Blocks of Mackey categories

Contents lists available atScienceDirect

Journal

of

Algebra

www.elsevier.com/locate/jalgebra

Blocks

of

Mackey

categories

Laurence Barker1

DepartmentofMathematics,BilkentUniversity,06800Bilkent,Ankara,Turkey

a r t i c l e i n f o a bs t r a c t

Article history:

Received5May2015

Availableonline29September2015 CommunicatedbyMichelBroué

MSC:

20C20

Keywords:

Mackeysystem

Blockofalinearcategory Locallysemisimple Bisetcategory

ForasuitablesmallcategoryF ofhomomorphismsbetween finite groups, we introduce two subcategories of the biset category,namely,thedeflationMackeycategoryM←_F andthe inflationMackeycategoryM→_F.LetG bethesubcategoryof

F consistingoftheinjectivehomomorphisms.Weshallshow that,forafieldK ofcharacteristiczero,theK-linearcategory KMG =KM←_G =KM→_G hasasemisimplicitypropertyand, in particular, every block of KM_G owns a unique simple functoruptoisomorphism.Ontheotherhand,weshallshow that, whenF is equivalentto thecategory offinite groups, the_K-linearcategories_KM←_F and_KM→_F eachhaveaunique block.

© 2015ElsevierInc.All rights reserved.

1. Introduction

Mackeyfunctorsarecharacterizedbyinductionandrestrictionmapsassociatedwith somegrouphomomorphisms.Forexample,thegroupsinvolvedcanbethesubgroupsof a fixed finite group and the homomorphisms canbe the composites of inclusionsand

E-mailaddress:[email protected].

1 _{ThisworkwassupportedbyTÜBITAK ScientificandTechnologicalResearchFundingProgram1001}

undergrantnumber114F078.Someofthisworkwascarriedoutin2010/11whentheauthorheldaVisiting AssociateProfessorshipattheUniversityofCaliforniaatSantaCruz.

http://dx.doi.org/10.1016/j.jalgebra.2015.09.002

conjugations. As another example, the groups can be arbitrary finite groups and the homomorphismscanbe arbitrary.

WeshalluseBouc’stheoryofbisets[4]torecastthetheoryofMackeyfunctorsinthe followingway. LetK be aset offinitegroupsthatisclosedundertakingsubgroups.(In applications,K canplaytheroleofaproperclass.Forinstance,ifK ownsanisomorphic copy of every finite group, then K can play the role of the class of all finite groups.) Generalizing the notion of afusion system on a finite p-group, we shall introduce the notion of a Mackey system on K, which is a category F such that the set of objects is Obj(F)= K and the morphisms inF are group homomorphisms subjectto certain axioms. In the case where all the homomorphisms in F are injective, we call F an

ordinaryMackey system.

ForanyMackeysystemF onK,weshalldefinetwosubcategoriesofthebisetcategory, namely,thedeflationMackeycategoryM←_F andtheinflationMackeycategoryM→_F.The category M←_F isgenerated byinductionsviahomomorphisms inF andrestrictionsvia inclusions.ThecategoryM→_F is generatedbyinductionsviainclusionsandrestrictions viahomomorphismsinF.WhenF isanordinaryMackeysystem,M←_F andM→_F coin-cide,andwewriteitas M_F,callingitanordinaryMackey category.

LetR beacommutativeunitalringandletRM←_F betheR-linearextensionofM←_F. ThenotionofaMackeyfunctoroverR willbereplacedbythenotionofanRM←_F-functor,

which is a functor from RM←_F to the category of R-modules. Our approach to the study of RM←_F-functors will be ring-theoretic. We shall introduce analgebra Π_RM←

F

over R, called the extended quiver algebra of RM←_F, whichhas the feature thatevery

RM←_F-functorisaΠ_RM←

F-module.Wedefinea block ofRM←F tobeablockofΠRM←F.

As in the blocktheory of suitable rings, everyindecomposable RM←_F-functor belongs toauniqueblockofRM←_F.SimilarconstructionscanbemadefortheinflationMackey categoryM→_F.

LetK beafieldofcharacteristiczero.RegardingtheblocksofKM←_F asapartitioning of thesimple KM←_F-functors, theblocks sometimespartition thesimple functors very finely. Corollary 4.7says that,forany ordinary MackeysystemG, eachblockof KM_G ownsauniquesimpleKMG-functor.Buttheblockscanalsopartitionthesimplefunctors verycoarsely.Ourmain result,Theorem 7.1,assertsthatifK ownsanisomorphiccopy ofeveryfinitegroupandF ownseveryhomomorphismbetweengroupsinK,thenKM←_F andKM→_F eachhaveauniqueblock.

Weshallbeneedingtwotheoremswhoseconclusionshavebeenobtainedbeforeunder different hypotheses. Theorem 4.6 asserts that the category KM_G, though sometimes infinite-dimensional,has a semisimplicityproperty. This result was obtainedby Webb [10, 9.5] in the special case where G is equivalent to the category of injective group homomorphisms. Thesame conclusionwas established by Thévenaz–Webb[8], [9,3.5] inadifferent scenario where thegroup isomorphismsthatcome into consideration are conjugationswithinafixedfinitegroup.Theirresultisnotaspecialcaseofoursbecause theirrelations[9,page1868]ontheconjugationmapsareweakerthanours.Theorem 5.2 assertsthat, taking G to be thelargest ordinaryMackeysystem thatis asubcategory

of F, restrictionandinflation yieldmutually inversebijectivecorrespondencesbetween thesimpleKM←_F-functorsandthesimpleKM_G-functors.Asimilarresultholdsforthe simple KM→_F-functors. A version of this resultwas obtained by Yaraneri [11, 3.10] in the scenario where the isomorphismsare conjugations within afixed finite groupand, again, therelationsontheconjugationmaps areasin[9,page 1868].

AscenariosimilartoourswasstudiedinBoltje–Danz[2].Weshallmakemuchuseof theirtechniques.TheyconsideredsomesubalgebrasofthedoubleBurnsidealgebrathat can be identified with endomorphism algebras of objects of Mackeycategories. Boltje andDanzobtainedanalogues[2,5.8,6.5]ofTheorems 4.6 and5.2fortheendomorphism algebras. Those analogues can be recovered from Theorems 4.6 and 5.2 by cutting by idempotents.

Thematerialisorganizedasfollows.Section2isanaccountofthegeneralnotionofa blockofanR-linearcategory.InSection3,weclassifythesimplefunctorsoftheR-linear

extension of aMackey category. InSection 4, we provethat the K-linear extension of an ordinary Mackeycategory hasa semisimplicityproperty. InSection 5, we compare theK-linear extensionofadeflationMackeycategorywiththeK-linearextension ofan ordinaryMackeycategory.Section6concernstheuniquenon-ordinarydeflationMackey category in the case where K consists only of a trivial group and a groupwith prime order. Section7proves atheoremontheuniquenessoftheblockof adeflationMackey category thatis, insomesense,maximal amongalldeflationMackeycategories.

The author would like to thank RobertBoltje for contributingsomeof the ideasin this paper.

2. Blocksof linearcategories

An R-linear category (also called anR-preadditive category)is definedto be a cat-egory whose morphism sets are R-modules and whose composition is R-bilinear. An

R-linear functor between R-linear categories is defined to be a functor which acts on morphism sets as R-linear maps. We shall define the notionof ablock of an R-linear

category, andweshallestablishsomeofitsfundamentalproperties.Itwill benecessary to giveabriefreview ofsomematerialfrom [1]on quiveralgebras andextendedquiver algebras ofR-linearcategories.

Let L be a small R-linear category. Consider the direct product Π = 

F,G∈Obj(L)L(F,G) where Obj(L) denotes the set of objects of L and L(F,G)

de-notestheR-moduleofmorphismsF ← G inL.Givenx∈ Π,wewritex= (FxG) where FxG ∈ L(F,G). Let ΠL be the R-submodule ofΠ consisting of those elements x such

that,foreachF ∈ Obj(L),thereexistonlyfinitelymanyG∈ Obj(L) satisfyingFxG= 0

or GxF = 0.We make ΠL become aunital algebra with multiplication operation such

that

F(xy)G=



G∈Obj(L) FxGyH

where F,H ∈ Obj(L) and x,y ∈ ΠL and FxGyH =FxG.GyH. The sum makes sense

becauseonly finitely manyof the termsare non-zero.We call ΠL the extended quiver algebra ofL.Therationaleforthetermwillbecomeapparentlaterinthis section.

Afamily (xi : i∈ I) of elements xi∈ΠL issaid to be summable provided, foreach F ∈ Obj(L), there are only finitely many i ∈ I and G ∈ Obj(L) such that F(xi)G =

0 or G(xi)F = 0. In that case, we define the sum ixi ∈ ΠL to be such that its

(F,G)-coordinate is F(ixi)G =i F(xi)G.Any element x∈ΠL canbe writtenas a

sum

x = 

F,G∈Obj(L) FxG.

TheunityelementofΠ_{L isthesum}

1_L= 

G∈Obj(L)

idG.

Proofofthenextremarkisstraightforward.

Remark2.1. Anyelementz ofthecentreZ(ΠL) canbeexpressed asasum

z = 

G∈Obj(L)

zG

where zG ∈ L(G,G). Conversely, given elements zG ∈ L(G,G) defined for each G ∈

Obj(L),then wecanformthesumz∈Π_{L asabove,whereupon z∈ Z(}Π_{L) ifand only}

if,forallF,G∈ Obj(L) and x∈ L(F,G),wehavezFx= xzG.

Wedefinea block ofaunitalringΛ tobeaprimitiveidempotentofZ(Λ).Letblk(Λ) denotethesetofblocksofΛ.ItiseasytoseethatZ(Λ) hasfinitelymanyidempotentsif andonlyifΛ hasfinitelymanyblocksandthesumoftheblocksistheunityelement 1Λ.

Inthatcase,we saythatΛ has a finiteblockdecomposition. Wedefinea block ofL to beablockofΠ_L.

Theorem2.2. If thealgebraL(G,G)= End_L(G) hasafinite block decompositionforall G∈ Obj(L),then

1_L= 

b∈blk(L) b .

Proof. Weadapttheproof ofBoltje–Külshammer[3,5.4].Let

E = 

G∈Obj(L)

Let ∼ be the reflexive symmetric relation on E such that, given F,G ∈ Obj(L) and d ∈ blk(L(F,F )) and e ∈ blk(L(G,G)), then d ∼ e provided dL(F,G)e = {0} or eL(G,F )f = {0}. Let ≡ be the transitive closure of ∼. We mean to say, ≡ is the equivalence relation suchthat d ≡ e if and only ifthere exist elements f0,. . . ,fn ∈ E

suchthatf0= d andfn= e andeachfi−1∼ fi.ThehypothesisonthealgebraL(G,G)

implies thatevery subset of E is summable. Plainly, 1_L = _e∈Ee. It suffices to show thatthereis abijectivecorrespondencebetweentheequivalence classesE under ≡ and

theblocksb ofL such thatE↔ b provided b=_e∈Ee.

Let E beanequivalence classunder≡ and letb=_e∈Ee.Wemust show thatb is

ablockofL.Plainly, b isanidempotentof Π_{L.Given F,G∈ Obj(L) andx∈ L(F,G),}

then bFx = bFx1L=   d∈EF dx  e∈blk(L(G,G)) e=  d∈EF, e∈EG dxe = xbG

where EF = E∩ blk(L(F,F )).So,byRemark 2.1,b∈ Z(ΠL).Supposethatb= b1+ b2

as a sumof orthogonal idempotents of Z(Π_{L) with b}

1 = 0. Since bb1 = 0, there exist

F ∈ Obj(L) andd∈ EF suchthatdb1= 0.Wehavedb1= d(b1)F = d because(b1)F is

acentralidempotentofL(F,F ).ForallG∈ Obj(L) and e∈ EG,wehave dL(F, G)b1e = db1L(F, G)e = dL(F, G)e .

So, if dL(F,G)e = {0}, then b1e = 0, whereupon, by an argument above, b1e = e.

Similarly,the conditioneL(G,F )d= {0} implies thatb1e= e. Wededuce thatb1e= e

foralle∈ E. Therefore,b1= b and b2= 0.Wehaveshownthatb isablockof L.

Conversely,givenablockb of L,lettingf ∈ E suchthatbf= 0 andlettingE be the equivalence classoff ,thenb_e∈Ee= 0,henceb coincideswiththeblock_e∈Ee. We haveestablishedthebijectivecorrespondenceE↔ b,asrequired. 2

As asubalgebraofΠ_L,wedefine

⊕_{L =} 

F,G∈Obj(L)

L(F, G) .

We call ⊕L the quiver algebra ofL. When no ambiguity can arise, we write L =⊕L. Plainly, the followingthree conditions areequivalent: Obj(L) isfinite;thealgebraL is unital;wehaveanequalityofalgebrasL=ΠL.

We define anL-functor to be anR-linear functor L → R–Mod.Given an L-functor

M , we can form a Π_{L-module M}

Π = _GM (G) where an element x ∈ L(F,G) acts

on MΠ as M (x),annihilating M (G) for allobjects G distinct from G.Byrestriction,

we obtain an L-module M_⊕. Note that LMΠ = MΠ, in other words, LM⊕ = M⊕.

Given anotherL-functor M, then eachnaturaltransformation M → M givesrise, in anevidentway,toaΠL-mapMΠ→ MΠ whichisalsoanL-mapM⊕→ M⊕.Conversely,

theL-mapsM⊕ → M⊕ coincidewiththeΠL-mapsMΠ→ MΠ and giverisetonatural

transformationsM→ M.Puttingtheconstructionsinreverse,givenanL-moduleM_⊕

suchthat LM_⊕ = M_⊕, we canextendM_⊕ to aΠ_{L-module M}

Π and we can alsoform

anL-functor M suchthat M (G)= idGM⊕ = idGMΠ. Henceforth, we shall neglectto

distinguishbetween M andMΠ andM⊕.Thatis to say,we identifythecategory ofL

functorswiththecategoryofΠ_{L-modulesM satisfyingLM = M andwiththecategory}

ofL-modulesM satisfyingLM = M.

AnL-functorM is saidto belongto ablockb ofL providedbM = M .Inthatcase, wealsosaythatb owns M .Theorem 2.2hasthefollowingimmediatecorollary.

Corollary2.3. IfL(G,G) hasafinite blockdecompositionforallG∈ Obj(L),thenevery indecomposableL-functorbelongs toauniqueblock ofL.

Proof. Let M be an indecomposable L-functor. Choose an object G of L such that

M (G) = 0. We have idG = _b∈blk(L)bG as a sum with only finitely many non-zero

terms.SobGM (G)= 0 forsomeb.Inparticular,bM = 0.ButM = bM⊕ (1− b)M and M isindecomposable,so M = bM . 2

The next three results describe how the simple L-functors and the blocks of L are relatedtothesimplefunctorsandblocks ofafullsubcategory ofL.

Proposition2.4.LetK beafullsubcategoryofL.Thenthereisabijectivecorrespondence betweentheisomorphismclassesof simpleK-functors S andtheisomorphism classesof simpleL functorsT suchthat1_KT = 0.ThecorrespondenceissuchthatS↔ T provided S ∼= 1KT .

Proof. We have Π_{K = 1}

K.ΠL.1K. So the assertion is a special case of Green [6, 6.2]

which says that, given an idempotent i of a unital ring Λ, then the condition S ∼=

iT characterizes abijective correspondencebetween the isomorphism classesof simple

iΛi-modulesS andtheisomorphismclassesofsimpleΛ-modulesT satisfyingiT = 0. 2

Proposition 2.5. Suppose that L(G,G) has a finite block decomposition for all G ∈

Obj(L).LetK beafullsubcategoryofL andletS andSbesimpleK-functors.LetT and T betheisomorphicallyuniquesimple L-functorssuchthatS ∼= 1KT andS∼= 1KT.If S andS belongtothesame blockof K, thenT andT belongtothesameblock of L.

Proof. Let a anda be theblocks ofK owning S andS, respectively. Letb and b be theblocksof L owningT and T, respectively.Thecentral idempotentb1_K of Π_{K acts}

as theidentity onS, so ab= a.Similarly,ab = a. If a= a then abb = a= 0, hence

bb= 0, whichimpliesthatb= b. 2

Proposition 2.6. Suppose that L(G,G) has a finite block decomposition for all G ∈

of L if and only if there exists a full R-linear subcategory K of L such that Obj(K) is

finite and the simple K-functors 1_KT and 1_KT are non-zero and belong to the same block of K.

Proof. In one direction, this is immediate from the previous proposition. Conversely, suppose thatT andT belongto the sameblockb ofL.Let G,G ∈ Obj(L) such that

T (G) = 0 and T(G)= 0. Let e∈ blk(L(G,G)) and e ∈ blk(L(G,G)) be such that

eT (G)= 0 andeT(G)= 0.SinceebT (G)= eT (G), wehaveeb= 0.Similarly,eb= 0.

Therefore e ≡ e where ≡ is the equivalence relation in the proof of Theorem 2.2. So there exist G0,. . . ,Gn ∈ Obj(L) and fi ∈ blk(L(Gi,Gi)) such that G0 = G, f0 = e,

Gn = G, fn = e and each fi−1 ∼ fi. Let K be the full subcategory of L such that

Obj(K)={G0,. . . ,Gn}.Thene ande arestillequivalentundertheequivalencerelation

associated withK. Bytheproof of Theorem 2.2, there exists ablocka ofK such that

ae = e and ae = e. We have ea1_KT = e1_KT = eT (G) = 0, hence a1_KT = 0 and,

similarly, a1_KT= 0. Therefore1_KT and1_KT bothbelongtoa. 2

3. Mackeycategoriesandtheirsimplefunctors

WeshallintroducethenotionsofaMackeysystemandaMackeycategory.Weshall also classifythesimplefunctorsoftheR-linearextension ofagivenMackeycategory.

First,letusbrieflyrecallsomefeaturesofthebisetcategoryC.Detailscanbefoundin Bouc[4,Chapters2,3].LetF ,G,H befinitegroups.ThebisetcategoryC isaZ-linear category whoseclassofobjectsistheclassoffinitegroups.TheZ-moduleofmorphisms

F ← G in C is

C(F, G) = B(F × G) = 

A≤GF×G

Z[(F × G)/A]

where B indicates theBurnside ring,the indexA runs overrepresentatives of the con-jugacy classes of subgroups of F × G and [(F × G)/A] denotes theisomorphism class of the F –G-biset (F × G)/A. The morphisms having the form [(F× G)/A] are called transitive morphisms. The composition operation for C is defined in[4, 2.3.11, 3.1.1]. A useful formulaforthecompositionoperation is

F× G A  G× H B  =  p2(A)gp1(B)⊆G F× H A∗(g,1)_B  .

Here,thenotationindicatesthatg runsoverrepresentativesofthedoublecosetsofp2(A)

and p1(B) in G. Foran account of the formula and for specificationof the rest of the

notation appearinginit,see[4,2.3.24].

Given agrouphomomorphismα : F ← G,wedefinetransitivemorphisms

called induction and restriction. The composite of two inductionsis an induction and thecompositeoftworestrictionsisarestriction.Indeed,usingtheaboveformulaforthe compositionoperation, itiseasy tosee that,givenagrouphomomorphismβ : G← H

then,

FindαGindβH=Find αβ H , Hres β GresαF =Hres αβ F .

Whenα isinjective,wecall_Findα_G an ordinaryinduction andwecall_Gresα_F an ordinary restriction. When α is aninclusion F ← G,we omit thesymbol α from the notation, justwriting_Find_G and_Gres_F.Whenα issurjective, wewrite

FdefGα=FindαG , Ginf

α

F =Gres

α F

whichwecall deflation and inflation.Notethat,forarbitraryα,wehavefactorizations

FindαG=Findα(G)def α

G , GresαF =Ginfα(G)α resF .

Whenα isanisomorphism,wewrite

FisoαG=FindαG =Fresα

−1

G

whichwecall isogation.InC,theidentity morphismonG istheisogationisoG=Giso1G.

Given g ∈ G,we letc(g) denoteleft-conjugation byg.Let V,V ≤ G. Againusing the aboveformulaforcomposition,werecover thefamiliarMackeyrelation

VresGindV =



V gV⊆G

VindV∩g_Visoc(g)Vg_∩VresV .

A transitive morphism τ : F ← G is said to be left-free provided τ is the isomor-phism classof anF -free F –G-biset.The left-freetransitive morphismsF ← G are the morphismsthatcanbeexpressed intheform

Find α

VresG =Findα(V )def α V resG= F× G S(α, V ) 

whereV ≤ G andα : F ← V and

S(α, V ) ={(α(v), v) : v ∈ V } .

Evidently, the left-free transitive morphisms are those transitive morphism which can be expressed as the composite of an ordinary induction, a deflation and an ordinary restriction. The right-free transitive morphisms, defined similarly, are those transitive morphismswhichcanbeexpressedasthecompositeofanordinaryinduction,aninflation andanordinaryrestriction.

Proposition 3.1 (Mackey relation forleft-freetransitive morphisms). LetF and V ≤ G and W ≤ H be finite groups.Letα : F ← V andβ : G← W be grouphomomorphisms. Then Find α VresGind β WresH =  V gβ(W )⊆G Find αgc(g)βg β−1(Vg₎res_H

where αg: F ← V ∩gβ(W ) andβg: Vg∩ β(W )← β−1(Vg) arerestrictions ofα andβ.

Proof. Usingthestar-productnotationofBouc [4,2.3.19],

{(v, v) : v ∈ V } ∗(g,1)_{S(β, W ) = S(c(g)β}

g, β−1(Vg)) .

Hence_Vres_Gindβ_W = 

V gβ(W ) Vind

c(g)βg

β−1(Vg₎res_W. 2

As inSection1,letK beasetoffinitegroupsthatisclosed undertakingsubgroups. Wedefine a Mackeysystem onK tobe acategory F suchthattheobjectsofF arethe groupsinK, everymorphisminF isagrouphomomorphism,composition istheusual compositionofhomomorphisms,and thefollowingfouraxioms hold:

MS1: For allV ≤ G∈ K,theinclusionG← V isinF.

MS2: For allV ≤ G∈ K andg∈ G,theconjugationmapg_V g_{v → v ∈ V isinF.}

MS3: For anymorphism α : F ← G in F,the associated homomorphismα(G)← G is

inF.

MS4: For anymorphismα inF suchthatα isagroupisomorphism,α−1 isinF. We call F an ordinary Mackey system provided all themorphisms in F are injective. As anexample,afusionsystemonafinite p-groupP is preciselythesamething as an ordinaryMackeysystemonthesetof subgroupsofP .

Remark3.2.Given aMackeysystemF onK,then:

(1) There exists a linear subcategory M←_F of C such that Obj(M←_F) = K and, for

F,G ∈ K, the morphisms F ← G in M←_F are the linear combinations of the left-free transitivemorphisms_Findα

VresG whereV ≤ G andα : F ← V isamorphism inF.

(2) There exists a linear subcategory M→_F of C such that Obj(M→_F) = K and, for

F,G∈ K, themorphisms F ← G in M→_F are thelinear combinations of theright-free transitivemorphisms_Find_Uresβ_G whereU ≤ F andβ : U → G isamorphisminF. Proof. In the notationof Proposition 3.1,supposing thatF,G,H ∈ K and thatα and β are morphisms inF then, by axioms MS1and MS3, each αg and βg are in F and,

byaxiomMS2,eachc(g) isinF.Part(1)is established.Part(2)canbedemonstrated similarly orbyconsideringduality. 2

We callM←_F the deflation Mackeycategory ofF. Therationalefortheterminology is thatM←_F is generated by inductions from subgroups, restrictions to subgroups and

deflations coming from surjections in F. We call M→_F the inflation Mackey category ofF.

Remark3.3. GivenanordinaryMackeysystemG,thenM←_G =M→_G. Proof. Thisfollows fromaxiomMS4. 2

ThecategoryM_G =M←_G =M→_G iscalled an ordinaryMackeycategory.

Fortherestof thissection, wefocus onthedeflationMackeycategory M←_F.Similar constructionsandargumentsyieldsimilarresultsfortheinflationMackeycategoryM→_F. Weshallneedsomenotationforextension tocoefficientsinR.Given aZ-moduleA,we writeRA= R⊗_ZA.GivenaZ-mapθ : A→ A,weabusenotation,writingtheR-linear

extension as θ : RA→ RA. Given aZ-linear category L, we write RL to denote the

R-linearcategory suchthat(RL)(F,G)= R(L(F,G)) forF,G∈ Obj(L).

Remark 3.4. Given a Mackey system F on K andF,G ∈ K, then the following three conditionsareequivalent:thatF andG areisomorphicinF;thatF andG areisomorphic inM←_F;thatF andG areisomorphicinRM←_F.

Proof. Givenanisomorphismγ : F ← G in F, then_Fisoγ_G : F ← G isanisomorphism inM←_F.Sothefirstconditionimpliesthesecond.Trivially,thesecondconditionimplies thethird.Assumethethirdcondition.Letθ : F ← G andφ: G← F bemutuallyinverse isomorphismsinRM←_F.Writingθ =_iλiθiandφ=



jμjφjaslinearcombinationsof

transitivemorphismsθi andφj,thenisoF = θφ=i,jλiμjθiφj.AnargumentinBouc

[4,4.3.2],makinguseof[4,2.3.22],impliesthatθiandφjareisogationsforsomei and j.

Wehavededuced thefirstcondition. 2

For F,G ∈ K, we write F(F,G) to denote the set of morphisms F ← G in F. We makeF(F,G) becomeanF×G-setsuchthat

(f,g)_{α = c(f ) α c(g}−1₎

for(f,g)∈ F × G andα∈ F(F,G).Since α c(g−1)= c(α(g−1))α,the F×G-orbits of

F(F,G) coincidewith theF -orbits.Let α denote theF -orbit of α.Wehave α β = αβ

forH ∈ K andβ ∈ F(G,H).Sowe canform aquotientcategory F of F suchthatthe setofmorphismsF ← G inF isF(F,G)={α : α ∈ F(F,G)}.InF,theautomorphism groupof G is

Out_F(G) = Aut_F(G)/Inn(G)

Remark3.5.LetF beaMackeysystemonK.GivenF,G∈ K andα,α∈ F(F,G),then thefollowingthreeconditionsareequivalent:that_Findα

G=Findα



G;thatGresαF =Gresα



F;

thatα = α.

Proof. Another equivalentconditionisS(α,G)=_F×GS(α,G). 2

Let P_F,GF denote the set of pairs (α,V ) where V ≤ G and α ∈ F(F,V ). We allow

F × G toactonP_F,GF suchthat

(f,g)_{(α, V ) = (}

(f,g)α,gV )

forf ∈ F andg∈ G.LetPF_F,Gdenote thesetofF×G-orbits inP_F,GF .Let[α,V ] denote

theF×G-orbitof(α,V ).

Proposition 3.6. LetF be aMackey systemon K.Then, forF,G∈ K, theR-moduleof morphisms F← G in RM←_F is

RM←_F(F, G) = 

[α,V ]∈PFF,G

R ._Findα_Vres_G.

Proof. For V,V ≤ G and α ∈ F(F,V ) and α ∈ F(F,V), we have _Findα

VresG =

Findα



VresG ifandonlyifS(α,V )= S(α,V),inotherwords,[α,V ]= [α,V]. 2

We define a seed for F over R tobe apair (G,V ) where G∈ K andV is a simple

ROut_F(G)-module.Twoseeds(F,U ) and(G,V ) forF overR aresaidtobe equivalent provided there exist an F-isomorphism γ : F ← G and an R-isomorphism φ: U ← V suchthat,givenη∈ OutF(G),then γηγ−1 ◦φ= φ◦η.

The next result is different in context but similar in form to the classifications of simple functors in Thévenaz–Webb [9,Section 2], Bouc [4, 4.3.10], Díaz–Park [5, 3.2]. It can be provedby similar methods.It is also aspecialcase of [1,3.7].Observe that, givenG∈ K andanRM←_F-functorM ,thenM (G) becomesanROut_F(G)-module such thatanelementη∈ Out_F(G) actsas_Gisoη_G.WecallG a minimalgroup forM provided M (G)= 0 andM (F )= 0 forallF ∈ K with|F |<|G|.

Theorem 3.7. Let F be a Mackey system on K and let M = M←_K. Given a seed

(G,V ) for F over R, then there is a simple RM-functor SRM_G,V determined up to iso-morphism by the condition that G is aminimal group for SRM

G,V and SG,VRM(G) ∼= V as ROut_F(G)-modules. The equivalence classes of seeds (G,V ) for F over R are in a bi-jective correspondencewiththeisomorphismclassesof simpleRM-functorsS suchthat

4. OrdinaryMackeycategories andsemisimplicity

Throughout this section, we let G be an ordinary Mackey system on K. We shall considertheordinaryMackeycategoryN = M_G.Recall,fromSection1,thatK isafield ofcharacteristiczero.WeshallprovethattheK-linearcategoryKN hasasemisimplicity property.As mentionedinSection1,this conclusionwasobtainedbyWebb[10, 9.5]in a special case and by Thévenaz–Webb [8], [9, 3.5] in scenario involving a fixed finite group. Another related result, with adifferent conclusionbut in asimilar scenario, is Boltje–Danz[2,5.8],whichsaysthatthealgebraKN (G,G) issemisimpleforallG∈ K.

Letusdiscuss,inabstract,thesemisimplicitypropertythatweshallbe establishing.

Remark4.1.GivenanR-linearcategoryL,thenthefollowingtwoconditionsare equiv-alent:

(a) ForeveryfulllinearsubcategoryL0 ofL withonlyfinitelymanyobjects,thequiver

algebraL0is semisimple.

(b) ThealgebraiLi issemisimpleforeveryidempotenti ofthequiveralgebraL. Proof. If each iLi is semisimple then, given L0, we have L0 = 1L0.L.1L0, which is

semisimple.Conversely,supposethateachL0issemisimple.Giveni,letL0bea

subcat-egoryofL suchthatObj(L0) isfiniteandi hastheformi=F,G∈Obj(L0) FiGwitheach FiG∈ L(F,G).Then1L0i= i= i1L0.SincethealgebraL0= 1L0.L.1L0 issemisimple,

thealgebraiLi= i1_L0.L.1L0i issemisimple. 2

Whentheequivalentconditionsintheremarkhold,wesaythatL is locallysemisimple. InTheorem 4.6,weshallprovethattheK-linear categoryKN islocally semisimple.

ForG,H ∈ K, letL(G,H) bethe Z-modulefreely generated bythe formalsymbols

Gind β

H whereβ runsovertheelementsofG(G,H).ItistobeunderstoodthatGind β

H =

Gind β

H ifandonlyifβ = β.Thus

L(G, H) =  β∈G(G,H) ZGind β H . WedefineaZ-module L =  G,H∈K L(G, H) .

Wedefine aZ-epimorphismπ :N → L suchthat,givenW ≤ H andβ∈ G(G,W ),then

π(_Gind_Wβres_H) = ⎧ ⎨ ⎩ Gind β W if W = H, 0 if W < H.

By Proposition 3.1, ker(π) is aleftideal ofN . Wemake L become anN -module with representationσ : N → End_Z(L) suchthatσ(m)π(x)= π(mx) form,x∈ N .Thenext lemma expressestheactionofN moreexplicitly.

Lemma 4.2.ForF,G,H ∈ K,V ≤ G, α∈ G(F,V ),β∈ G(G,H), wehave σ(_Findα_Vres_G)_Gind_Hβ = 

V gβ(H)⊆G : V ≥g_β(H)

Find αc(g)β

H .

Proof. Thisfollowsfrom Proposition 3.1. 2

LetI bethelinearsubcategoryofN generatedbytheisogations.Thatistosay,the quiver ring I isthesubringof M generatedbythe isogations.Infact,I istheZ-span of theisogations and

I(J, K) = δ

ZJiso δ K

where J,K ∈ K andδ runs over theG-isomorphisms J ← K.Note that,viathe corre-spondence _Hisoγ_H↔ γ,wehaveanalgebraisomorphism

I(H, H) ∼=ZOutG(H) .

Wemake L becomeanI-module withrepresentationτ :I → EndZ(L) suchthat

τ (KisoγJ)Gind β H= ⎧ ⎨ ⎩ Gind βγ−1 K if J = H, 0 otherwise.

SincetheactionsofN andI commutewitheachother,σ andτ areringhomomorphisms

σ : N → End_I(L) , τ : I → End_N(L) . As anN -submoduleof L,wedefine L(–, H) = τ (isoH)L =  G∈K L(G, H) .

Each L(–,H) isanI(H,H)-moduleandbecomesapermutationZOut_G(H)-modulevia theisomorphismI(H,H)∼=ZOutG(H).TheactionofZOutG(H) onL(–,H) issuchthat

anelementγ∈ Out_G(H) sendsthebasiselement_Gind_Hβ tothebasiselement_Gind_Hβγ−1. Letusrecallthenotionofasuborbitmaponapermutationmodule.LetΓ beafinite group andΩ afinite Γ-set. Forω1,ω2 ∈ Ω,let(ω1,ω2) betheZ-linearendomorphism

(ω1, ω2)ω =



ω1 if ω = ω2,

0 if ω= ω2.

TheendomorphismringEnd_ZΓ(ZΩ) has aZ-basisconsistingof themaps $(ω1, ω2) =



(ω1,ω2)∈Ω×Ω : (ω1,ω2)=Γ(ω1,ω2)

(ω₁, ω₂) .

We call $(ω1,ω2) a suborbit map on ZΩ. Since $(ω1,ω2) = $(ω1,ω2) if and only if

(ω1,ω2)=Γ(ω1,ω2),wehave

End_ZΓ(ZΩ) = 

(ω1,ω2)∈ΓΩ×Ω

Z $(ω1, ω2)

where the notation indicatesthat (ω1,ω2) runsover representatives of the Γ-orbits of

Ω× Ω.

Proposition4.3. LetH ∈ K.Thenthereisabijectivecorrespondence between:

(a) thetransitivemorphisms_Findα

VresG in N suchthat V ∼=G H,

(b) thesuborbitmaps$ on thepermutationOut_G(H)-module L(−,H). Thecorrespondence_Findα

VresG↔ $ ischaracterized bytheconditionthat FindαVresG actsonL(–,H) asapositiveinteger multipleof $.

Proof. Fix F,G ∈ K. Two transitive morphisms _Findα

VresG and Findα



Vres_G coincide

provided[α,V ]= [α,V], inotherwords, there existf ∈ F and g∈ G suchthatV =

g_{V and α} _{= c(f )αc(g}−1_{). Twosuborbit maps $(}

Find

μ

H,GindνH) and $(Find μ H,Gindν



H)

coincide provided there exists γ ∈ OutG(H) such that μ = μγ−1 and ν = νγ−1, in other words, there exist f ∈ F and g ∈ G and γ ∈ Aut_G(H) such thatμ = c(f )μγ−1 andν = c(g)νγ−1.

Given=_Findα

VresG,we defineasuborbitmap $= $(Find μ

H,GindHν) asfollows. We

chooseaG-isomorphismν0: V ← H andextendν0 toahomomorphismν : G← H by

composingwiththeinclusionG← V .Wedefine μ= αν.Thesuborbitmap$ does not dependonthechoiceofν0 because,ifwereplaceν0 withν0γ−1 forsomeγ∈ AutF(H),

then μ and ν arereplaced by μγ−1 and νγ−1. To complete the demonstration that$ depends only on , we must show independence from the choice of α and V . Suppose that  = _Findα

VresG. Let f and g be such that V = gV and α = c(f )αc(g−1). Let ν0 = c(g)ν0. Extending ν0 to a homomorphism ν : G ← H and defining μ = αν,

thenν = c(g)ν and μ = c(f )μ.So$(_Find_Hμ,_Gind_Hν)= $. Wehaveestablishedthat$ dependsonlyon.

Conversely,given asuborbit map $= $(_Find_Hμ,_Gindν

H),we define atransitive

mor-phism  = _Findα

VresG as follows. Let V = ν(H), letν0 : V ← H be the isomorphism

restrictedfrom ν and letα = μν₀−1 : F ← V .We mustshow that dependsonly on$ and noton thechoiceof μ and ν.Supposethat $= $(_Find_Hμ,_Gind_Hν). Let f ,g, γ be

such thatμ = c(f )μγ−1 and ν = c(g)νγ−1. Letting V = ν(H), then V =gV . The isomorphismV ← H restrictedfrom ν isν₀ = c(g)ν0γ−1.Definingα= μν0−1,then

α= c(f )μγ−1γν₀−1c(g−1) = c(f )αc(g−1) .

So_Findα_Vres_G= . Wehaveestablishedthat depends onlyon$.

It is easy to check that the abovefunctions  → $ and $ →  are mutual inverses. Now suppose that  ↔ $. It remains only to show that the action of  is a positive integer multiple of $. Since the action of N on L(–,H) commutes with the action of ZOutG(H),theactionof  isaZ-linearcombination ofsuborbitmaps. ByLemma 4.2,

any suborbitmap with non-zerocoefficient has apositive integercoefficient. Let $1 =

$(_Findμ1

H ,GindHν1) be a suborbit map with non-zero coefficient. We are to show that

$1 = $. Since σ()Gind ν1

H = 0, Lemma 4.2 implies thatV = xν1(H) for some x∈ G.

Replacing ν1 withc(x)ν1 doesnotchangeGind ν1

H, so wemayassumethatV = ν1(H).

Then ν1 = νγ−1 for some γ ∈ OutG(H). That is to say, Gind ν1

H belongs to the same

Out_G(H)-orbit as _Gindν

H. So we may assume that Gind ν1

H = GindHν. By Lemma 4.2

again, _Findμ1

H =Find

αc(g)ν

H for someg ∈ NG(V ). Theproof of the well-definednessof

thefunction→ $ nowshows that→ $1,inother words,$1= $. 2

Proposition 4.4. Therepresentation σ :N → End_I(L) isinjective.

Proof. Letκ∈ N .Recallthatκ=_{F,G F}κ_G asasumwithonlyfinitelymanynon-zero terms.Each term_Fκ_G∈ N (F,G) actsonL asamap

σ(_Fκ_G) : 

H∈K

L(F, H)←  H∈K

L(G, H) .

Suppose thatκ= 0.We must show thatσ(κ) = 0.Wemayassume thatκ=_Fκ_G for someF,G∈ K.Write κ = n  j=1 λj.Find αj VjresG

as aZ-linear combination ofdistinct transitivemorphisms inN witheach λj = 0.Let V be maximalamong theVj.Replacing someoftheVj withG-conjugatesifnecessary,

we can choosetheenumeration suchthatVj = V forj ≤ m andVj G V for j > m.

Invoking Proposition 4.3, let $j be the suborbit map corresponding to _Findα_Vjres_G for j ≤ m.Notethatthe$jaremutuallydistinct.ByLemma 4.2,σ(Find

αj

VjresG) annihilates

L(–,V ) forj > m. So,byProposition 4.3, thereexist non-zerointegersz1,. . . ,zm such

thattherestrictionofσ(κ) toL(–,V ) ism_j=1λjzj$j.Perforce,σ(κ)= 0. 2

Proof. Bythepreviousproposition,the K-linearmap σ is injective.Weargueby com-parisonofdimensions.Summingoverrepresentatives H oftheG-isomorphismclassesin K,wehave KI = H KIH , KIH =  H1,H2∈K : H1∼=GH2∼=GH KI(H1, H2) , KL = H KLH , KLH=  H1∈K : H1∼=GH KL(–, H1) .

ThesubalgebraKIHisisomorphictoafullmatrixalgebraoverKI(H,H)∼=KOutG(H).

So

End_KI(KL) ∼= End _HKOutG(H)

 

H

KL(–, H) ∼=

H

End_KOutG(H)(KL(–, H)) .

ThenumberofsuborbitmapsonthepermutationZOut_G(H)-module L(–,H) is

dim_K(End_KOutG(H)(KL(–, H))) =



F,G∈K nF,G_H

wherenF,G_H isthenumberofsuborbitmapsL(F,H)← L(G,H).ByProposition 4.3,the numberoftransitivemorphismsF ← G in N is

dimK(KN (F, G)) = H nF,G_H . Sodim_K(KN ) =  F,G∈K   H nF,G_H = H   F,G∈K nF,G_H = dim_K(End_KI(KL)). 2

Theorem 4.6. The K-linear category KN is locally semisimple. In particular, if K is

finite,then thequiver algebraKN is semisimple.

Proof. Firstsuppose thatK isfinite.AswesawintheproofofProposition 4.5,each al-gebraKIH isisomorphictoafullmatrixalgebraoverthesemisimplealgebraKOutG(H).

SoKI issemisimple.Therefore,End_KI(KL) issemisimple.AnappealtoProposition 4.5 nowcompletestheargumentinthiscase.

Now letK bearbitrary. Leti be anidempotent of KN .Let K0 be afinite subset of

K such that K0 is closed under taking subgroups and i can be expressed in the form

i = _F,G∈K0Fi_G with _Fi_G ∈ KN (F,G). Let N0 be the full subcategory of N such

thatObj(N0)= K0.Since K0 isfinite,thealgebra1N0.KN .1N0 =KN0 issemisimple.

ArguingasintheproofofRemark 4.1,wededucethatiKN i is semisimple. 2 Corollary 4.7. There is a bijective correspondence between the isomorphism classes of simple KN -functorsS and theblocksb ofKN suchthat S↔ b providedS belongs tob.

Proof. We aretoshowthat,givensimpleKN -functors S andS belongingtothesame block b of KN , then S ∼= S. By Theorem 4.6, this is already clear when K is finite. Generally,byProposition 2.6,thereexistsafullsubcategoryK ofKN suchthatObj(K) is finite and theK-functors 1_KS and 1_KS are non-zero and belong to the sameblock ofK.LetK0bethesetofsubgroupsofelementsofObj(K).LetN0bethefullsubcategory

ofN withObj(N0)= K0.Proposition 2.5,appliedtothesubcategoryK ofKN0,tellsus

that1_N0S and1N0SbelongtothesameblockofKN0.ButK0isfinite,so1N0S ∼= 1N0S.

ByProposition 2.4,S ∼= S. 2

5. SimplefunctorsofdeflationMackeycategories

Let F be a Mackeysystem on K. Let G be the ordinary Mackey system such that the morphisms inG are the injective morphisms in F. Consider the deflation Mackey category M=M←_F and theordinary Mackeycategory N = M_G.We shall show that the simple KM-functorsrestrict to and are inflated from the simple KN -functors. By similar arguments, asimilar resultholds for theK-linear extension KM→_F of the infla-tion Mackeycategory M→_F. A variant ofthis result, inadifferent scenario, appearsin Yaraneri [11, 3.10]. Another related result is Boltje–Danz [2, 6.5], which asserts that, for G ∈ K,the simpleKM(G,G)-modules restrict to and are inflated from the simple KN (G,G)-modules.

For F,G ∈ K, let V ≤ G and let α : F ← V be a morphism in F. Following Boltje–Danz[2,4.2], wedefineaK-linearmap

ρF,G_α,V : KM(F, G) → K

suchthat,givenanF –G-bisetX whoseisomorphismclass[X] belongstoM(F,G),then

ρF,G_α,V[X] =|XS(α,V )_|/|C

G(V )|

where XS(α,V ) _{denotes the set of elements of X fixed by S(α,V ). Let} KJ (F,_{G) be}

the K-submodule of KM(F,G) consisting of those elements x∈ KM(F,G) such that

ρF,G_α,V(x)= 0 whenever α isinjective.AsaK-submoduleofKM,wedefine KJ = 

F,G∈K

KJ (F, G) .

Proposition 5.1. Wehave KM=KN ⊕ KJ , furthermore,KJ isan idealof KM. IfK

is finite,thenKJ = J(KM), theJacobsonradical.

Proof. Following[2,Section4],weshallconstructanisomorphiccopyKM of thealgebra KM. ForF,G∈ K,we introduceaK-moduleKM(F, G) withabasisconsisting of the symbols (α,V )F,Gwhere (α,V )∈ PF,GF .Wemakethedirectsum



KM = 

F,G∈K

 KM(F, G) becomeanalgebrawithmultiplicationgiven by

(α, V )F,G(β, W )G,H =



(αβ, W )F,H|CG(V )|/|G| if G = G and V = β(W ),

0 otherwise.

TheactionofF× G onP_F,GF givesrisetoapermutationactionofF× G onKM(F, G).

Let



KM(F, G) = KM(F, G)F×G_.

As an element of KM(F, G), let[α,V ]+_F,G denote the sum of the F×G-conjugates of (α,V )F,G. Theorbitsums[α,V ]+F,GcompriseabasisforKM(F, G),indeed,

 KM(F, G) =  [α,V ]∈PFF,G K . [α, V ]+ F,G. Asasubalgebra ofKM, wedefine  KM =  F,G∈K  KM(F, G) .

Itisshownin[2,4.7]thatthereisanalgebraisomorphismρ:KM→ KM givenbythe mapsρF,G:KM(F,G)→ KM(F,G) suchthat,forx∈ KM(F,G),wehave

ρF,G(x) =  (α,V )∈PF,GF ρF,G_α,V(x)(α, V )F,G=  [α,V ]∈PFF,G ρF,G_α,V(x)[α, V ]+_F,G.

Let KJ be the ideal of KM spanned by those elements (α,V )F,G such that α is

non-injective.LetKJ =  KM ∩ KJ ,whichisanidealofKM. Thus,KJ is spannedby those orbit sums [α,V ]+_F,G such that α isnon-injective. Bythe definitions of KJ and 

KJ ,wehaveKJ = ρ( ⊕KJ ).ThereforeKJ isanidealofKM.

Given (α,V ) ∈ P_F,GF with α non-injective then, for all (F,G)-bisets X such that [X]∈ N (F,G),wehaveXS(α,V ) _=∅,henceρF,G

α,V[X]= 0.Soρ

F,G

α,V(KN (F,G))={0}.It

followsthatρ(KN (F,G))∩ KJ (F,G)={0}.Byconsideringdimensions,KM(F, G)=

ρ(KN (F,G)) ⊕ KJ (F,G). So KM = ρ(KN ) ⊕ KJ = ρ(KN )⊕ ρ(KJ ). Therefore, KM=KN ⊕ KJ .

NowsupposethatK isfinite.Givenanon-zeroproduct(α1,V1)F1,G1. . . (αn,Vn)Fn,Gn

of basis elements of KJ , then each Vj = αj+1(Vj+1), which is smaller than Vj+1



KJ is nilpotent. ThereforeKJ is nilpotent, inother words, KJ ≤ J(KM). But The-orem 4.6 implies that KN is semisimple. So KN ∩ J(KM) = {0}. We deduce that KJ = J(KM). 2

Theorem5.2.Let(G,V ) beaseedforF overK.ThenthesimpleKM-functorS_G,VKM and thesimple KN -functorS_G,VKN arerelated by

S_G,VKN ∼=_KNRes_KM(S_G,VKM) , S_G,VKM∼=_KMInf_KN(S_G,VKN)

wheretheinflationisviathecanonicalalgebraepimorphismKM→ KN withkernelKJ .

Proof. By the latest proposition, the description of the inflation functor _KMInf_KN makes sense.The KM-functorS =_KMInf_KN(S_G,VKN) issimpleand S(G)∼= S_G,VKN(G) ∼=

V as FOutG(G)-modules. By Theorem 3.7, S ∼= S_G,VKM. It follows that, S_G,VKN ∼=

KNResKM(S). 2

Theorem 5.3. Every idempotent of Z(Π_{KM) belongs to Z(}Π_{KN ). In particular, every}

block of KM isacentralidempotent ofΠ_{KN .}

Proof. AlemmainBoltje–Külshammer[3,5.2]assertsthat,givenasubringΓ ofaringΛ suchthatΛ= Γ⊕ J(Λ),theneveryidempotentofZ(Λ) belongstoZ(Γ).This,together with Proposition 5.1, immediately implies therequired conclusioninthe casewhere K is finite.For arbitrary K, let e be an idempotent of Z(ΠKM).Let G ∈ K and letKG

be the set ofsubgroups ofG. LetMG andNG be thefull subcategories of M andN ,

respectively, suchthatObj(MG)= Obj(NG)= KG. Since KG is finite,theidempotent

1_MGe ofZ(

Π_KM

G) mustbelongtoZ(ΠKNG).The(G,G)-coordinateeG ofe coincides

with the(G,G)-coordinate of 1_MGe.SoeG ∈ N (G,G).ByRemark 2.1,e=



G∈KeG,

hencee∈Π_{KN .Bute iscentralin}Π_{KM,soe iscentralin}Π_{KN . 2}

6. Multipleblocks

In Corollary 4.7, we found that,for an ordinary Mackeycategory N ,each blockof KN owns a uniqueisomorphism class ofsimple KN -functors. In this section, weshall giveanexampleofanon-ordinaryMackeycategorysuchthatmostof theblocksofthe K-linear extensionstillown auniqueisomorphismclassofsimplefunctors.

LetF
(K) denotetheMackeysystemonK suchthatthemorphismsinF
(K) arethe homomorphisms between groupsinK. The deflationMackeycategory M
_K =M←_F
(K)

is calledthe complete deflationMackeycategory onK. LetFΔ_{(K) denotetheordinary}

Mackey system on K such that the morphisms in F
(K) are the injective homomor-phisms betweengroups inK. Theordinary Mackeycategory MΔ

K =MF
(K) is called

the complete ordinary Mackey category on K. Weshall giveanexample ofacomplete deflationMackeycategorywhoseK-linearextensionhasp− 1 blocksandp isomorphism

Lemma6.1.ConsiderthecompleteordinaryMackeycategoryMpΔ=MΔ_{1,Cp}.Thereare

exactlyp isomorphism classesofsimpleCMΔ_p-functors.ThecategoryCMΔ_p hasexactly p blocks.

Proof. The first part follows from Theorem 3.7. The second part then follows from Corollary 4.7. 2

AsasteptowardsfindingtheblocksofCM_p
,weshallfirstfindtheblocksofCMΔ

p.

Write c = C = Cp. For 1≤ j ≤ p− 1, letσj be the automorphism of C suchthat c→ cj_{. Let}

α =₁iso₁, τ =_Cind₁, ρ =₁res_C, αj=Ciso σj

C .

ObservethatCMΔ

p hasaC-basis consisting ofthe elements α,τ , ρ,τ ρ, α1,. . . ,αp−1.

Let

e1,1 = α + τ ρ/p .

WeidentifyOut(C) withAut(C).WealsoidentifyOut(C) withtheunitgroup(Z/p)× of the ring Z/p of integers modulo p. Let Irr(COut(C)) denote the set of irreducible COut(C)-characters.Forχ∈ Irr(COut(C)),wedefineeC,χsuchthat,writing1 todenote

thetrivialcharacter,

eC,1 =−τρ/p + 1 p− 1 p−1  j=1 αj

and,whenχ is non-trivial,

eC,χ= 1 p− 1 p−1  j=1 χ(j−1)αj.

Lemma6.2. The blocksof CMΔ

p are e1,1 andeC,χ with χ∈ Irr(COut(C)).

Proof. For G∈ {1,C},letA_C(G) denote thecharacterring ofCG.Since G isabelian, thecharacteralgebraCAC(G) canbeidentifiedwiththeC-moduleoffunctionsG→ C.

Lete1 be theelement ofCA_C(1) such thate1(1)= 1.Let eC₀,. . . ,eC_p−1 betheelements ofCA_C(C) suchthateC

i (ci)= 1 andeCi vanishesoff{ci}.Then{e1} and{eC0,. . . ,eCp−1}

arebasesforCA_C(1) andCA_C(C),respectively. Weshall make use ofthe representation CMΔ

p → EndC(CAC) of theCMΔp-functor

CAC.TheC-moduleCAC=CAC(1)⊕ CAC(C) hasabasisconsistingoftheelementse1

andeC

i for0≤ i≤ p− 1.Wehave

and α,τ , ρ,αj annihilatetheotherbasiselementsofCAC. Letting sC,χ= p−1  i=1 χ(i−1)eC_i

then αj(sC,χ)= χ(j)sC,χ andρ(sC,χ)= τ (sC,χ)= α(sC,χ)= 0.Itis noweasyto check

that,as adirectsumofsimpleCMΔ

p-functors,

CAC= S1,1⊕



χ∈Irr(COut(C)) SC,χ

where S1,1 = span_C{e1,eC0} and SC,χ = span_C{sC,χ}.This is adirect sumof p

mutu-ally distinct simple CMΔ

p-functors. (It is also easy to check thatthe notation here is

compatible with thatwhich appearedin the classification of simple functors in Theo-rem 3.7,butweshallnotbemakinguseofthatfact.)ByLemma 6.1,everyisomorphism class of simple CMΔ

p -functor occurs exactly once in CAC. So the blocksof CMΔp are

precisely theelementsofCMΔ_p thatactastheprojectionstothesimplesummands.By direct calculation, e1,1 actsas theprojection to S1,1, while eC,χ acts as theprojection

to SC,χ. 2

Proposition6.3.TheblocksofCM
_p aree1,1+eC,1andeC,χwithχ∈ Irr(COut(C))−{1}. The blocke1,1+ eC,1 ownsexactly2 isomorphismclassesofsimpleCM
p-functors.Each of theotherp− 1 blocks ownsaunique isomorphismclassof simpleCM
_p-functors.

Proof. By Theorem 5.3, everycentralidempotent of the algebraCM
p =ΠCM
p is a

centralidempotentofthealgebraCMΔ

p =ΠCMΔp.WehaveCM
p =CMΔp ⊕ Cδ where δ = ₁def_C.Sothecentralidempotents ofCM
_p areprecisely thosecentralidempotents ofCMΔ

p whichcommutewithδ.UsingaformulaforcompositioninSection3,weobtain

thecommutation relations

δα = αjδ = τ ρδ = 0 , αδ = δαj = δ , δτ ρ = ρ .

We find that δ does not commute with e1,1 nor with eC,1, but δ does commute with e1,1+ eC,1 andwitheC,χ forχ= 1. SotheblocksofCM
p areasasserted.

ByTheorem 5.2and theproof ofLemma 6.2,there exist simpleCM
_p-functorsS_1,1

and S_C,χ
thatrestrict to the simpleCMΔ

p functors S1,1 and SC,χ, respectively, where χ∈ Irr(COut(C)).Furthermore,everysimpleCM
_p-functorisisomorphictoS_1,1
orone of the S_C,χ
. Since e1,1+ eC,1 actsas the identity onS1,1 and SC,1, the CM
p-functors S_1,1
andS_C,1
belongto e1,1+ eC,1.Similarly,SC,χ
belongstoeC,χforχ= 1. 2

7. Auniqueblock

Throughout this section, we shall assumethat everyfinite group isisomorphic to a groupinK.Weshallprovethefollowing theorem.

Theorem7.1. Considerthecomplete deflationMackeycategoryM=M
_K.TheK-linear extensionKM has auniqueblock.

Weshallmakeuseofthetheorem ofHartley–Robinson[7], whichimpliesthat,given afinitegroupG andaprimep notdividing|G|,thenthereexistsafinitep-groupP and

asemidirectproductF = G P suchthatOut(F )= 1.Inparticular,everyfinitegroup isaquotient ofafinite groupwithatrivialouter automorphismgroup.

Letb betheblockofKM owningthesimpleKM-functorS_1,1KM.ToproveTheorem 7.1, wemustshowthatb= 1_M.ConsiderthecompleteordinaryMackeycategoryN = MΔ

K.

ByTheorem 5.3, b ∈ Z(ΠKN ). By Remark 2.1, we canwrite b =_G∈KbG with each bG ∈ KN (G,G).Sinceb ownsS1,1KM,the(1,1)-coordinateofb is b1= iso1.

LetP_G,G
denotethesetofpairs(α,V ) suchthatV ≤ G andα : F ← V isa homomor-phism.LetPΔ

G,G denotethesubsetofPG,G
consistingofthose pairs(α,V ) suchthatα

isinjective.InthenotationoftheproofofProposition 5.1,ρ(bG) isalinearcombination

ofelements(α,V )G,G∈ KM(G,G) where(α,V ) runsovertheelements ofPG,G
.Aswe

sawintheproofofProposition 5.1, whenα isnon-injective,ρG,G_α,V(KN (G,G))={0}.In particular,whenα isnon-injective,ρG,G_α,V(bG)= 0.Therefore,

ρ(bG) =



(α,V )∈PΔ

G,G

ρG,G_α,V(b) (α, V )G,G.

Lemma7.2. Let H,K ∈ K andlet π : H← K be a surjectivehomomorphism. Then, in thenotation oftheproof of Proposition 5.1,

[π, K]+_H,K = 

hZ(H)⊆H

(c(h)π, K)H,K.

Proof. EveryH×K-conjugateof(π,K) hastheform(c(h)π,K) forsomeh∈ H. 2

Lemma7.3. ForallG∈ K,wehave  α∈Out(G)

ρG,G_α,G(b) = 1.

Proof. Let π be the homomorphism 1← G. By Lemma 7.2, [π,G]+_1,G = (π,G)1,G. In

particular,(π,G)1,GbelongstoKM and commuteswithρ(b).Therefore

(π, G)1,G= ρ(b1)(π, G)1,G= (π, G)1,Gρ(bG) =



α∈Out(G)

Lemma 7.4.ForallG∈ K and α∈ Out(G),wehave ρG,G_α,G(bG) =



1 if α = 1, 0 otherwise.

Proof. By the theorem of Hartley and Robinson mentioned at the beginning of this section, there exists a group F ∈ K such that Out(F ) = 1 and G is isomorphic to a quotient ofF .Letπ : G← F beasurjectivehomomorphism.Wehave

[π, F ]+_G,Fρ(bF) = [π, F ]+G,Fρ(b) = ρ(b)[π, F ] + G,F = ρ(bG)[π, F ]+G,F . Using Lemma 7.2, [π, F ]+_G,Fρ(bF) =  (β,W )∈PΔ F,F, gZ(G)⊆G ρF,F_β,W(bF)(c(g)π, F )G,F(β, W )F,F = ρF,F_1,F(bF)  gZ(G)⊆G (c(g)π, F )G,F .

Ontheother hand,using Lemma 7.2again,

ρ(bG)[π, F ]+G,F =  (α,V )∈PΔ G,G, gZ(G)⊆G ρG,G_α,V(bG)(α, V )G,G(c(g)π, F )G,F =  α∈Out(G), gZ(G)⊆G ρG,G_α,G(bG)(αc(g)π, F )G,F .

Comparingcoefficients,wededucethatρG,G_α,G(bG)= 0 whenα= 1.Lemma 7.3nowyields ρG,G_1,G(bG)= 1. 2

Lemma 7.5. For G∈ K,let KN<(G,G) be theideal of KN (G,G) spanned by the tran-sitive morphisms that have the form _Gind_Wβ res_G with W < G. Then bG ≡ 1 modulo

KN<(G,G).

Proof. ByProposition 3.1,KN<(G,G) isindeedanidealofKN (G,G).Wecanwrite bG= cG+



α∈Out(G)

bα,G.GisoαG

where cG∈ KN<(G,G) andeachbα,G∈ K.SinceρG,Gα,G(bG)= bα,G,therequired

conclu-sionfollows fromLemma 7.4. 2

The latest lemma implies that, for every seed (G,V ) of KM, the idempotent bG

acts as the identity on S_G,VKM. So b owns S_G,VKM. By Theorem 3.7, b owns every simple KM-functor.Therefore b= 1_M.Theproofof Theorem 7.1iscomplete.

Wementionthat,ifweweretoassumethattheisomorphismclassesinK arethoseof the finitesolvable groups,then theconclusionofTheorem 7.1wouldstillhold because,

intheproofofLemma 7.4,wecouldtakeF tobesolvable.Wedonotknowwhetherthe conclusionofthetheoremstillholdswhentheisomorphismclassesinK arethoseofthe finitep-groups.

References

[1] L.Barker,M.Demirel,Simplefunctorsofadmissiblelinearcategories,preprint,2015.

[2]R.Boltje,S.Danz,Aghostringfortheleft-freedoubleBurnsideringandanapplicationtofusion systems,Adv.Math.229(2012)1688–1733.

[3]R.Boltje, B.Külshammer,Central idempotents ofthebifreeandleft-free doubleBurnside ring, IsraelJ.Math.202(2014)161–193.

[4]S.Bouc,BisetFunctorsforFiniteGroups,LectureNotesinMath.,vol. 1990,Springer,Berlin,2010. [5]A.Díaz,S.Park,Mackeyfunctorsandsharpnessoffusionsystems,Homology,HomotopyAppl.17

(2015)147–164.

[6]J.A.Green,PolynomialRepresentationsofGLn,2nded.,LectureNotesinMath.,vol. 830,Springer,

Berlin,2007.

[7]B.Hartley,D.J.S.Robinson,Onfinitecompletegroups,Arch.Math.(Basel)35(1980)67–74. [8]J.Thévenaz,P.Webb,SimpleMackeyfunctors,in:Proc.of2ndInternationalGroupTheory

Con-ference,Bressanone,1989,Rend.Circ.Mat.PalermoSuppl.23(1990)299–319.

[9]J.Thévenaz,P.J. Webb,Thestructure ofMackeyfunctors,Trans.Amer. Math.Soc.347(1995) 1865–1961.

[10]P.J.Webb,StratificationsofMackeyfunctors,II,J.K-Theory6(2010)99–170.