Equivalences between blocks of p-local Mackey algebras
aa r X i v : . [ m a t h . R T ] J un Equivalences between blocks of p -local Mackey algebras. Baptiste RognerudAugust 18, 2018
Abstract
Let G be a finite group and ( K, O , k ) be a p -modular system. Let R = O or k . There is a bijection between the blocks of the group algebra and the blocks ofthe so-called p -local Mackey algebra µ R ( G ). Let b be a block of RG with abeliandefect group D . Let b ′ be its Brauer correspondant in N G ( D ). It is conjectured byBrou´e that the blocks RGb and RN G ( D ) b ′ are derived equivalent. Here we look atequivalences between the corresponding blocks of p -local Mackey algebras. We provethat an analogue of the Brou´e’s conjecture is true for the p -local Mackey algebrasin the following cases: for the principal blocks of p -nilpotent groups and for blockswith defect 1. We also point out the probable importance of splendid equivalencesfor the Mackey algebras. Key words: Modular representation. Finite group. Mackey functor. Block theory.A.M.S. subject classification: 20C05, 18E30,16G10,16W99.
Let G be a finite group and ( K, O , k ) be a p -modular system. Let R = O or k . In theirpaper, Th´evenaz and Webb proved that there is a bijection b b µ between the blocks of RG and the primitive central idempotents of µ R ( G ), called the blocks of µ R ( G ), where µ R ( G ) is the so called p -local Mackey algebra.This bijection preserves the defect groups (which are the same as the defect groups of thecorresponding block of RG , see Section 5 . b be a block of RG with defect group D and b ′ bethe Brauer correspondent of b in RN G ( D ). b ∈ Z ( RG ) / / (cid:15) (cid:15) b µ ∈ Z ( µ R ( G )) (cid:15) (cid:15) b ′ ∈ Z ( RN G ( D )) / / b ′ µ ∈ Z ( µ R ( N G ( D ))) . If D is abelian, it is conjectured by Brou´e that the block algebras RGb and RN G ( D ) b ′ are derived equivalent. In [21], the Author proved that if two blocks of group algebras1re splendidly derived equivalent, then the corresponding blocks of the so-called coho-mological Mackey algebras are derived equivalent. The cohomological Mackey algebrais a quotient of the p -local Mackey algebra. So, it is a very natural question to ask ifthe same can happen for the corresponding p -local Mackey algebras which contain muchmore informations than their cohomological quotient (see Proposition 3.5 e.g.). Question 1.1 (Bouc) . Let G be a finite group and b be a block of O G with abelian defectgroup D . Let b ′ be its Brauer correspondent in O N G ( D ) . Is there a derived equivalence D b ( µ O ( G ) b µ ) ∼ = D b ( µ O ( N G ( D )) b ′ µ ) ? Unfortunately, the tools which was developed for the cohomological Mackey algebrascannot be used here. Moreover Question 1.1 is much harder than its analogue for thecohomological Mackey algebras and is connected to deep questions of representation offinite groups such as the knowledge of the indecomposable p -permutation modules (seeSection 6 and 7 of [4]). Here, we will not answer this question in general, but we considerit in the following two cases: first for the Mackey algebra of the principal block of p -nilpotent groups, and then for groups with Sylow p -subgroups of order p .The main result of this paper is the following theorem: Theorem 1.2.
The answer to Question 1.1 is affirmative in the following cases: • For principal blocks of p -nilpotent groups. Here we do not assume the Sylow p -subgroups to be abelian. Moreover it is a Morita equivalence. • When the p -local Mackey algebra is symmetric. That is for the blocks with defect . For a non principal block of a p -nilpotent group, or more generally for a nilpotentblock it is not expect for the corresponding block of the Mackey algebra to be Moritaequivalent to the Mackey algebra of its defect group.Let b and c be two blocks of group algebras. Then it is probable that the correspondingblocks of p -local Mackey algebras will be equivalent only if there is an equivalence be-tween b and c which respect the p -permutation modules ( splendid equivalences e.g.). Wedo not have a proof of this fact, but there are some clues in this direction. Example 4.9is exactly an example of blocks which are Morita equivalent but not ‘splendidly’ Moritaequivalent. In this case, the corresponding blocks of p -local Mackey algebras are notMorita equivalent but derived equivalent. Other example can be found in Chapter 4 of[19]. Moreover the decomposition matrix of a block of p -local Mackey algebras involves p -permutation modules and Brauer quotient (see Proposition 3.5).In the first part, we recall some basic definitions and results on Mackey functors. Sincewe will use several points of view on Mackey functors, we give a full proof of the well-known fact that all these points of view are equivalent. Then we recall the bijectionbetween the blocks of the group algebra and the blocks of the p -local Mackey algebra.At the end of the second section, we explain how the decomposition matrix of the p -localMackey algebra of a finite group G can be computed from the knowledge of the ordi-nary characters of some p -local subgroups of G and the indecomposable p -permutation2 G -modules.In the last part, we investigate on some basic properties of equivalences between blocksof p -local Mackey algebras. We show that naive thoughts about these questions areseldom true. For example an equivalence between blocks of p -local Mackey algebrasdo not need to induce an equivalence for the cohomological quotient. If two blocks ofgroup algebras are isomorphic (resp. Morita equivalent), then the corresponding blocksof p -local Mackey algebras do not need to be isomorphic (resp. Mortia equivalent).We show that there is a Morita equivalence between the principal block of the p -localMackey algebra of a p -nilpotent group G and the Mackey algebra of a Sylow p -subgroupof G . Moreover, we determine the structure of algebra of the block of the p -local Mackeyalgebra in this case. This can be view as an analogue of Puig’s theorem (see [16] or [15])for the Mackey algebras.Then we will answer Question 1 . p . The resultuses the fact that the Mackey algebras are symmetric algebras in this situation. Moreprecisely over the field k , they are Brauer tree algebras and over the valuation ring O they are Green orders .It should be notice that all the equivalences between blocks of groups algebras whichinduce equivalences between the corresponding p -local Mackey algebras that we wereable to construct are in fact splendid . Notation:
Let R be a commutative ring with unit. We denote by R - M od the categoryof (all) R -modules.Let G be a finite group. Let p be a prime number. We denote by ( K, O , k ) a p -modularsystem, i-e O is a complete discrete valuation ring with maximal ideal p , such that O / p = k is a field of characteristic p and F rac ( O ) = K is a field of characteristic zero.If R = O or k , then the direct summands of the permutation RG -modules are called p - permutation modules. We denote by G - set the category of finite G -sets. If H is asubgroup of G then, we denote by N G ( H ) its normalizer in G . The quotient N G ( H ) /H will be, sometimes, denoted by N G ( H ). We denote by Ω G the disjoint union of alltransitive G -sets.Let V be an RG -module, for R = O or k . Let Q be a p -subgroup of G . Since we aredealing with Mackey functor we use the notation V [ Q ] instead of the usual V ( Q ) for theBrauer quotient of V (see Section 1 [7]).If F : A → B and G : B → A are two functors, we denote by F ⊣ G the fact that F is aleft adjoint of G .N.B. We will denote by the same letter the block idempotents for the ring O and thefield k . Let G be a finite group, and R be a commutative ring. There are several definitions ofMackey functors for G over a ring R , the first one was introduced by Green in [11]: Definition 2.1.
A Mackey functor for G over R consists of the following data:3 For every subgroup H of G , an R -module denoted by M ( H ). • For subgroups H ⊆ K of G , a morphism of R -modules t KH : M ( H ) → M ( K ) called transfer, or induction, and a morphism of R -modules r KH : M ( K ) → M ( H ) called restriction. • For every subgroup H of G , and each element x of G , a morphism of R -modules c x,H : M ( H ) → M ( x H ) called conjugacy map.such that:1. Triviality axiom : For each subgroup H of G , and each element h ∈ H , the mor-phisms r HH , t HH et c h,H are the identity morphism of M ( H ).2. Transitivity axiom : If H ⊆ K ⊆ L are subgroups of G , then t LK ◦ t KH = t LH and r KH ◦ r LK = r LH . Moreover if x and y are elements of G , then c y, x H ◦ c x,H = c yx,H .3. Compatibility axioms: If H ⊆ K are subgroups of G , and if x is an element of G ,then c x,K ◦ t KH = t x K x H ◦ c x,H et c x,H ◦ r KH = r x K x H ◦ c x,K .4. Mackey axiom: If H ⊆ K ⊇ L are subgroups of G , then r KH ◦ t KL = X x ∈ [ H \ K/L ] t HH ∩ x L ◦ c x,H x ∩ L ◦ r LH x ∩ L . where [ H \ K/L ] is a set of representatives of the double cosets H \ K/L .In particular, for each subgroup H of G , the R -module M ( H ) is an N G ( H ) /H -module.A morphism f between two Mackey functors M and N is the data of a R -linear morphism f ( H ) : M ( H ) → N ( H ) for every subgroup H of G . These morphisms are compatiblewith transfer, restriction and conjugacy maps. We denote by M ack R ( G ) the categoryof Mackey functors for G over R . Example . Let V be an RG -module. The fixed point functor F P V is the Mackeyfunctor for G over R defined as follows:For H G , then F P V ( H ) = V H := { v ∈ V ; h · v = v ∀ h ∈ H } . If H K G ,we have V K ⊆ V H , so we define the restriction map r KH as the inclusion map. Thetransfer map t KH : V H → V K is defined by t KH ( v ) = P k ∈ [ K/H ] k · v where [ K/H ] is a setof representatives of
K/H . The conjugacy maps are induced by the action of G on V .It is not hard to see that the construction V F P V is a functor from RG - M od to M ack R ( G ).Conversely we have an obvious functor ev : M ack R ( G ) → RG - M od given by the eval-uation at the subgroup { } . Proposition 2.3. [22] The functor ev is a left adjoint to the functor V F P V . I.e.there is a natural isomorphism: Hom
Mack R ( G ) ( M, F P V ) ∼ = Hom RG ( M (1) , V ) for a Mackey functor M and an RG -module V . emark . The full subcategory of
M ack R ( G ) consisting of fixed point functors F P V is equivalent to the category RG - M od (see proof of Theorem 18.1 [22]).An other definition of Mackey functors was given by Dress in [10]:
Definition 2.5.
A bivariant functor M = ( M ∗ , M ∗ ) from G - set to R - M od is a pairof functors from G - set to R - M od such that M ∗ is a contravariant functor, and M ∗ isa covariant functor. If X is a finite G -set, then the image by the covariant and by thecontravariant part coincide. We denote by M ( X ) this image. A Mackey functor for G over R is a bivariant functor from G - set to R - M od such that: • Let X and Y be two finite G -sets, i X and i Y the canonical injection of X (resp. Y )in X ⊔ Y . Then M ∗ ( i X ) ⊕ M ∗ ( i Y ) and ( M ∗ ( i X ) , M ∗ ( i Y )) are inverse isomorphisms,from M ( X ) ⊕ M ( Y ) to M ( X ⊔ Y ). • If X a / / b (cid:15) (cid:15) Y c (cid:15) (cid:15) Z d / / T is a pull back diagram of G -sets, then the diagram M ( X ) M ∗ ( b ) (cid:15) (cid:15) M ( Y ) M ∗ ( a ) o o M ∗ ( c ) (cid:15) (cid:15) M ( Z ) M ( T ) M ∗ ( d ) o o is commutative.A morphism between two Mackey functors is a natural transformation of bivariant func-tors. Example . If X is a finite G -set, the category of G -sets over X is thecategory with objects ( Y, φ ) where Y is a finite G -set and φ is a morphism from Y to X . A morphism f from ( Y, φ ) to (
Z, ψ ) is a morphism of G -sets f : Y → Z such that ψ ◦ f = φ .The Burnside functor at X , written B ( X ), is the Grothendieck group of the categoryof G -sets over X , for relations given by disjoint union. This is a Mackey functor for G over R by extending scalars from Z to R . We will denote RB the Burnside functor afterscalars extension.If X is a G -set, the Burnside group RB ( X ) has a ring structure. A G -set Z over X × X is the data of a G -set Z and a map ( b × a ) from Z to X × X , denoted by ( X b ← Y a → X ).The product of (the isomorphism class of ) ( X α ← Y β → X ) and (the isomorphism class5f )( X γ ← Z δ → X ) is given by (the isomorphism class of) the pullback along β and γ . P ~ ~ Y α ~ ~ ⑥⑥⑥⑥⑥⑥⑥ β ❆❆❆❆❆❆❆ Z γ ~ ~ ⑦⑦⑦⑦⑦⑦⑦⑦ δ ❅❅❅❅❅❅❅❅ X X X
The identity of this ring is (the isomorphism class) X ❆❆❆❆❆❆❆❆ ❆❆❆❆❆❆❆❆⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥ X X
In the rest of the paper, we will use the same notation for a G -set over X × X and itsisomorphism class in RB ( X × X ).We need a last definition of Mackey functors which was given by Th´evenaz and Webbin [22], and uses the Mackey algebra. Definition 2.7.
The Mackey algebra µ R ( G ) for G over R is the unital associative algebrawith generators t KH , r KH and c g,H for H K G and g ∈ G , with the following relations: • P H G t HH = 1 µ R ( G ) . • t HH = r HH = c h,H for H G and h ∈ H . • t LK t KH = t LH , r KH r LK = r LH for H ⊆ K ⊆ L . • c g ′ , g H c g,H = c g ′ g,H , for H G and g, g ′ ∈ G . • t g K g H c g,H = c g,K t KH and r g K g H c g,K = c g,H r KH , H K , g ∈ G . • r HL t HK = P h ∈ [ L \ H/K ] t LL ∩ h K c h,L h ∩ K r KL h ∩ K for L H > K . • All the other products of generators are zero.
Definition 2.8.
A Mackey functor for G over R is a left µ R ( G )-module. Proposition 2.9.
The Mackey algebra is a free R -module, of finite rank independent of R . The set of elements t HK xr LK x , where H and L are subgroups of G , where x ∈ [ H \ G/L ] ,and K is a subgroup of H ∩ x L up to ( H ∩ x L ) -conjugacy, is an R -basis of µ R ( G ) .Proof. Section 3 of [22].Let us recall that the Mackey algebra is isomorphic to a Burnside algebra:
Proposition 2.10 (Proposition 4.5.1 [3]) . The Mackey algebra µ R ( G ) is isomorphic to RB (Ω G ) , where Ω G = ⊔ L G G/L . roof. Let H K be two subgroups of G , then we denote by π KH the natural surjectionfrom G/H to G/K . If g ∈ G , then we denote by γ H,g the map from G/ g H to G/H defined by γ H,g ( xgHg − ) = xgH . The isomorphism β is defined on the generators of µ R ( G ) by: β ( t KH ) = G/H π KH y y rrrrrrrrrr ▲▲▲▲▲▲▲▲▲▲ ▲▲▲▲▲▲▲▲▲▲ Ω G ⊃ G/K G/H ⊂ Ω G β ( r KH ) = G/H π KH & & ▲▲▲▲▲▲▲▲▲▲rrrrrrrrrrrrrrrrrrrr Ω G ⊃ G/H G/K ⊂ Ω G β ( c g,H ) = G/ g H γ H,g & & ▼▼▼▼▼▼▼▼▼▼♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣ Ω G ⊃ G/ g H G/H ⊂ Ω G We will make an intensive use of the connection between the different categories ofMackey functors, so let us recall the following well known result:
Proposition 2.11. [22] The different definitions of Mackey functors for G over R areequivalent.Sketch of proof. The idea of the proof can be found in various places such as [22], or[3]. Here we propose a quite different, and more conceptual approach where we do notneed to choose representatives. For this proof, we denote by
M ack ( a ) R ( G ) the categoryof Mackey functors for G over R in the sense of Green and by M ack ( b ) R ( G ) the categoryof Mackey functors for Dress. The most cost-effective way seems to prove the followingequivalences: M ack ( a ) R ( G ) ∼ = ' ' ❖❖❖❖❖❖❖❖❖❖❖❖ M ack ( b ) R ( G ) ∼ = w w ♥♥♥♥♥♥♥♥♥♥♥♥ µ R ( G )- M od ∼ = / / RB (Ω G )- M od
For the equivalence
M ack aR ( G ) ∼ = µ R ( G )- M od , if M ∈ M ack ( a ) R ( G ), then L H G M ( H )is a µ R ( G )-module. Conversely, if V is a µ R ( G )-module, then one can define a Mackeyfunctor M by: for H G , then M ( H ) := t HH V . The Mackey functor structure of M isinduced by the module structure of V .The equivalence between µ R ( G )- M od and RB (Ω G )- M od follows from the isomorphismof Proposition 2.10. 7he equivalence
M ack ( b ) R ( G ) ∼ = RB (Ω G )- M od , is a bit more technical. Let M be aMackey functor for G over R , using Dress definition. Then M (Ω G ) is a RB (Ω G )-module.The action of a G -set Z = (Ω G b ← Z a → Ω G ) over Ω G × Ω G on M (Ω G ) is defined by: let m ∈ M (Ω G ), Z · m = M ∗ ( b ) ◦ M ∗ ( a )( m ) . Let V be an RB (Ω G )-module, let X be a finite G -set. Then RB X = RB (Ω G × X ) is an RB (Ω G )-module. The action of RB (Ω G ) on RB X is given by left multiplication. Let f : X → Y be a morphism between two finite G -sets. Let us denote by ˆ f the morphism Id Ω G × f . Let φ : Z → X × Ω G be a G -set over X × Ω G and let ψ : W → Y × Ω G be a G -setover Y × Ω G . Then there is a morphism of RB (Ω G )-modules: RB ( f ) ∗ : RB X → RB Y defined by: RB ∗ ( f )( φ ) = φ ◦ ˆ f . On the other direction, there is a morphism of RB (Ω G )-modules defined by: RB ∗ ( f )( ψ ) = π, where π : P → X × Ω G is a G -set over X × Ω G such that the following diagram is apullback diagram: P / / π (cid:15) (cid:15) W ψ (cid:15) (cid:15) X × Ω G ˆ f / / Y × Ω G then one can defined a Mackey functor Y V for G over R by: Y V ( X ) = Hom RB (Ω G ) ( RB X , V ) . The Mackey functor structure of Y V is induced by the covariant and the contravariantpart of RB .The functor from M ack ( b ) R ( G ) to RB (Ω G )- M od sending M to M (Ω G ) is fully faithful,indeed, if X is a finite G -set, then let X = F H G n H G/H be the decomposition of X assum of transitive G -sets. If a transitive G -set G/H appears with a positive coefficient in X , then we denote by i XH the canonical injection from G/H to X . Moreover, we denoteby i G/H the canonical injection from
G/H to Ω G . Let φ : M → N be a morphismof Mackey functors, then since φ is a natural transformation of bivariant functors, andsince P H n H M ∗ ( i XH ) M ∗ ( i XH ) = Id M ( X ) , we have: φ X = X H G n H N ∗ ( i XH ) φ G/H M ∗ ( i XH )= X H G n H N ∗ ( i XH ) N ∗ ( i G/H ) φ Ω G M ∗ ( i G/H ) M ∗ ( i XH ) . So φ is characterized by its evaluation at Ω G . Conversely with this formula, if we have f : M (Ω G ) → N (Ω G ) a morphism of RB (Ω G )-modules, then one can defined a morphism8f Mackey functors φ : M → N such that φ Ω G = f . The fact that this construction givesa natural transformation of bivariant functors follows from the fact that f is a morphismof RB (Ω G )-modules.Moreover the functor M M (Ω G ) is dense. Indeed let V be an RB (Ω G )-module, then Y V (Ω G ) = Hom RB (Ω G ) ( RB Ω G , V )= Hom RB (Ω G ) ( RB (Ω G ) , V ) ∼ = V. Now, this isomorphism is natural in V . So, the categories M ack ( b ) R ( G ) and RB (Ω G )- M od are equivalents, and since Y V (Ω G ) ∼ = V , by unicity of the adjoint, the functor V Y V is a quasi-inverse equivalence to the functor M M (Ω G ).In the rest of the paper, if no confusion is possible, we denote by M ack R ( G ) thecategory of Mackey functors for G over R for one of these three definitions. Let G be a finite group and ( K, O , k ) be a p -modular system for G which is “large enough”for all the N G ( H ) /H for H G . In [22], Th´evenaz and Webb proved that there is abijection between the blocks of the group algebra O G and the blocks of M ack O ( G, M ack O ( G,
1) is the full subcategory of
M ack O ( G ) consisting of Mackey functorswhich are projective relatively to the p -subgroups of G .The category M ack O ( G,
1) is equivalent to the category of µ O ( G )-modules where µ O ( G ) is the subalgebra of µ O ( G ) generated by the r HQ , t HQ , c Q,x where Q H G , x ∈ G and Q is a p -group. This subalgebra is called the p -local Mackey algebra of G over O . The same definitions hold for K or k . Theorem 3.1.
The set of central primitive idempotents of the p -local Mackey algebra µ O ( G ) is in bijection with the set of the blocks of O G .Remark . The bijection is moreover explicit. If b is a block of O G then Bouc gavean explicit formula for the corresponding central idempotent of µ O ( G ) denoted by b µ (Theorem 4.5.2 [5]).Using the equivalence of categories M ack O ( G, ∼ = µ O ( G )- M od , we have a decom-position of
M ack O ( G,
1) into a product of categories, which were called the blocks of
M ack O ( G,
1) by Th´evenaz and Webb in [22] Section 17. The formula of the idempotentsof µ R ( G ) (Remark 3.2) is rather technical but it is immediate to check that the actionof a block idempotent b µ on the evaluation at { } of a Mackey functor M (using Green’snotation for evaluation) is given as follows: let m ∈ M (1), writing the idempotent b = P x ∈ G b ( x ) x where b ( x ) ∈ O , we have b µ · m = X x ∈ G b ( x ) c x, ( m ) . (1)9 roposition 3.3. Let R = k or O . The set of isomorphism classes of projective in-decomposable Mackey functors in a block b µ of M ack R ( G, is in bijection with the setof isomorphism classes of indecomposable p -permutation modules contained in the block RGb .Proof.
We use Green’s notation for the evaluation of Mackey functors. By Corollary 12.8of [22], we know that the projective indecomposable Mackey functors of
M ack k ( G,
1) arein bijection with the indecomposable p -permutation kG -modules: if P is an indecompos-able projective Mackey functor, then P (1) is an indecomposable p -permutation module.Let Q be an other indecomposable projective Mackey functor. Then P ∼ = Q if and onlyif P (1) ∼ = Q (1). The same holds for the projective Mackey functors of M ack O ( G, P is in the block b µ if and only if b µ · P = 0,but b µ · P is projective so: b µ · P = 0 ⇔ ( b µ · P )(1) = 0 ⇔ b · ( P (1)) = 0 ⇔ P (1) is in the block b of RG .We will use the following notation: M ack R ( b ) (resp. µ R ( b )) for the category ofMackey functors which belong to the block b µ (resp. the algebra b µ µ R ( G )) for R = O or k . If H is a subgroup of G , then there is an induction functor from M ack R ( H ) to M ack R ( G )denoted by Ind GH . There is also a restriction functor from M ack R ( G ) to M ack R ( H )denoted by Res GH .If N is a normal subgroup of G , there is an inflation functor Inf
GG/N from
M ack R ( G/N )to
M ack R ( G ). For the definition of these three functors using Green’s point of view seeSection 4 and Section 5 of [23]. These three functors, using the Dress point of view,arrive from a more general setting, using the fact that a G - H -biset produces a functor,by ‘pre-composition’, from M ack R ( H ) to M ack R ( G ) (see Chapter 8 [3] for more details).Let D be a finite G -set. The Dress construction (see [10] or 1 . D is an endo-functor of the Mackey functors category given by the pre-composition by the cartesianproduct with D : if M is a Mackey functor for G in the sense of Dress, and if X is afinite G -set, then the Dress construction of M , denoted by M D is: M D ( X ) = M ( X × D ) . Let R be a commutative ring. Let Q be a p -subgroup of G . The Brauer constructionfor Mackey functors is a functor M ack R ( G ) → M ack R ( N G ( Q )) denoted by M M Q .10f M ∈ M ack R ( G ), then for N/Q a subgroup of N G ( Q ), M Q ( N/Q ) = M ( N ) / X Q ≮ R 1. The functor M M Q is left adjoint to the functor Ind GN G ( Q ) Inf N G ( Q ) N G ( Q ) : M ack R ( N G ( Q )) → M ack R ( G ) .2. The functor M M Q sends projective functors to projective functors.3. Let H be a subgroup of G . Then ( Ind GH ( M )) Q = 0 if Q is not conjugate to asubgroup of H .Let ( K, O , k ) be a p -modular system, and R = O or k .4. If M ∈ M ack R ( G, , then M Q ∈ M ack R ( N G ( Q ) , .5. Let P be a projective Mackey functor of M ack k ( G, and L be a projective func-tor of M ack O ( G, which lifts P . Then L Q is a projective Mackey functor of M ack O ( G, which lifts P Q .6. Let P be a projective Mackey functor of M ack k ( G, . Then P Q ( Q/Q ) ∼ = P (1)[ Q ] .7. Let P be an indecomposable projective Mackey functor of M ack R ( G, . Then thevertices of P are the maximal p -subgroups Q of G such that P Q = 0 .Sketch of proof. 1. Theorem 5.1 of [23] with a different notation.2. Since M M Q is left adjoint to an exact functor, it sends projective objects toprojective objects.3. By successive adjunction: for L ∈ M ack R ( N G ( Q )) and M ∈ M ack R ( H ), andusing the Mackey formula for Mackey functors, we have: Hom Mack R ( N G ( Q )) (( Ind GH M ) Q , L ) ∼ = Hom Mack R ( H ) ( M, Res GH Ind GN G ( Q ) Inf N G ( Q ) N G ( Q ) L ) ∼ = M g ∈ [ H \ G/N G ( Q )] Hom Mack R ( H ) (cid:18) M, Ind HH ∩ g N G ( Q ) Iso ( g ) Res N G ( Q ) N G ( Q ) ∩ H g Inf N G ( Q ) N G ( Q ) L (cid:19) . Iso ( g ) the functor from the category M ack R ( N G ( Q ) ∩ H g ) tothe category M ack R ( H ∩ g N G ( Q )) induced by the conjugacy by g . The resultnow follows from the fact that Res N G ( Q ) N G ( Q ) ∩ H g Inf N G ( Q ) N G ( Q ) L = 0 if Q is not a subgroupof H .4. A Mackey functor M is in M ack k ( G, 1) if and only if there exist a p -group P anda Mackey functor N for P such that M | Ind GP ( N ). Moreover, it is not difficult tocheck that ( Ind GP N ) Q is a direct sum of Mackey functors induced from subgroupsof conjugates of P ( ⋆ ), so M Q ∈ M ack k ( N G ( Q ) , ⋆ ) In order to prove this, one may continue the proof of the point 3 (or, in a moregeneral setting, look at Lemme 3 of [2]).5. Let M be a Mackey functor for N G ( Q ). Then using successive adjunctions, wehave: Hom Mack k ( N G ( Q )) ( L Q / p ( L Q ) , M ) ∼ = Hom Mack k ( N G ( Q )) ( k ⊗ O L Q , M ) ∼ = Hom Mack O ( N G ( Q )) ( L Q , Hom k ( k, M )) ∼ = Hom Mack O ( G ) ( L, Ind GN G ( Q ) Inf N G ( Q ) N G ( Q ) Hom k ( k, M )) . However, Ind GN G ( Q ) Inf N G ( Q ) N G ( Q ) Hom k ( k, M ) ∼ = Hom k ( k, Ind GN G ( Q ) Inf N G ( Q ) N G ( Q ) ( M )), so Hom Mack k ( N G ( Q )) ( L Q / p ( L Q ) , M ) ∼ = Hom Mack O ( G ) ( L, Hom k ( k, Ind GN G ( Q ) Inf N G ( Q ) N G ( Q ) ( M ))) ∼ = Hom Mack k ( G ) ( L/ p L, Ind GN G ( Q ) Inf N G ( Q ) N G ( Q ) ( M )) ∼ = Hom Mack k ( N G ( Q )) (( L/ p L ) Q , M ) . 6. Lemme 5.10 of [4].7. This is the first assertion of Theorem 3 . . p -local projective Mackey functor is non zero if and onlyif its evaluation at 1 is non zero.Let F be a field and let G be a finite group. By Theorem 8 . G over the field F is inbijection with the set of pairs ( H, V ) where H runs through a set of representativesof conjugacy classes of subgroups of G and V through a set of isomorphism classes of F N G ( H )-simple modules. We denote by S H,V the simple Mackey functor correspondingto the pair ( H, V ). Let us recall that: S H,V = Ind GN G ( H ) Inf N G ( H ) N G ( H ) S N G ( H )1 ,V , S G ,V ( K ) = Im ( t K : V → V K ) . Here t K is the relative trace map, that is t K ( v ) = P k ∈ K k · v , for v ∈ V .If ( K, O , k ) is a p -modular system, then a simple Mackey functor S H,V over k is in M ack k ( G, 1) if and only if H is a p -group (see the discussion after Proposition 9 . P H,V the projective cover of S H,V . Proposition 3.5 (Decomposition matrix of µ O ( G )) . Let G be a finite group, and ( K, O , k ) be a p -modular system which is “large enough” for the groups N G ( L ) where L runs through the p -subgroups of G .The decomposition matrix of µ O ( G ) has rows indexed by the isomorphism classes ofindecomposable p -permutation kG -modules, the columns are indexed by the ordinary ir-reducible characters of all the N G ( L ) where L runs through the p -subgroups of G up toconjugacy.Let χ be an ordinary character of the group N G ( L ) and let W be an indecomposable p -permutation kG -module. Then the decomposition number d χ,W is equal to d χ,W = dim K Hom KN L ( Q ) ( K ⊗ O [ W [ L ] , K χ ) , where [ W [ L ] is the (unique) p -permutation O G -module which lifts W [ L ] and K χ is thesimple KG -module afforded by the character χ .Proof. Since ( K, O , k ) is a splitting system for µ O ( G ), and since µ K ( G ) is a semi-simplealgebra, the Cartan matrix is symmetric (Proposition 1 . . µ k ( G )-modules, and the columns areindexed by the simples µ K ( G )-modules. Instead of working with the p -local Mackey al-gebras, we use the Mackey functors point of view. Since the indecomposable projectiveMackey functors of M ack k ( G, 1) are in bijection with the indecomposable p -permutation kG -modules, we can index the rows of this matrix by the (isomorphism classes of) in-decomposable p -permutation modules. The bijection send a p -local projective Mackeyfunctor P to its evaluation P (1) (with Green’s notation for the evaluation). Moreover,the (isomorphism classes of) simple Mackey functors of M ack K ( G, 1) are in bijectionwith the pairs ( L, V ), where L runs through the conjugacy classes of p -subgroups of G and V runs through the isomorphism classes of simple KN G ( L ) /L -modules. So we canindexed the columns of the matrix by the set of ordinary characters of N G ( L ) /L when L runs through the conjugacy classes of p -subgroups of G .Let L be a p -subgroup of G and χ be an ordinary character of KN G ( L ). The cor-responding simple Mackey functor for G over K , denoted by S L,K χ is isomorphic to Ind GN G ( L ) Inf N G ( L ) N G ( L ) F P K χ (see Lemma 9 . K χ is the simple KN G ( L )-moduleafforded by the character χ .Let H be a p -subgroup of G and V be a simple kN G ( H )-module. We denote by P H,V the corresponding projective indecomposable Mackey functor for G over k , and by [ P H,V G over O which lifts P H,V .We denote by M a Mackey functor for G over O such that M is O -free and K ⊗ O M ∼ = S L,K χ . Then, the decomposition number indexed by S L,K χ and P H,V is: d S L,Kχ ,P H,V = dim k Hom Mack k ( G, ( P H,V , k ⊗ O M )= rank O Hom Mack O ( G, ( [ P H,V , M )= dim K Hom Mack K ( G, ( K ⊗ O [ P H,V , S L,K χ )= dim K Hom Mack K ( G, ( K ⊗ O [ P H,V , Ind GN G ( L ) Inf N G ( L ) N G ( L ) F P K χ )= dim K Hom Mack K ( N G ( L ) /L, ( (cid:0) K ⊗ O [ P H,V (cid:1) L , F P K χ )= dim K Hom KN G ( L ) /L ( (cid:0) K ⊗ O [ P H,V (cid:1) L ( L/L ) , K χ ) . The two last equalities come from the fact that the two following pairs are pairs ofadjoint functors: ev ⊣ F P − and ( − ) L ⊣ ind GN G ( L ) Inf N G ( L ) N G ( L ) .Moreover, we have ( K ⊗ O [ P H,V ) L ( L/L ) ∼ = K ⊗ O (( [ P H,V ) L ( L/L )). By Lemma 3.4,the O N G ( L )-module ( [ P H,V ) L ( L/L ) is the unique (up to isomorphism) p -permutationmodule which lifts ( P H,V ) L ( L/L ) ∼ = P H,V (1)[ L ], so: d S L,Kχ ,P H,V = dim K Hom KN L ( Q ) ( K ⊗ O [ W [ L ] , K χ ) , where W is the indecomposable p -permutation kG -module P H,V (1). Remark . By Section 4.4 of [6], the sub-matrix indexed by the ordinary characters of G , and the (isomorphism classes of) indecomposable p -permutation kG -modules is thedecomposition matrix of the cohomological Mackey algebra coµ O ( G ). The sub-matrixindexed by the ordinary characters of G and the isomorphism classes of indecomposableprojective kG -modules is the decomposition matrix of O G . p -local Mackey algebras. In this section we give some examples of equivalences between blocks of p -local Mackeyalgebras. We look at the case of p -nilpotent groups. We prove that the p -local algebraof the principal block of such a group is Morita equivalent to the Mackey algebra of itsSylow p -subgroup. But as in [17] the case of the non-principal block seems unexpectedlymuch more difficult. We give an example which proves that in general there is no Moritaequivalence between the p -local Mackey algebra of the block and the Mackey algebra ofthe defect. Then we look at the case of a finite group with Sylow p -subgroups of order p .In this case the p -local Mackey algebras are symmetric algebras, they are more preciselyBrauer tree algebras, so we can use the tools which were developed for Brou´e’s conjecturefor blocks with cyclic Sylow p -subgroups. 14 .1 Basic results. Since the p -local Mackey algebra and the cohomological Mackey algebra share a lot ofproperties, for example, they have the same number of simple modules in each block andthe projective cohomological Mackey functors are the biggest cohomological quotients ofthe p -local projective Mackey functors, one may ask if an equivalence between blocks of p -local Mackey algebras induces in some sense, an equivalence between the correspondingblocks of the cohomological Mackey algebras. Such a result would allow us to use [21].The following example shows that the situation is unfortunately not that simple. Example . Let R be a commutative ring and G = C be the (cyclic) group of order 2.Then a basis of µ R ( C ) is given by: t C C , t C r C , t C , r C , t and t x where x ∈ C and t x means t c ,x . Then, there is an automorphism φ of µ k ( C ) where φ is defined on the basiselements by: φ ( t C C ) = t , φ ( t C r C ) = t + t x , φ ( t C ) = r C , φ ( r C ) = t C , φ ( t ) = t C C and φ ( t x ) = t C r C − t C C . This gives an unitary automorphism of µ R ( C ). By generalresults of Morita theory, the bimodule µ R ( C ) φ induces a Morita self-equivalence of µ R ( C ). Using the usual equivalence of categories between µ R ( C )- M od and M ack R ( C )(in the sense of Green), it is not hard to check that this Morita self-equivalence inducesan self-equivalence F of M ack R ( C ) which has the following property. Let M be aMackey functor for C . Then F ( M )(1) = M ( C ) and F ( M )( C ) = M (1). The maps t C and r C are exchanged. It is now easy to check that this functor F does not preservethe cohomological structure.If R = k is a field of characteristic 2, then the functor F exchanges the simples S ,k and S C ,k , so it exchanges the indecomposable projective Mackey functors P ,k and P C ,k .Moreover P ,k is in the full subcategory of M ack R ( G ) equivalent to kC - M od and P C ,k is not. So we have that F does not restrict to a self-equivalence of he full subcategoryof cohomological Mackey functors or of the full subcategory equivalent to kC - M od . kC - M od (cid:31) (cid:127) / / × (cid:15) (cid:15) Comack k ( C ) (cid:31) (cid:127) / / × (cid:15) (cid:15) M ack k ( C ) F (cid:15) (cid:15) kC - M od (cid:31) (cid:127) / / Comack k ( C ) (cid:31) (cid:127) / / M ack k ( C )This example proves the following Proposition: Proposition 4.2. Let G and H be two finite groups, and k be a field of characteristic p > . An equivalence between M ack k ( G, and M ack k ( H, 1) does not have to sendcohomological Mackey functors to cohomological Mackey functors and does not have torespect the vertices of the projective indecomposable Mackey functors. In particular itdoes not have to induce an equivalence between the full-subcategories equivalent to kG - M od and kH - M od .Remark . This Proposition does not state that there exist finite groups G and H andblocks b and c of RG and RH such that M ack R ( b ) and M ack R ( c ) are equivalent andthe category RGb - M od and RHc - M od are not equivalent. We do not have any resultsin this direction. This Proposition state that we have to be careful when we choose an15quivalence between categories of Mackey functors if we want some results for the fullsubcategories of cohomological Mackey functors or of modules over the group algebra. p -nilpotent groups. Lemma 4.4. Let M and M ′ be two projective Mackey functors of M ack k ( G, , and f : M → M ′ be a morphism. The morphism f is an isomorphism if and only if themorphism f (1) : M (1) → M ′ (1) is an isomorphism of kG -modules.Proof. Lemme 6 . Theorem 4.5. Let G = N ⋊ P be a p -nilpotent group, where P is a Sylow p -subgroupof G . Let b be the principal block of kG . Then M ack k ( b ) is equivalent to M ack k ( P ) .Remark . If b is a nilpotent block, for some m ∈ N , we have an isomorphism of algebras kGb ∼ = M at ( m, kP ), where P is a defect group of the block b . This is not the case forthe Mackey algebras. Example if G = S and k is a field of characteristic 2. Let b bethe principal block of kS , one can check that dim k ( µ k ( C )) = 6 and dim k ( µ k ( b )) = 56(for a proof see Lemme 4 . . Proof. Let us remark that we do not assume the Sylow p -subgroups to be abelian.Recall that the principal block idempotent of kG is b = | N | P n ∈ N n , so we have: kGb ∼ = kP . In consequence, there is a Morita equivalence kGb - M od ∼ = kP - M od . Thisequivalence is given by the following pair of adjoint functors Res GP ⊣ b Ind GP .We use the fact that, in our situation, there are three pairs of adjoint functors res GP ⊣ Ind GP . As we saw, there is one pair for the restriction and induction between the cate-gories of modules over the group algebra. There is an other one between the categoriesof sets with a group action. Here, we denote by ǫ and η are the unit and co-unit of theusual adjunction Ind GP ⊣ Res PG between the categories of finite G -sets and finite P -sets(see Section 4 [23] for an explicit definition of these natural transformations).Finally, at the level of Mackey functors, we also have a restriction and an inductionfunctors. That is: Res GP : M ack k ( b ) → M ack k ( P ) , and Ind GP : M ack k ( P ) → M ack k ( G ) . Applying the block idempotent b µ , we have a functor b µ Ind GP : M ack k ( P ) → M ack k ( b ) . . The functor Res GP is both left and right adjoint to the functor b µ Ind GP , since it is thecase for Ind GP and Res GP . Recall that the unit and the co-unit of the above adjunctionare given by: 16 The unit of the adjunction Res GP ⊣ b µ Ind GP is the natural transformation δ definedby: if M ∈ M ack k ( G ), then δ M = b µ M ∗ ( ǫ ) : M → b µ Ind GP Res GP M. The co-unit is the natural transformation γ defined by: if M ∈ M ack k ( P ), then γ M = b µ M ∗ ( η ) : Res GP b µ Ind GP M → M. • For the adjunction b µ Ind GP ⊣ Res GP , the unit is the natural transformation δ ′ defined by: if M ∈ M ack k ( P ), then δ ′ M = M ∗ ( η ) : M → Res GP b µ Ind GP M. The co-unit is the natural transformation γ ′ defined by: if M ∈ M ack k ( G ), then γ ′ M = b µ M ∗ ( ǫ ) : b µ Ind GP Res GP M → M. . Let M be a projective Mackey functor in M ack k ( b ), and let N be a projectiveMackey functor in M ack k ( P ). Here we use Green’s notation for the evaluation. Wehave a natural isomorphism of kG -modules( Ind GP N )(1) ∼ = Ind GP ( N (1)) , where the first induction functor is the induction of Mackey functors and the second isthe induction of kP -modules. As well, we have an isomorphism of kP -modules Res GP ( N )(1) ∼ = Res GP ( N (1)) , where the first restriction is a restriction of Mackey functors and the second is therestriction of kG -modules. The proof is straightforward but can be found in Proposition5 . δ M , γ N and δ ′ N , γ ′ M are theunits and co-units of the adjunction res GP ⊣ Ind GP between the categories of modulesover group algebras. Since the last adjunction is an equivalence of categories, we havethat the following composition is equal to Id M (1) : M (1) δ M (1) / / b µ Ind GP Res GP ( M )(1) γ ′ M (1) / / M (1) . Moreover, the map γ N (1) ◦ δ ′ N (1) is equal to Id N (1) . By Lemma 4.4 the maps δ M and γ ′ M are two inverse isomorphisms and the maps γ N and δ ′ N are two inverse isomorphismsof Mackey functors.3 . If M ∈ M ack k ( b ), let P • be a projective resolution of M in M ack k ( b ), then we havethe following commutative diagram, · · · / / P / / δ P (cid:15) (cid:15) P / / δ P (cid:15) (cid:15) M δ M (cid:15) (cid:15) / / · · · / / b Ind GP Res GP ( P ) / / b Ind GP Res GP ( P ) / / b Ind GP Res GP ( M ) / / δ P i for i > δ M is an isomorphism. By the samemethod, if N ∈ M ack k ( P ), then γ N is an isomorphism.17 emark . One can see that we did not use the fact that the group G is p -nilpotentin the proof. Instead we use the fact that the restriction functor is an equivalencebetween the categories kGb - M od and kP - M od . More generally, let G be a finite groupand let H be a subgroup of G . Let b be a block of kG and c be a block of kH . Ifthe functor c · Res GH : kGb - M od → kHc - M od is an equivalence of categories, then c µ Res GH : M ack k ( b ) → M ack k ( c ) is an equivalence of categories. Corollary 4.8. There is an isomorphism of algebras µ k ( b ) ∼ = kB ( X ) for the P -set X ∼ = Iso PG/N Def GG/N Ω G , and kB ( X ) is the evaluation of the Burnside functor at X (see example 2.6).Proof. By Theorem 4.5, we have an equivalence of categories µ k ( b )- M od ∼ = µ k ( P )- M od ,so by Morita Theorem, there is an isomorphism of algebras µ k ( b ) ∼ = End µ k ( P ) ( T ), where T is the bimodule Res GP ( µ k ( b )). We will denote by B the Mackey functor, in the senseof Dress, which corresponds to the µ k ( b )-module µ k ( b ) under the usual equivalenceof categories. Since the finitely generated projective Mackey functors of M ack k ( P ) areexactly the Dress constructions kB X of the Burnside functor where X is a finite P -set,we have Res GP ( B ) ∼ = kB X , for some P -set X . In particular, using Green’s notation for the evaluation, we have: kX ∼ = Res GP ( B )(1). But in the equivalence of categories between µ k ( G )- M od and M ack k ( G ), the evaluation at 1 correspond to the multiplication by the idempotent t .So we have: Res GP ( B (1)) = b t µ k ( G ) . A basis of µ k ( G ) is given by t AB xR CB x , where A and C are subgroups of G , the elements x ∈ [ A \ G/C ] and B is a p -subgroup of A ∩ x C up to A ∩ x C -conjugacy . So a basisof t µ k ( G ) is the set of t xr H for x ∈ G and H G . Set γ H,x = t b xR H . Wehave that γ H,nx = γ H,x and γ H,xh = γ H,x for x ∈ G , n ∈ N and h ∈ H . The set { γ H,x ; H G, x ∈ G/N H } is a µ k ( P )-basis of t µ k ( b ). The action of y ∈ P on aelement γ H,x is given by y.γ H,x = γ H,yx . So, kX ∼ = M H G Res GP ( kG/N H ) , but for a p -group P , two permutation modules are isomorphic if and only if the corre-sponding P -sets are isomorphic by [9]. Hence X ∼ = ⊔ H G Res GP ( G/N H ) ∼ = ⊔ H G Res GP Ind GG/N Def GG/N ( G/H ) ∼ = Res GP ( Inf GG/N Def GG/N (Ω G )) ∼ = Iso PG/N Def GG/N (Ω G ) . So Res GP ( B ) ∼ = kB X , where X = Iso PG/N Def GG/N (Ω G ), and finally, we have: µ k ( b ) ∼ = End Mack k ( P ) ( kB X ) ∼ = kB ( X ) , 18y adjunction property of kB X see (Proposition 3 . . kGb ∼ = M at ( n, kP ) for nilpo-tent blocks.For non principal blocks of p -nilpotent groups, or more generally for nilpotent blocks,the situation is much more complicate. The next example shows that, in general, if b is a nilpotent block with defect group D , then the Mackey algebra µ k ( b ) is not Moritaequivalent to the Mackey algebra µ k ( D ). Example . Let G = SL ( F ) ∼ = Q ⋊ C , and k be a field of characteristic 3. Thegroup G is 3-nilpotent. Let b be the block idempotent such that the block kGb containsthe simple kG -module W of dimension 2. Then kGb - M od ∼ = kC - M od . One can ask ifthe same happens to the corresponding blocks of the Mackey algebras. But the Cartanmatrix of µ k ( C ) is (cid:18) (cid:19) and the Cartan matrix of µ k ( b ) is (cid:18) (cid:19) . So there isno Morita equivalence between these two algebras. One can compute these matrices byusing Proposition 3.5 or see below.By Theorem 4.11 they are in fact derived equivalent. In this example it is not diffi-cult to make everything explicit: the projective indecomposable Mackey functors in theblock b µ are in bijection with the indecomposable p -permutation modules in the cor-responding blocks. These indecomposable p -permutation modules are: Ind GQ ( W ) and Ind GC Inf C C k ǫ where C ∼ = Z ( Q ) × C , and k ǫ is the non trivial simple kC -module. Sothe projective indecomposable Mackey functors are P = Ind GQ F P W and Q = Ind GC kB ǫ ,where kB ǫ is a direct summand of the Dress construction of the Burnside functor for C : kB C /C .More precisely, for each C -sets, if Y is any finite C -set, then kB C /C ( Y ) = kB ( Y × C /C ) is a right kC -module, since C /C ∼ = End C − set ( C /C ) End C − set ( Y × C /C ) . So kB ǫ ( Y ) = kB ( Y × C /C ) ⊗ k ( C /C ) k ǫ . Now this it is not difficult to compute theCartan matrix of the block.The derived equivalence can be given by a two terms complex:0 / / P π, / / Q / / π is the morphism of maximal rank between P and Q .It should be notice that in this example the Morita equivalence between kGb and kC is not given by a p -permutation bimodule. If we want to find an equivalence betweenthese two algebras which respects the subcategories of p -permutation modules, we needto replace the bimodule which gives the Morita equivalence by a splendid tilting complex(see [17] Section 7 . .3 Groups with Sylow p -subgroup of order p . Lemma 4.10. Let b be a block of RG with defect group D . Then µ R ( b ) is a symmetricalgebra if and only if | D | < p .Proof. By Corollary 4 . µ R ( D ) is symmetric if and onlyif | D | < p . Now, we just have to prove that µ R ( b ) is a symmetric algebra if andonly if µ R ( D ) is a symmetric algebra. Recall that a finite dimensional R -algebra A is symmetric if and only if for every finitely generate projective A -module P and forevery finitely generated A -module which is R -projective, we have an isomorphism of R -modules: Hom A ( P, M ) ∼ = Hom R ( Hom A ( M, P ) , R ) , and this isomorphism is natural in P and M (see Proposition 2.7 [8]).Suppose that µ R ( b ) is symmetric. Let L be a finitely generated projective µ R ( b )-modulei.e. by using the usual equivalence of categories L is a finitely generated projectiveMackey functor. Let M be an R -projective finitely generated Mackey functors for D .Then Hom Mack R ( D ) ( Res GD ( L ) , M ) ∼ = Hom Mack R ( G ) ( L, b µ Ind GD ( M )) ∼ = Hom R ( Hom Mack R ( G ) ( L, b µ Ind GP ( M )) , R ) ∼ = Hom R ( Hom Mack R ( D ) ( Res GD ( L ) , M ) , R ) . But every finitely projective Mackey functor for the group D is direct summand of therestriction of a finitely generated Mackey functor for G . Since the previous isomorphismis natural, it implies an isomorphism between the corresponding direct summands. Con-versely, since every projective Mackey functor for G is direct summand of the inductionof a projective Mackey functor for D , by similar arguments, we have the result. Theorem 4.11. Let G and H be two finite groups. Let b be a block of kG with cyclicdefect group of order p , and c be a block of kH with cyclic defect group of order p . Then, D b ( kGb - M od ) ∼ = D b ( kHc - M od ) if and only if D b ( µ k ( b ) - M od ) ∼ = D b ( µ k ( b ′ ) - M od ) .Proof. By Theorem 20.10 of [22], in this situation, the blocks of Mackey algebras areBrauer tree algebras. Let T Mack be the tree of this algebra. And T Mod be the tree ofthe corresponding block of the group algebra. The tree T Mod is isomorphic to a subtreeof T Mack , still denoted by T Mod . Some properties of the tree T Mack are determined bythe knowledge of the tree T Mod . If e is the number of edges of T Mod , then the numberof edges of T Mack is 2 e . The exceptional vertex of T Mack is the same as the exceptionalvertex of the tree T Mod . Each edge of T Mack which is not in T Mod is a twig. By generalresults on derived equivalences for Brauer tree algebras, two Brauer tree algebras withsame exceptional multiplicity, over the same field, are derived equivalent if and only ifthey have the same number of edges.Even if the tree T Mack seems to be determined by the group algebra kG , if two blocksof group algebras are Morita equivalent, it is not always true that the corresponding20locks for the Mackey algebras are Morita equivalent (see Example 4.9). The tree T Mack is in fact determined by the corresponding block of kG and its Brauer correspondent in N G ( P ) where P is the defect of the block. Remark . In [22], all the results are given for blocks of groups with a Sylow p -subgroup of order p . One can check that these results can be generalized quite easily togeneral blocks with defect group of order p (see Chapter 6 of [19] for the proof). Proposition 4.13. Let G and H be two finite groups. Let b (resp. c ) be a block of kG (resp. kH ) with a defect p -group of order p . If kGb and kHc are splendidly Moritaequivalent, then there is an equivalence of categories: µ k ( e ) - M od ∼ = µ k ( c ) - M od .Proof. By Theorem 20.10 of [22] and Theorem 4.11, the block algebras µ k ( e ) and µ k ( f )are derived equivalent Brauer tree algebras. Since two Brauer tree algebras over the samefield are Morita equivalent if and only if they have isomorphic trees and same exceptionalmultiplicity, it is enough too prove that they have the same Cartan matrices. We willprove that the decomposition matrices of µ O ( b ) and µ O ( c ) are the same. By Proposition3.5, the decomposition matrices of µ O ( b ) can be computed from the knowledge of the p -blocks O Gb and the Brauer correspondent of b in O N G ( C ).Suppose that there are e simple kG -modules in the block b of kG , then there are e + 1simple KG -modules in this block, one of this simple module may be exceptional. Thenumber of simple kN G ( C )-modules W in a block b ′ which is the Brauer correspondentof b is e . Since N G ( C ) is a p ′ -group, each simple kN G ( C )-module gives rise to a uniquesimple KN G ( C )-module. Thus the decomposition matrix of µ O ( b ) is the following blockmatrix with 2 e + 1 columns and 2 e rows: D ( coµ O ( b )) 0 e × e Id e × e Where D ( coµ O ( b )) is the decomposition matrix of coµ O ( b ).So if two blocks kGb and kHc , with cyclic defect group of order p are splendidlyMorita equivalent, the blocks O Gb and O Hc are splendidly Morita equivalent by Section5 of [17]. By Proposition 4 . coµ O ( b ) and coµ O ( c ) are Morita equivalent, so the Cartan matrices of µ k ( b ) and µ k ( c )are the same. Corollary 4.14. Let G and H be two finite groups. Let b be a block of O G and let c bea block of O H . Let us suppose that the blocks b and c have a defect group of order p .Then1. D b ( µ O ( b )) ∼ = D b ( µ O ( c )) if and only if D b ( O Gb ) ∼ = D b ( O Hc ) .2. If O Gb and O Hc are splendidly Morita equivalent, then µ O ( b ) - M od ∼ = µ O ( c ) - M od .Proof. For this proof, we suppose that the reader is familiar with the notion of Greenorders . For the definition and properties of Green orders see [14] and [13].21he Mackey algebra µ k ( b ) over k of a block with defect of order p is a Brauer tree algebra,so there is a Green walk on this tree. One can lifts this Green walk for µ O ( b ) exactlyas Green did in [12]. This shows that the Mackey algebra over O is a Green orderin the sense of [18]. K¨onig and Zimmermann in [13] proved that two generic Greenorders with trees having same number of vertices and same exceptional vertex plussome local properties (the orders attached to the vertices have to coincide) are derivedequivalent. In our situation, that is: group algebras or p -local Mackey algebras over a p -modular system which is ‘large enough’, one can prove that, the order attached to a nonexceptional vertex is O and the order attached to the exceptional vertex depends only onthe exceptional multiplicity (for more details see Lemme 6 . . 24 of [19]). The first part ofthe theorem follows. The second part follows from the fact that two generic Green ordersare Morita equivalent if and only if they have isomorphic trees, with same exceptionalmultiplicity and the local datas of the two trees coincide. 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