groups, by e and f respectively two central idempotents of OG and OH. We set A := OGe and B := OHf.

3.A. Preliminaries : bimodules and adjunctions.

We first recall in this context well known properties of functors induced by bimodules. Let M be an (A, B)–bimodule. Let X (resp. Y) be an A–module (resp. a B–module).

1. We have Hom_{A}(M ⊗_{B} Y, X) ≃ Hom_{B}(Y, Hom_{A}(M, X)) through the maps

(α: M ⊗_{B} Y → X) ↦ (α̂: Y → Hom_{A}(M, X) , y ↦ (m ↦ α(m ⊗ y)))
(β: Y → Hom_{A}(M, X)) ↦ (β̂: m ⊗ y ↦ β(y)(m))

2. Let us set M^{∨} := Hom_{A}(M, A) viewed as an object of _{B}mod_{A}. We denote by <,>: M × M^{∨} → A the natural A–pairing between M and M^{∨}. Suppose that M ∈ _{A}mod_{B} ∩_{A}proj. Then the map

M^{∨} ⊗_{A} X → Hom_{A}(M, X)
(m^{∨} ⊗_{A} x) ↦ (m ↦ <m, m^{∨}>x)

is an isomorphism in _{B}mod.

3. Let us set M^{∗} := Hom_{O}(M, O), viewed as an object of _{B}mod_{A}. Since the linear form t_{A}: A → O , ∑_{g∈G} a(g)g ↦ a(1) is a symmetrizing form for A, the maps

m^{∨} ↦ t_{A} · m^{∨} and m^{∗} ↦ (m ↦ ∑_{g∈G} m^{∗}(g^{−1}m)g)

are inverse isomorphisms (in _{B}mod_{A}) between M^{∨} and M^{∗}.

Suppose given a (B, A)–bimodule N and a duality M × N → O which is (A, B)– compatible, i.e., <m, bna> = <amb, n> for a ∈ A , b ∈ B , m ∈ M , n ∈ N. From what preceds, we deduce two explicit isomorphisms (respectively in _{A}mod_{A} and in Bmod_{B}) :

Hom_{B}(N, N)⟶^{∼}Hom_{A}(M ⊗_{B} N, A)
Hom_{A}(M, M)⟶^{∼}Hom_{B}(B, N ⊗_{A} M) .

We denote by ε_{M,N} the image of Id_{M} through the first isomorphism, and by η_{M,N} the image of Id_{N} through the second isomorphism. The maps ε_{M,N} and η_{M,N} are called the adjunctions associated with the pair (M, N).

3.1. Proposition. With the previous hypothesis and notation, the maps ε_{M,N} and η_{M,N} are computed as follows :

ε_{M,N}: M ⊗_{B} N → A , m ⊗_{B} n ↦ ∑_{g∈G} <g^{−1}m, n>g

η_{M,N}: B → N ⊗_{A} M , b ↦ ∑_{i∈I} n_{i} ⊗_{A} m_{i}

where ∑_{i∈I} n_{i} ⊗_{A} m_{i} is the element of N ⊗_{A} M such that, for all m ∈ M,

∑_{g∈G} ∑_{i∈I} <n_{i}g^{−1}, m>gn_{i} = bm .

3.B. Morita theorem and block invariants.

The following statement is a variation on Morita’s theorem, applied in the particular case of symmetric algebras.

3.2. Theorem. The following assertions are equivalent :

(i) The categories _{A}mod and _{B}mod are equivalent.

(ii) There exist

• an (A, B)–bimodule M which is projective both as an A–module and as a module–B,

• a (B, A)–bimodule N which is projective both as a B–module and as a module–A,

• an (A, B)–compatible O–duality between M and N

such that M ⊗_{B} N ≃ A in _{A}mod_{A} and N ⊗_{A} M ≃ B in _{B}mod_{B}.

Moreover, if the preceding statements are satisfied, then all of the adjunctions ε_{M,N}, η_{M,N}, ε_{N,M}, η_{N,M} are isomorphisms.

Morita equivalence and triangle invariants. If (M, N), as above, defines an equivalence between _{A}mod and _{B}mod, the pairs (K ⊗_{O} M, K ⊗_{O} N) and (k ⊗_{O} M, k ⊗_{O} N) define equivalences respectively between _{KA}mod and _{KB}mod and between _{kA}mod and _{kB}mod. So the Morita equivalence defined by (M, N) induces bijections

Irr(KA) ≃ Irr(KB) , Irr(kA) ≃ Irr(kB) , Pim(kA) ≃ Pim(kB) .

Moreover, by construction of the c-d-triangles (see §1 above), it is clear that the induced isomorphisms R(KA) ≃ R(KB), R(kA) ≃ R(kB), R^{pr}(kA) ≃ R^{pr}(kB), commute with the Cartan and the decomposition maps. To summarize :

3.3. Proposition. A Morita equivalence between A and B induces an isomorphism between the c-d-triangles T (A) and T (B), which preserves the canonical basis.

[figure: R(KB); ^{dec^{B}}; R(kB); R(KA); _{dec_{A}}; R(kA); R^{pr}(kB); _{Car_{B}}; R^{pr}(kA); _{_{t}dec_{A}}; _{Car_{A}}]

As a consequence, a Morita equivalence between A and B preserves all the invariants determined by the c-d-triangles and their canonical basis :

k(A) = k(B) , l(A) = l(B) , C^{A} = C^{B} , D^{A} = D^{B} , d(A) = d(B) .

Morita equivalence and centers. As it is well known, a Morita equivalence between A and B induces an algebra isomorphism between the centers Z(A) and Z(B), since Z(A) may be viewed as the center of the category _{A}mod. It also results from what follows (see [Ri2]), which also proves the preservation of projective centers.

We denote by A^{op} the opposite algebra of A, and A^{en} := A⊗_{O} A^{op} the “enveloping algebra”. Assume that (M, N) induces a Morita equivalence between A and B. Then M ⊗_{O} N is endowed with a natural structure of (A^{en}, B^{en})–bimodule defined by

(a ⊗ a′)(m ⊗ n)(b ⊗ b′) := amb ⊗ b′na′

and similarly N ⊗_{O} M is endowed with a natural structure of (B^{en}, A^{en})–bimodule. The map

M ⊗_{O} N × N ⊗_{O} M → O , (m ⊗ n, n′ ⊗ m′) ↦ <m, n′><m′, n>

is an (A^{en}, B^{en})–compatible duality.

3.4. Proposition. With the previous notation, (M ⊗_{O} N, N ⊗_{O} M) defines a Morita equivalence between A^{en} and B^{en} which exchanges A and B.

3.5. Corollary. A Morita equivalence between A and B induces an algebra isomorphism between Z(A) and Z(B) which restricts to an isomorphism between Z^{pr}(A) and Z^{pr}(B).

Morita equivalence and defects of irreducible KA–modules. By 3.3, a Morita equivalence induces a bijection I: Irr(KB)⟶^{∼}Irr(KA), and it results from 2.2, (2), that I preserves the defects : for all Y ∈ Irr(KB), d(I(Y)) = d(Y). This is also a consequence of what follows, which will be generalized later on to the case of a Rickard equivalence.

For X ∈ Irr(KA) (resp. Y ∈ Irr(KB)) we denote by e_{X} (resp. f_{Y}) the corresponding primitive idempotent of ZKA (resp. of ZKB). We denote by e (resp.

f) the unity element of A (resp. B), so A = OGe and e = ∑_{X∈Irr(KA)} e_{X} (resp. B = OHf and f = ∑_{Y ∈Irr(KB)} f_{Y}). The set {e_{X}}_{X∈Irr(KA)} is a K–basis of ZKA, and if ∑_{X} λ_{X}e_{X} ∈ Z(A), then λ_{X} ∈ O for all X ∈ Irr(KA).

Let (M, N) as in 3.2 which induces an equivalence between _{A}mod and _{B}mod. It induces a bijection Irr(KA)⟶^{∼}Irr(KB) which we denote by X ↦ Y_{X}. The adjunctions η_{N,M} and ε_{M,N} are isomorphisms in _{A}mod_{A}, hence η_{N,M} · ε_{M,N} is an automorphism of A in _{A}mod_{A} and it restricts to an automorphism of Z(A) viewed as an O–module.

3.6. Proposition. With the previous notation,

η_{N,M} · ε_{M,N}: e ↦ ∑_{X∈Irr(KA)} \frac{|G|/dim X}{|H|/dim Y_{X}} e_{X} .

In particular, \frac{|G|/dim X}{|H|/dim Y_{X}} is invertible modulo p and the defects are preserved :

d(X) = d(Y_{X}).

3.7. Corollary. If e is primitive (i.e., if A is a block), then \frac{|G|/dim X}{|H|/dim Y_{X}} is constant

modulo p for X ∈ Irr(KA).

3.C. Examples of Morita equivalences.

Clifford Theory. “Clifford theory” is the name of a set of theorems relating representations of a group G with representations of a normal subgroup N of G. It can be viewed as a series of Morita equivalences. We present here the first (and easy) part of Clifford theory : the “reduction to the inertial group”.

Let N be a normal subgroup of G. Let f be a central primitive idempotent of ON. We denote by H the stabilizer of f in G (which acts by conjugation on the set of central idempotents of ON). Then f is a central idempotent of OH. We set B := OHf.

We set e := ∑_{g∈[G/H]} gfg^{−1}, where [G/H] denotes a set of representatives of the cosets of G modulo H. Then e is a central idempotent of OG. We set A := OGe.

Let M := eOGf = OGf, endowed with left multiplication by A and right multiplication by B. Let N := fOGe = fOG, viewed similarly as a (B, A)–bimodule. Since M ⊗_{B} N ≃ A in _{A}mod_{A} and N ⊗_{A} M ≃ B in _{B}mod_{B}, and since the functors

M ⊗_{B} ▪ and N ⊗_{A} ▪ are respectively f · Res_{H}^{G} and Ind_{H}^{G}, we get :

3.8. Proposition. The functors f ·Res_{H}^{G} and Ind_{H}^{G} are inverse equivalences between Amod and _{B}mod.

p–nilpotent groups. Let G be a p–nilpotent group, i.e., G ≃ S ⋊ P where P is a p–group and S is a normal p′–subgroup (group with order prime to p) of G.

For every irreducible KS–module X, we denote by f_{X} the corresponding central primitive idempotent of KS (since S is a p′–group, f_{X} ∈ ZOS) and by P_{X} the stabilizer of f_{X} in P.

3.9. Proposition. We have

OGmod ≃ ⨁_{X∈Irr(KS) mod P} _{OP_{X}} mod .

Sketch of proof. Set e_{X} := ∑_{[P/P_{X}]} gf_{X}g^{−1}, and G_{X} := S ⋊P_{X}. By 3.8 above, we see that _{OGe_{X}} mod is equivalent to _{OG_{X}f_{X}} mod. Since OG = ⨁_{X∈Irr(KS) mod P} OGe_{X}, it suffices to prove that _{OG_{X}f_{X}} mod ≃ _{OP_{X}} mod.

We may assume P_{X} = P. Let X_{0} be an O–free OS–module such that X = K⊗_{O}X_{0} (note that X_{0} is unique up to isomorphism). Because dim X and |P | are relatively prime, there is an action of S ⋊ P on X_{0} which extends the action of S. This allows us to define on M := OG ⊗_{OS} X_{0} a structure of (OGe_{X}, OP)–bimodule as follows :

g.(g_{1} ⊗_{OS} x).π := gg_{1} ⊗_{OS} π^{−1}(x) for g, g_{1} ∈ G , x ∈ X_{0} , π ∈ P .

Similarly, the module N := X_{0}^{∗} ⊗_{OS} OG is endowed with a structure of (OP, OGe_{X})–

bimodule, and it is not difficult to check that (M, N) induces a Morita equivalence between _{OGe_{X}} mod and _{OP} mod. □

Remark. There are lots of Morita equivalences in the theory of blocks of p–solvable groups, analogous (although sometimes far deeper) to the ones just described for the case of a p–nilpotent group. The classical “Fong reduction theorem” (see [Fo]) may be viewed as the description of a Morita equivalence between two blocks of two p–solvable groups (see for example [Pu1]). The description of blocks of groups of p–length one relies on some highly non trivial Morita equivalences (see [Da]).

On the other hand, Morita equivalences between blocks seem far less frequent for non abelian simple groups. In this case, the equivalence must be weakened to what we call a Rickard equivalence.

4. Rickard equivalences between blocks

4.A. Complexes : Notation and conventions.

As in the previous section, we denote by G and H two finite groups, by e and f respectively two central idempotents of OG and OH, and we set A := OGe, B := OHf.

Homomorphisms and tensor product of two complexes. The definitions we use here for the differentials of the homomorphisms and the tensor product of two complexes are slightly different from the usual ones (although they provide complexes isomorphic to the usual ones).

1. Let X := (⋯ → X^{m} →^{d_{X}^{m}} X^{m+1} → ⋯) and Y := (⋯ → Y^{m} →^{d_{Y}^{m}} Y^{m+1} → ⋯)

be complexes in mod A. We set

Hom_{A}^{▪}(X, Y) := (⋯ → Hom_{A}^{m}(X, Y) →^{d^{m}} Hom_{A}^{m+1}(X, Y) → ⋯)

where

Hom_{A}^{m}(X, Y) := ∏_{i,j,j−i=m} Hom_{A}(X^{i}, Y^{j}) and

d^{m}: Hom_{A}(X^{i}, Y^{j}) → Hom_{A}(X^{i}, Y^{j+1}) × Hom_{A}(X^{i−1}, Y^{j})
α ↦ ((−1)^{m}d_{Y}^{j} · α, −α · d_{X}^{i−1})

We set X^{∗} := Hom_{O}^{▪}(X, O), viewed as a complex in mod_{A}, and d_{X^{∗}}^{m} = −^{t}d_{X}^{−(m+1)}. 2. Assume now that Y is a complex in mod_{A}. We set

Y ⊗_{A} X := (⋯ → (Y ⊗_{A} X)^{m} →^{d^{m}} (Y ⊗_{A} X)^{m+1} → ⋯)

where

(Y ⊗_{A} X)^{m} := ⨁_{i,j,i+j=m}(Y^{j} ⊗_{A} X^{i}) and

d^{m}: (Y^{j} ⊗_{A} X^{i}) → (Y^{j+1} ⊗_{A} X^{i}) ⊕ (Y^{j} ⊗_{A} X^{i+1})
y ⊗ x ↦ ((−1)^{m}d_{Y}^{j} (y) ⊗ x) ⊕ (y ⊗ d_{X}^{i}(x))

Some classical maps. We denote now by M := (⋯ → M^{i} →^{d_{M}^{i}} M^{i+1} → ⋯) a bounded complex of (A, B)–bimodules.

Let N := (⋯ → N^{j} →^{d_{N}^{j}} N^{j+1} → ⋯) be a complex in _{B}mod_{A}. An (A, B)–

compatible O–duality between M and N is the following datum :

(1) for all i, an (A, B)–compatible duality <,>: N^{i} × M^{−i} → O ,

(2) such that the maps d_{N}^{i} and −d_{M}^{−(i+1)} are adjoint for this duality.

• We denote by ε_{M^{i},N^{−i}}: M^{i} ⊗_{B} N^{−i} → A the map defined by

ε_{M^{i},N^{−i}}(m ⊗ n) := ∑_{g∈G} <n, g^{−1}m>g ,

and we denote by ε_{M,N}: M ⊗_{B} N → A the chain map defined by

ε_{M,N} := ∑_{i}(−1)^{i}ε_{M^{i},N^{−i}} .

• We denote by τ_{N^{−j},M^{i}}: N^{−j} ⊗_{A} M^{i} → Hom_{A}(M^{j}, M^{i}) the map defined by

τ_{N^{−j},M^{i}}(n ⊗ m): m′ ↦ ε_{M^{j},N^{−j}}(m′ ⊗ n)m ,

and we denote by τ_{N,M}: N ⊗_{A}M → Hom_{A}^{▪}(M, M) the chain map defined by the family

(τ_{N^{−j},M^{i}})_{i,j}.

• Finally, we denote by σ_{M^{i}}^{B}: B → Hom_{A}(M^{i}, M^{i}) the morphism which defines the structure of module–B on M, and we denote by σ_{M}^{B}: B → Hom_{A}^{▪}(M, M) the chain map defined by

b ↦ ∏_{i} σ_{M^{i}}^{B}(b) .

The adjunctions. Let _{A}com be the category of complexes in _{A}mod with chain maps as morphisms, and let _{A}com_{B} be the category of complexes in _{A}mod_{B} with chain maps as morphisms.

Assume that, as above, M is a bounded complex of (A, B)–bimodules, N is a bounded complex of (B, A)–bimodules with a given (A, B)–compatible duality with M. Assume moreover that each component M^{i} of M is projective both as an A– module and as a module–B. Then the functors

M ⊗_{B} ▪: _{B}com → _{A}com and N ⊗_{A} ▪: _{A}com → _{B}com

are adjoint one to the the other on both sides.

The chain map τ_{N,M}: N ⊗_{A} M → Hom_{A}^{▪}(M, M) is an isomorphism in _{B}com_{B}. We

denote by η_{M,N}: B → N ⊗_{A} M the chain map defined by η_{M,N} := τ_{N,M}^{−1} · σ_{M}^{B} .

Definition. We call adjunctions the two pairs of chain maps of complexes of bimodules

ε_{M,N}: M ⊗_{B} N → A and η_{M,N}: B → N ⊗_{A} M
ε_{N,M}: N ⊗_{A} M → B and η_{N,M}: A → M ⊗_{B} N

which define respectively adjunctions for the adjoint pairs (M ⊗_{B} ▪, N ⊗_{A} ▪) and (N ⊗_{A}

▪, M ⊗_{B} ▪).

4.B. Rickard equivalences and block invariants.

Definition. We say that A and B are Rickard equivalent if there exists

• a bounded complex M in _{A}com_{B}, each component of which is both projective as an A–module and as a module–B,

• a bounded complex N in _{B}com_{A}, each component of which is both projective as an B–module and as a module–A,

• an (A, B)–compatible O–duality between M and N,

such that \{ \begin{matrix}M ⊗_{B} N & is homotopy equivalent to A in _{A}com_{A} \\ N ⊗_{A} M & is homotopy equivalent to B in _{B}com_{B} . \end{matrix}

In this case, the complexes M and N are called “Rickard tilting complexes” for A and B.

We denote by D^{b}(A) the derived bounded category of _{A}mod.

The following theorem is due to J. Rickard ([Ri1], [Ri3]). Note that it may be viewed as a generalization of Morita theorem 3.2.

4.1. Theorem. The following assertions are equivalent :

(i) The derived bounded categories D^{b}(A) and D^{b}(B) are equivalent as triangulated categories.

(ii) The algebras A and B are Rickard equivalent. Moreover, in this case, the adjunctions ε_{M,N}, η_{M,N}, ε_{N,M}, η_{N,M} are all homotopy equivalences between the corresponding complexes of bimodules.

Like in the case of a Morita equivalence (see §3 above), any pair of complexes (M, N) which induces a Rickard equivalence between A and B defines pairs (K ⊗_{O} M, K ⊗_{O} N) and (k ⊗_{O} M, k ⊗_{O} N) which induce Rickard equivalences between respectively KA and KB, kA and kB.

An object of D^{b}(A) is called perfect (see [Gro]) if it is isomorphic to a bounded complex of projective A–modules. Let D_{perf}^{b}(A) be the full subcategory of D^{b}(A) consisting of perfect complexes. A Rickard equivalence between A and B induces an equivalence of categories between D_{perf}^{b}(A) and D_{perf}^{b}(B).

Rickard equivalences and triangle invariants. The Grothendieck groups of the triangulated categories (see [Gro], and also [Ha]) D^{b}(KA), D^{b}(kA), D_{perf}^{b}(kA) are respectively the groups R(KA), R(kA), R^{pr}(kA). Hence a Rickard equivalence induces isomorphisms between these groups. By construction, these isomorphisms commute with the maps of the c-d-triangle (see §1).

The metric structure on the c-d-triangle may be defined in terms of the derived category. For example, the duality between R(kA) and R^{pr}(kA) is defined as follows. For P := (⋯ → P^{i} → P^{i+1} → ⋯) a bounded complex of projective A–modules, object of D_{perf}^{b}(kA), and X := (⋯ → X^{i} → X^{i+1} → ⋯) an object of D^{b}(kA), we have

<[P], [X]> := ∑_{i}(−1)^{i}dim Hom_{D^{b}(kA)}(P [i], X) .

Hence the isomorphisms between triangles defined by a Rickard equivalence preserve the natural ℤ–dualities.

Since Irr(KA) is an orthonormal basis of the ℤ–module R(KA), it follows that the isomorphism R(KA)⟶^{∼}R(KB) sends an element of Irr(KA) onto an element of {±Y | (Y ∈ Irr(KB))} (while a Morita equivalence induces a bijection between Irr(KA) and Irr(KB)). There is no analogous property for Irr(kA) or Pim(kA).

4.2. Proposition. A Rickard equivalence between A and B induces an isomorphism between the c-d-triangles T (A) and T (B) (viewed as endowed with their natural “metric”).

[figure: R(KB); ^{dec^{B}}; R(kB); R(KA); _{dec_{A}}; R(kA); R^{pr}(kB); _{Car_{B}}; R^{pr}(kA); _{_{t}dec_{A}}; _{Car_{A}}]

As a consequence, a Rickard equivalence between A and B preserves all the invariants determined by the c-d-triangles and their metrics :

• k(A) = k(B) , l(A) = l(B) , d(A) = d(B) ,

• the Cartan matrices C^{A} and C^{B} are equivalent as matrices of quadratic forms over ℤ (in particular they have the same invariant factors, and l^{pr}(A) = l^{pr}(B)),

• the decomposition matrices are equivalent as follows : there exists an orthonormal matrix U in Mat_{k(A)}(ℤ) and an invertible matrix V in Mat_{l(A)}(ℤ) such that D^{B} = UD^{A}V (and in particular D^{A} and D^{B} have the same invariant factors).

Rickard equivalence and centers. A Rickard equivalence between A and B induces an algebra isomorphism between the centers Z(A) and Z(B), since Z(A) may be viewed as the center of the category D^{b}(A).

Like in the case of Morita equivalences (see §3), it also results from the following proposition.

4.3. Proposition. If (M, N) induces a Rickard equivalence between A and B, then (M ⊗_{O} N, N ⊗_{O} M) defines a Rickard equivalence between A^{en} and B^{en} which exchanges A and B.

Since a morphism between two modules factorizes through a projective module if and only if it factorizes through a perfect complex, we get as a consequence :

4.4. Corollary. A Rickard equivalence between A and B induces an algebra isomorphism between Z(A) and Z(B) which restricts to an isomorphism between Z^{pr}(A) and Z^{pr}(B).

Rickard equivalence and defects of irreducible KA–modules. Like in §3, for X ∈ Irr(KA), we denote by e_{X} the corresponding primitive idempotent of ZKA. We denote by e (resp. f) the unity element of A (resp. B).

Let (M, N) be a pair of complexes which induces a Rickard equivalence between D^{b}(A) and D^{b}(B). It sends an element X ∈ Irr(KA) onto ε_{X}Y_{X} where ε_{X} = ±1 and Y_{X} ∈ Irr(KB). The adjunctions η_{N,M} and ε_{M,N} are homotopy equivalences between objects of _{A}com_{A}, hence η_{N,M} · ε_{M,N} is an automorphism of A in _{A}mod_{A} and it restricts to an automorphism of Z(A) viewed as an O–module.

4.5. Proposition. With the previous notation,

η_{N,M} · ε_{M,N}: e ↦ ∑_{X∈Irr(KA)} \frac{|G|/dim X}{|H|/ε_{X}dim Y_{X}} e_{X} .

In particular, \frac{|G|/dim X}{|H|/ε_{X}dim Y_{X}} is invertible modulo p and the defects are preserved :

d(X) = d(Y_{X}).

4.6. Corollary. If e is primitive in Z(A) (i.e., if A is a block), \frac{|G|/dim X}{|H|/ε_{X}dim Y_{X}} is

constant modulo p for X ∈ Irr(KA).

4.C. Examples of Rickard equivalences. Although derived equivalences are conjecturally very frequent in block theory, only a very small number of them is actually proved.

Groups with cyclic Sylow p–subgroups. The following result is a particular case of results proved by Rickard and Linckelmann (see [Ri2] and [Li]) as a consequence of the structure theorem of blocks with cyclic defect groups.

4.7. Theorem. Assume that G has a cyclic Sylow p–subgroup P. Let us denote by A_{0} and B_{0} respectively the principal blocks of G and of the normalizer of P in G. Then A_{0} and B_{0} are Rickard equivalent.

Remark. At this date (december 1992), no explicit construction of a Rickard tilting complex is known.

The principal 2–block of 𝔄_{5} and 𝔄_{4}. Let us denote by G the alternating group 𝔄_{5} on 5 letters, and by H the normalizer of a Sylow 2-subgroup of G, isomorphic to the alternating group 𝔄_{4}. Let p = 2. Let A_{0} (resp. B_{0}) be the principal block of G (resp. H).

It is easy to check that A_{0} and B_{0} are not Morita equivalent (since, for example, they have different decomposition matrices). Nevertheless, the functors Ind_{H}^{G} and Res_{H}^{G} induce a stable equivalence between A_{0} and B_{0}. The following unpublished result of Rickard shows that one can “twist” this stable equivalence to get a Rickard equivalence.

Let AOG be the augmentation ideal of OG and let P (AOG) →^{d} AOG be the projective cover of AOG viewed as a (OG, OH)–bimodule. We set

M := (⋯ → 0 → P (AOG) →^{d} OG → 0 → ⋯)

and N := M^{∗}.

4.8. Theorem. The pair (M, N) defines a Rickard equivalence between A_{0} and B_{0}.

A conjecture. The preceding two examples are particular cases of a conjectural general result (see [Br1]).

4.9. Conjecture. Let G be a finite group whose Sylow p–subgroups are abelian. Let H be the normalizer of one of the Sylow p–subgroups of G. Then the principal blocks of G and H are Rickard equivalent.

4.D. Perfect isometries.

The preceding conjecture seems hard to prove (or even to check on examples) at the moment. Nevertheless, one of its non–trivial consequences, which should be viewed as the “shadow”, at the level of characters, of a Rickard equivalence, has already been checked on a long series of cases.

For the definitions and properties stated in this paragraph, see [Br1].

As in §3, we denote by G and H two finite groups, by e and f respectively two central idempotents of OG and OH, and we set A := OGe, B := OHf. From now on, we identify R(KG) with the group of virtual characters of KG–representations, and Irr(KG) with the set of irreducible characters.

Let μ be a virtual character of G×H, element of R(K[G×H]). Then μ corresponds to a linear map I_{μ}: R(KH) → R(KG) as follows : for ζ ∈ Irr(KH), the function

I_{μ}(ζ) is defined by I_{μ}(ζ)(g) := \frac{1}{|H|} ∑_{h∈H} μ(g, h^{−1})ζ(h) .

Definition. We say that a virtual character μ of G × H is perfect if : (pe.1) for all g ∈ G and h ∈ H, |C_{G}(g)|_{p} and |C_{H}(h)|_{p} both divide μ(g, h), (pe.2) if μ(g, h) ≠ 0, then either g and h are both p–regular, or g and h are both

p–singular. If moreover the map I_{μ} defined by μ induces an isometric bijection from R(KB) to R(KA), we say that I_{μ} is a perfect isometry between B and A, and that A and B are perfectly isometric.

The connection with Rickard equivalences is made by the following statement.

4.10. Proposition. Assume that M is a Rickard tilting complex for A and B. Let μ_{M} be the virtual character of G × H defined by

μ_{M}(g, h) := ∑_{i}(−1)^{i}tr((g, h^{−1}); M^{i}) .

Then μ_{M} defines a perfect isometry between B and A.

The point is that, if A and B are perfectly isometric, their invariants behave “as if” they were Rickard equivalent (see [Br1]) — compare with assertions 4.2, 4.4, 4.5 above.

4.11. Theorem. Suppose that A and B are perfectly isometric.

(1) There is an isomorphism between the c-d-triangles T (A) and T (B) (viewed as endowed with their natural “metric”). In particular,

• k(A) = k(B) , l(A) = l(B) , d(A) = d(B) ,

• the Cartan matrices C^{A} and C^{B} are equivalent as matrices of quadratic forms over ℤ,

• the decomposition matrices are equivalent as follows : there exists an orthonormal matrix U in Mat_{k(A)}(ℤ) and an invertible matrix V in Mat_{l(A)}(ℤ) such that D^{B} = UD^{A}V.

(2) There is an algebra isomorphism between Z(A) and Z(B) which restricts to an isomorphism between Z^{pr}(A) and Z^{pr}(B).

(3) There is an automorphism of (A, A)–bimodules of Z(A) such that, if I_{μ_{M}}^{−1} (X) = ε_{X}Y_{X}, then

e ↦ ∑_{X∈Irr(KA)} \frac{|G|/dim X}{|H|/ε_{X}dim Y_{X}} e_{X} .

In particular, \frac{|G|/dim X}{|H|/ε_{X}dim Y_{X}} is invertible modulo p and the defects are preserved :

d(X) = d(Y_{X}).

The following conjecture is a weaker form of 4.9.

4.12. Conjecture. Let G be a finite group whose Sylow p–subgroups are abelian. Let H be the normalizer of one of the Sylow p–subgroups of G. Then the principal blocks of G and H are perfectly isometric.

The preceding conjecture is known to be true in the following cases :

• for all p, if G is p–solvable ;

• for p = 2, in all cases ([FoHa]) ;

• for all p, if G is a sporadic simple group ([Rou]) ;

• for all p, if G is a symmetric group ([Rou]) or an alternating group (Fong, private communication) ;

• if G is the group of rational points of a connected reductive algebraic group G defined over 𝔽_{q} and p is a prime number which does not divide q and which is good for G ([BMM], [BrMi]).

4.E. The case of the finite reductive groups.

In the case where G is a “finite reductive group”, the conjecture 4.9 can be made more precise and closely linked with the underlying algebraic geometry (for more details, see [BrMa]).

Notation. In this paragraph, we temporarily change our notation to fit with the usual notation of finite reductive groups : our prime p (the characteristic of our field k := O/𝔭) is now denoted by ℓ, and q denotes a power of another prime p ≠ ℓ.

Let G be a connected reductive algebraic group over 𝔽̄_{q}, endowed with a Frobenius endomorphism F which defines a rational structure on 𝔽_{q}. Let P be a parabolic subgroup of G, with unipotent radical U, and with F –stable Levi subgroup L. We denote by Y(U) the associated Deligne–Lusztig variety defined (cf. for example [Lu]) by

Y(U) := {g(U ∩ F (U)) ∈ G/U ∩ F (U) ; g^{−1}F (g) ∈ F (U)} ,

and we recall that G^{F} acts on Y(U) by left multiplication while L^{F} acts on Y(U) by right multiplication. It is known (cf. [Lu]) that Y(U) is an L^{F} –torsor on a variety X(U), which is smooth of pure dimension equal to dim(U/U ∩ F (U)), and which is affine (at least if q is large enough). In particular X(U) is endowed with a left action of G^{F}. If O is a commutative ring, the image of the constant sheaf O on Y(U) through the finite morphism π: Y(U) → X(U) is a locally constant constant sheaf π_{∗}(O) on X(U). We denote this sheaf by F_{OL^{F}}.

Let ℓ be a prime number which does not divide q and let O be the ring of integers of a finite extension of the field ℚ_{ℓ} of ℓ–adic numbers. For any G^{F} –equivariant torsion free O–sheaf F on X(U), we denote by H_{O}(X(U), F) the algebra of endomorphisms of the “ℓ–adic cohomology” complex RΓ_{c}(X(U), F) viewed as an element of the derived bounded category D^{b}(OG^{F}) of the category of finitely generated OG^{F} –modules.

We set RΓ_{c}(Y(U)) := RΓ_{c}(X(U), F_{OL^{F}}) and H_{O}(Y(U)) := H_{O}(X(U), F_{OL^{F}}) . Note that the algebra H_{O}(Y(U)) contains the group algebra OL^{F} as a subalgebra. For K an extension of O, we set H_{K}(X(U), F) := K ⊗_{O} H_{O}(X(U), F) .

The data.

• Let ℓ be a prime number, ℓ ≠ p, which does not divide |Z(G)/Z^{o}(G)| nor |Z(G^{∗})/Z^{o}(G^{∗})|, and which is good for G. We assume that the Sylow ℓ– subgroups of G^{F} are abelian.

• Let O be the ring of integers of a finite unramified extension k of the field of ℓ–adic numbers ℚ_{ℓ}, with residue field k, such that the finite group algebra kG^{F} is split.

• Let A = OG^{F}e be the principal block of OG^{F}. Let S be a Sylow ℓ–subgroup of G^{F}, let L := C_{G}(S), and let f be the principal block idempotent of OL^{F}.

The group L is a rational Levi subgroup of G. We have N_{G^{F}} (S) = N_{G^{F}} (L), and we set W_{G^{F}} (L) := N_{G^{F}} (L)/L^{F}. The group S is a Sylow ℓ–subgroup of Z(L)^{F}, and ℓ does not divide |W_{G^{F}} (L)|.

Conjectures. There exist

• a parabolic subgroup of G with unipotent radical U and Levi complement L,

• a finite complex Υ = (⋯ → Υ^{n−1} → Υ^{n} → Υ^{n+1} → ⋯) of (OG^{F} , OL^{F})– bimodules, which are finitely generated projective as OG^{F} –modules as well as OL^{F} – modules, with the following properties. (C1) Viewed as an object of the category D^{b}(_{OG^{F}} mod_{OL^{F}}), the complex Υ is

isomorphic to RΓ_{c}(Y(U)) . In particular, for each n, the n-th homology group of Υ is isomorphic, as an (OG^{F} , OL^{F})–bimodule, to O ⊗_{ℤ_{ℓ}} H_{c}^{n}(Y(U), ℤ_{ℓ}) .

(C2) The idempotent e acts as the identity on the complex Υ.f, (C3) • the structure of complex of (A, OL^{F} f)–bimodules of Υ.f extends to a struc-

ture of complex of (A, fH_{O}(Y(U))f)–bimodules, all of which are projective as right fH_{O}(Y(U))f–modules,

• the complexes (Υ.f ⊗_{fH_{O}(Y(U))f} f.Υ^{∗}) and A are homotopy equivalent as complexes of (A, A)–bimodules,

• the complexes (f.Υ^{∗}⊗_{OG^{F} e}Υ.f) and fH_{O}(Y(U))f are homotopy equivalent as complexes of (fH_{O}(Y(U))f, fH_{O}(Y(U))f)–bimodules.

(C4) The algebra fH_{O}(Y(U))f is isomorphic to the principal block ON_{G^{F}} (S)f.

5. Stable equivalences of Morita type

5.A. Definition and first remarks.

An example. Some blocks may look very similar without being Morita equivalent nor even Rickard equivalent. This is often the case in the following situation : (p–t.i.) We assume that the Sylow p–subgroups of G are t.i., i.e., for S a Sylow

subgroup, for all g ∈ G, one has S ∩ gSg^{−1} = {1} or S. We set H := N_{G}(S) and we denote respectively by A and B the principal blocks of G and H.

It is easy to see that the functors Ind_{H}^{G} and Res_{H}^{G} induce inverse stable equivalences between _{A}mod and _{B}mod. Such a stable equivalence has certain properties we shall formalize below : it is a “stable equivalence of Morita type”.

The known examples show that, under hypothesis (p–t.i.), if S is non abelian and G is a non abelian simple group, the algebras A and B are not necessarily Morita nor Rickard equivalent, although they have the same numbers of irreducible characters : k(A) = k(B) and l(A) = l(B), according to Alperin’s conjecture ([Al]).

For example, if G = Sz(8) and p = 2, the algebras A and B have non isomorphic centers (G. Cliff, private communication) and their Cartan matrices are not quadratically equivalent ([Br]).

Stable equivalences, Morita and Rickard equivalences. As before, we denote by G and H two finite groups, by e and f two central idempotents of OG and OH respectively, and we set A := OGe, B := OHf. We denote by _{A}stab and _{B}stab the stable categories of A and B respectively. The stable categories have a natural structure of triangulated categories (see for example [Ha]).

Definition. We say that A and B are “stably equivalent à la Morita” (or that there is a stable equivalence of Morita type between A and B) if there exist

• an (A, B)–bimodule M which is projective as an A–module and as a module– B,

• a (B, A)–bimodule N which is projective as a B–module and as a module–A,

• an (A, B)–compatible O–duality between M and N,

such that \{ \begin{matrix}M ⊗_{B} N & is stably equivalent to A in _{A}mod_{A} \\ N ⊗_{A} M & is stably equivalent to B in _{B}mod_{B} . \end{matrix}

The following statement is trivial.

5.1. Proposition. A stable equivalence of Morita type between A and B induces an equivalence of triangulated categories between the stable categories _{A}stab and _{B}stab

It is obvious that if A and B are Morita equivalent, then they are stably equivalent à la Morita. Since the stable category _{A}stab is equivalent to the quotient category D^{b}(A)/D_{perf}^{b}(A) (see [Ri2]), we see that if A and B are Rickard equivalent, then they are stably equivalent. In fact, Rickard proved a more precise result ([Ri3], 5.5)^{1}:

5.2. Proposition. Assume that A and B are Rickard equivalent. Then A and B are stably equivalent à la Morita.

5.B. Stable equivalences of Morita type and block invariants.

Stable triangle invariants. Let us set

R^{st}(KA) := R(KA)/im(^{t}dec^{A}) = “R(KA)/R^{pr}(kA)”
R^{st}(kA) := R(kA)/im(Car^{A}) = “R(kA)/R^{pr}(kA)” .

The triangle T (A) defines by quotient a morphism of abelian groups :

(T^{st}(A))  R^{st}(KA) ⟶^{dec_{st}^{A}} R^{st}(kA)

Since R^{st}(kA) is the Grothendieck group of the stable category of kA (viewed as triangulated category), a stable equivalence of Morita type induces an isomorphism between R^{st}(kA) and R^{st}(kB). More precisely :

5.3. Proposition. A stable equivalence of Morita type between A and B induces (1) an isomorphism between T^{st}(A) and T^{st}(B),

(2) an isometry between ker dec^{A} and ker dec^{B}.

1Strictly speaking, this result concerns group algebras over a field ; but it can easily be extended to our context.

[figure: R^{st}(KB); ^{dec{B \atop st}}; R^{st}(kB); R^{st}(KA); _{dec{A \atop st}}; R^{st}(kA)]

As a consequence, a stable equivalence of Morita type preserves the numerical invariants attached to the “stable triangles” T^{st}. For example :

k(A) − l(A) = k(B) − l(B) , l(A) − l^{pr}(A) = l(B) − l^{pr}(B) , d(A) = d(B) ,

and the Cartan matrices C^{A} and C^{B} have the same non trivial invariant factors.

Stable equivalences of Morita type and centers. We set Z^{st}(A) := Z(A)/Z^{pr}(A) and we call this algebra the stable center of A.

It is not true in general that the stable center of an algebra is the center of its stable category. Nevertheless :

5.4. Proposition. A stable equivalence of Morita type between A and B induces an algebra isomorphism between Z^{st}(A) and Z^{st}(B).

Proof. Let_{A}stab_{A} denote the stable category of A^{en}, and let_{A}stab_{A}^{pr} denote the full subcategory of _{A}stab_{A} whose objects are the (A, A)–bimodules which are projective as A–module and as module-A. Assume that (M, N) induces a stable equivalence of Morita type between A and B. Then the pair (M ⊗_{O} N , N ⊗_{O} M) (where M ⊗_{O} N is

viewed as an (A^{en}, B^{en})–bimodule and N ⊗_{O} M is viewed as a (B^{en}, A^{en})–bimodule,

as in §3 above) induce inverse equivalences between _{A}stab_{A}^{pr} and _{B}stab_{B}^{pr} which exchange A and B. The assertion follows from the fact that Z^{st}(A) is the algebra of endomorphisms of A in _{A}stab_{A}^{pr}. □

Example. Let ΩA denote the kernel of the multiplication map

A ⊗_{O} A → A , a ⊗ a′ ↦ aa′ .

Then the pair of (A, A)–bimodules (ΩA, (ΩA)^{∗}) induces a self stable equivalence of Morita type of A. Let X be an A–module and let π: P → X be a surjective morphism, where P is a projective A–module. Then there is a unique isomorphism ker π →^{∼}

ΩA ⊗_{A} X in _{A}stab.

Remark. It is not known at the moment whether the existence of a stable equivalence of Morita type between A and B implies that k(A) = k(B), and, if so, if there is a bijection between Irr(KA) and Irr(KB) which preserves the defects.

6. Inputting the group action

In all what has been stated so far, A and B might as well have been symmetric algebras over O — the groups themselves did not play an essential role. In what follows, we give brief indications on the actual methods of group representation theory.

6.A. Defect groups and source algebras.

Let Δ: G → G×G be the diagonal morphism. As G×G–module, OG is isomorphic to Ind_{ΔG}^{G×G}O.

6.1. Theorem–Definition. ( [Gre], [Al2], [Pu2]) Let A be a block of OG.

(1) The vertices of the O[G×G]–module A are the G×G–conjugates of ΔD, where D is a p–subgroup of G. The G–conjugates of D are called the defect groups of A.

(2) Let S be an indecomposable summand of Res_{G×D}^{G×G} A with vertex ΔD. Such an S, viewed as an A–module, is unique up to isomorphism, and is a progenerator of _{A}mod. We call source algebra of A the algebra Sce(A) := End_{OG}(S), viewed as endowed with the natural morphism D → Sce(A)^{×}.

Thus in particular a source algebra of A is Morita equivalent to A. But the source algebra contains much more information than the Morita type of A. One can prove^{2} that it contains all the “local information” of the block A, such as the category of subpairs ([AlBr]), the vertices and sources of indecomposable A–modules, and the generalized c-d-triangles (see below). The source algebra may be seen as the “group representation version” of the basic algebra.

6.B. Generalized c-d-triangles.

Definition. For x an element of finite order of a group, we let ζ_{x} ∈ \overline{ℚ} be a root of unity of the same order as x. If A is any ring, we set R_{x}(A) := ℤ[ζ_{x}] ⊗_{ℤ} R(A).

Let G be a finite group. As in [Se], chap. 18, we identify now R(KG), R(kG) and R^{pr}(kG) with various subgroups of the group of O–valued class functions on G.

Let x be a p–element of G. We denote by C_{G}(x) its centralizer in G. The generalized decomposition map dec^{G,x}: R_{x}(KG) → R_{x}(kC_{G}(x)) is defined as follows :

For χ ∈ R_{x}(KG), dec^{G,x}(χ) is the class function on C_{G}(x) defined by

dec^{G,x}(χ)(y) := \{ \begin{matrix}χ(xy) & if y is a p′–element , \\ 0 & if not . \end{matrix}

The generalized c-d-triangle associated with x is

(T (G, x))  [figure: R_{x}(KG); ^{dec^{G,x}}; R_{x}(kC_{G}(x)); R_{x}^{pr}(kC_{G}(x)); _{_{t}dec_{G,x}}; _{Car_{C_{G}(x)}}]

Notice that T (G, 1) = T (G).

The generalized decomposition matrix is the matrix of the map dec^{G,x} on the natural basis Irr(KG) and Irr(kC_{G}(x)).

2see for example [Br4] for a brief account and some bibliographical references of Puig’s work along these lines.

The triangle of a block. To simplify the exposition, we assume from now on that A is the principal block of OG^{3}. For x a p–element of G, we denote by A_{x} the principal block of OC_{G}(x). Then the combination of Brauer’s Second and Third Main Theorems (see for example [Fe]) implies that the image of R_{x}(KA) through dec^{G,x} is contained in R_{x}(kA_{x}), from which one defines the corresponding triangle :

(T (A, x))  [figure: R_{x}(KA); ^{dec^{A,x}}; R_{x}(kA_{x}); R_{x}^{pr}(kA_{x}); _{_{t}dec_{A,x}}; _{Car_{Ax}}]

Of course one has T (A, 1) = T (A).

6.C. Equivalences “with groups”.

Puig equivalences.

Definition. We say that two blocks A and B of two finite groups G and H are Puig equivalent^{4} if, denoting by D (resp. E) a defect group of A (resp. B), there exist a group isomorphism D →^{∼} E and an algebra isomorphism Sce(A) →^{∼} Sce(B) such that the following diagram is commmutative

[figure: OD; ∼; OE; Sce(A); ∼; Sce(B)]

Example. Let us use again the notation introduced in §4.E above : our prime p (the characteristic of our field k := O/𝔭) is now denoted by ℓ, and q denotes a power of another prime p ≠ ℓ.

Let G be a connected reductive algebraic group over 𝔽̄_{q}, endowed with a Frobenius endomorphism F which defines a rational structure on 𝔽_{q}. We assume for simplicity that (G, F) is split. Let W be the Weyl group of G.

Assume that ℓ does not divide |W | and divides (q−1). Then the Sylow ℓ–subgroups of G^{F} are abelian, and the centralizer in G of a Sylow ℓ–subgroup is a Levi subgroup of an F –stable parabolic subgroup of G. Let H be the normalizer in G of a Sylow ℓ–subgroup of G^{F}.

Then ([Pu4]) the principal ℓ–blocks of G^{F} and of H^{F} are Puig equivalent.

6.2. Puig Conjecture. ^{5} Given a finite p–group D, there exists only a finite number of interior D–algebras over O which are the source algebras of some block of some finite group.

The validity of this conjecture would imply in particular that there is only a finite number of Morita types for blocks with a given defect group.

3otherwise we would have to introduce the subpairs and the Brauer elements as in [AlBr].

4Puig says “isomorphic”.

5Stated in the conference on representation of finite group, Oberwolfach, 1982.

Puig equivalences as “equivalences with groups”.

Let us first recall the definition of the “Brauer functor” (see [Br3]). For V an OG–module and P a p–subgroup of G, we set

Br_{P} (V) := V^{P} /( ∑_{Q<P} Tr_{Q}^{P}(V^{Q}) + 𝔭V^{P}) ,

where V^{P} denotes the set of fixed points of V under P, and where Tr_{Q}^{P}(v) := ∑_{x∈[P/Q]} x(v) for Q a subgroup of P and v ∈ V^{Q}. It defines a functor

Br_{P} : _{OG}mod → _{k[N_{G}(P)/P]}mod .

In particular, if V is a permutation P –module, Br_{P} (V) is a permutation (N_{G}(P)/P)– module.

From now on, to simplify the exposition, the following hypothesis will be in force : (A1) G and H are two finite groups with a common Sylow p–subgroup D, and D

is abelian, (A2) N_{G}(D)/C_{G}(D) ≃ N_{H}(D)/C_{H}(D) — note that this implies that the Frobenius

categories 𝔉𝔯_{p}(G) and 𝔉𝔯_{p}(H) (see for example [Br4]) are equivalent. We denote by A and B the principal blocks of OG and OH. Whenever P is a subgroup of D, we denote by A_{P} and B_{P} the principal blocks of OC_{G}(P) and OC_{H}(P).

Assume that A and B are Puig equivalent. Then there exists a family (M_{P} , N_{P}) (P runs over the set of subgroups of D) where, for each P, M_{P} is an (A_{P} , B_{P})–bimodule and N_{P} is a (B_{P} , A_{P})–bimodule such that : (pu1) (M_{P} , N_{P}) induces a Morita equivalence between A_{P} and B_{P}. (pu2) As an O[C_{G}(P) × C_{H}(P)]–module, M_{P} is a summand of Ind_{Δ(D)}^{C_{G}(P)×C_{H}(P)}O,

where O is the trivial D–module. (pu3) k ⊗ M_{P} ≃ Res_{C_{G}(P)×C_{H}(P)}^{N_{G×H}(Δ(P))} Br_{Δ(P)}(M_{{1}}) .

Such a family (M_{P} , N_{P}) induces in particular an isomorphism between all generalized “local” c-d-triangles

T (A_{P} , x) ⟷^{∼} T (B_{P} , x) (for all x ∈ D)

which preserves the canonical basis.

Rickard equivalences with groups. As just seen, a Puig equivalence may be seen as a “Morita equivalence with groups” The preceding formulation of a Puig equivalence allows us to define (still under the hypothesis (A1) and (A2)) what is a “Rickard equivalence with groups”.

We still denote by A and B the principal blocks of OG and OH and, for P a subgroup of D, by A_{P} and B_{P} the principal blocks of OC_{G}(P) and OC_{H}(P).

We say that A and B are ”Rickard equivalent with groups” if there exists a family (M_{P} , N_{P}) (P runs over the set of subgroups of D) where, for each P, M_{P} is a bounded complex of (A_{P} , B_{P})–bimodules and N_{P} is a bounded complex of (B_{P} , A_{P})–bimodules such that :

(ri1) (M_{P} , N_{P}) induces a Rickard equivalence between A_{P} and B_{P}.

(ri2) As O[C_{G}(P) × C_{H}(P)]–modules, M_{P}^{n} is a summand of Ind_{Δ(D)}^{C_{G}(P)×C_{H}(P)}X_{P}^{n},

where X_{P}^{n} is a permutation D–module. (ri3) k ⊗ M_{P} ≃ Res_{C_{G}(P)×C_{H}(P)}^{N_{G×H}(Δ(P))} Br_{Δ(P)}(M_{{1}}) .

Such a family (M_{P} , N_{P}) induces in particular an isometry between all generalized “local” c-d-triangles

T (A_{P} , x) ⟷^{∼} T (B_{P} , x) (for x ∈ D)

corresponding to what is called an “isotypie” in [Br1].

Some unpublished work of J. Rickard shows the relevance of the preceding definition. In particular, complexes with properties (ri2) and (ri3) above occur naturally for finite reductive groups in the context of étale cohomology (see [Ri5]).

On stable equivalences. Let us end with a result which has been often used in applications to structure of finite groups. Consider a slighly more general situation than (A1) and (A2). Now G and H are two finite groups with a common Sylow p–subgroup D. The group D is not necessarily abelian, but we still assume that G and H have “the same fusion” on p–subgroups, i.e., the embedding of D in both G and H defines an equivalence between the Frobenius categories 𝔉𝔯_{p}(G) and 𝔉𝔯_{p}(H).

Let e and f be central idempotents of OG and OH respectively. We set A := OGe and B := OHf. For P a subgroup of D, we set \overline{e}_{P} := Br_{P} (e), \overline{f}_{P} := Br_{P} (f), and \overline{A}_{P} := kC_{G}(P)\overline{e}_{P}, \overline{B}_{P} := kC_{H}(P)\overline{f}_{P}.

Let M be an (A, B)–bimodule and N be a (B, A)–bimodule. For each subgroup P of D, we set \overline{M}_{P} := Br_{Δ(P)}(M) and \overline{N}_{P} := Br_{Δ(P)}(N).

6.3. Theorem. Assume that (st1) M is a summand of Ind_{Δ(D)}^{G×H}X, where X is a permutation D–module. (st2) For each non trivial subgroup P of D, (\overline{M}_{P} , \overline{N}_{P}) induces a Morita equivalence

between \overline{A}_{P} and \overline{B}_{P}. Then (M, N) induces a stable equivalence of Morita type between A and B.

Example. The following situation is a direct generalization of the (p–t.i.)–case mentioned in §5.

6.4. Assume that H is a subgroup of G with index prime to p, and with the following property : (p–s.c.) whenever P is a p–subgroup of H, we have N_{G}(P) = N_{H}(P)O_{p′}C_{G}(P) . Let A and B be the principal blocks of OG and OH respectively, with unity elements e and f. Then the functors e.Ind_{H}^{G} and f. Res_{H}^{G} induce inverse stable equivalences of Morita type between A and B.

The preceding statement has several applications to some “non–simplicity criteria” for finite groups. In this spirit, an important open question is to find a direct and “representation theoretic” proof to the Z_{p}^{∗}–theorem for p odd, which would provide a significant simplification in the classification of finite simple groups.

6.5. Theorem. Let H be a subgroup of G which controls the fusion of p–subgroups in G (i.e., the inclusion of H in G induces an equivalence between the Frobenius categories 𝔉𝔯_{p}(G) and 𝔉𝔯_{p}(H)). Assume that H is the centralizer in G of a p–subgroup of G. Then G = HO_{p′}(G).

For p = 2, the preceding theorem is due to Glauberman ([Gl]). For p odd, it is a consequence of the classification of finite simple groups. An important work of G. Robinson ([Ro1], [Ro2]) makes plausible to find a direct proof using representation theory.

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L.M.E.N.S.–D.M.I. (C.N.R.S., U.A. 762), 45 rue d’Ulm, F–75005 Paris, France E-mail address: broue@dmi.ens.fr