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 KMG owns a unique simple functoruptoisomorphism.Ontheotherhand,weshallshow that, whenF is equivalentto thecategory offinite groups, theK-linearcategoriesKM←F andKM→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:barker@fen.bilkent.edu.tr.
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 MF,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 ΠRM←
F
over R, called the extended quiver algebra of RM←F, whichhas the feature thatevery
RM←F-functorisaΠRM←
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 KMG 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 KMG, 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-functorsandthesimpleKMG-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.
TheunityelementofΠL isthesum
1L=
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∈ΠL asabove,whereupon z∈ Z(ΠL) 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 beablockofΠL.
Theorem2.2. If thealgebraL(G,G)= EndL(G) hasafinite block decompositionforall G∈ Obj(L),then
1L=
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, 1L = 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 ΠL.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(ΠL) 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 ,thenbe∈Ee= 0,henceb coincideswiththeblocke∈Ee. We haveestablishedthebijectivecorrespondenceE↔ b,asrequired. 2
As asubalgebraofΠL,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 ΠL-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 aΠL-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
functorswiththecategoryofΠL-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 suchthat1KT = 0.ThecorrespondenceissuchthatS↔ T provided S ∼= 1KT .
Proof. We have ΠK = 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 idempotentb1K of ΠK 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 1KT and 1KT 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 ea1KT = e1KT = eT (G) = 0, hence a1KT = 0 and,
similarly, a1KT= 0. Therefore1KT and1KT 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,wecallFindαG an ordinaryinduction andwecallGresαF an ordinary restriction. When α is aninclusion F ← G,we omit thesymbol α from the notation, justwritingFindG andGresF.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∩gVisoc(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)resH
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)) .
HenceVresGindβW =
V gβ(W ) Vind
c(g)βg
β−1(Vg)resW. 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,theconjugationmapgV gv → 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 transitivemorphismsFindα
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 transitivemorphismsFindUresβG 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
ThecategoryMG =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, thenFisoγG : 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
OutF(G) = AutF(G)/Inn(G)
Remark3.5.LetF beaMackeysystemonK.GivenF,G∈ K andα,α∈ F(F,G),then thefollowingthreeconditionsareequivalent:thatFindα
G=Findα
G;thatGresαF =Gresα
F;
thatα = α.
Proof. Another equivalentconditionisS(α,G)=F×GS(α,G). 2
Let PF,GF denote the set of pairs (α,V ) where V ≤ G and α ∈ F(F,V ). We allow
F × G toactonPF,GF suchthat
(f,g)(α, V ) = (
(f,g)α,gV )
forf ∈ F andg∈ G.LetPFF,Gdenote thesetofF×G-orbits inPF,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 .FindαVresG.
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
ROutF(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) becomesanROutF(G)-module such thatanelementη∈ OutF(G) actsasGisoηG.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 SRMG,V determined up to iso-morphism by the condition that G is aminimal group for SRM
G,V and SG,VRM(G) ∼= V as ROutF(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 = MG.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= i1L0.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(GindWβresH) = ⎧ ⎨ ⎩ 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 → EndZ(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(FindαVresG)GindHβ =
V gβ(H)⊆G : V ≥gβ(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 HisoγH↔ 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 → EndI(L) , τ : I → EndN(L) . As anN -submoduleof L,wedefine L(–, H) = τ (isoH)L = G∈K L(G, H) .
Each L(–,H) isanI(H,H)-moduleandbecomesapermutationZOutG(H)-modulevia theisomorphismI(H,H)∼=ZOutG(H).TheactionofZOutG(H) onL(–,H) issuchthat
anelementγ∈ OutG(H) sendsthebasiselementGindHβ tothebasiselementGindHβγ−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.
TheendomorphismringEndZΓ(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
EndZΓ(ZΩ) =
(ω1,ω2)∈ΓΩ×Ω
Z $(ω1, ω2)
where the notation indicatesthat (ω1,ω2) runsover representatives of the Γ-orbits of
Ω× Ω.
Proposition4.3. LetH ∈ K.Thenthereisabijectivecorrespondence between:
(a) thetransitivemorphismsFindα
VresG in N suchthat V ∼=G H,
(b) thesuborbitmaps$ on thepermutationOutG(H)-module L(−,H). ThecorrespondenceFindα
VresG↔ $ ischaracterized bytheconditionthat FindαVresG actsonL(–,H) asapositiveinteger multipleof $.
Proof. Fix F,G ∈ K. Two transitive morphisms Findα
VresG and Findα
VresG coincide
provided[α,V ]= [α,V], inotherwords, there existf ∈ F and g∈ G suchthatV =
gV 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 ∈ AutG(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$(FindHμ,GindHν)= $. Wehaveestablishedthat$ dependsonlyon.
Conversely,given asuborbit map $= $(FindHμ,Gindν
H),we define atransitive
mor-phism = Findα
VresG as follows. Let V = ν(H), letν0 : V ← H be the isomorphism
restrictedfrom ν and letα = μν0−1 : F ← V .We mustshow that dependsonly on$ and noton thechoiceof μ and ν.Supposethat $= $(FindHμ,GindHν). Let f ,g, γ be
such thatμ = c(f )μγ−1 and ν = c(g)νγ−1. Letting V = ν(H), then V =gV . The isomorphismV ← H restrictedfrom ν isν0 = c(g)ν0γ−1.Definingα= μν0−1,then
α= c(f )μγ−1γν0−1c(g−1) = c(f )αc(g−1) .
SoFindαVresG= . Wehaveestablishedthat depends onlyon$.
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
OutG(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 → EndI(L) isinjective.
Proof. Letκ∈ N .Recallthatκ=F,G FκG asasumwithonlyfinitelymanynon-zero terms.Each termFκG∈ N (F,G) actsonL asamap
σ(FκG) :
H∈K
L(F, H)← H∈K
L(G, H) .
Suppose thatκ= 0.We must show thatσ(κ) = 0.Wemayassume thatκ=FκG 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 $j be the suborbit map corresponding to FindαVjresG for j ≤ m.Notethatthe$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 ) ismj=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
EndKI(KL) ∼= End HKOutG(H)
H
KL(–, H) ∼=
H
EndKOutG(H)(KL(–, H)) .
ThenumberofsuborbitmapsonthepermutationZOutG(H)-module L(–,H) is
dimK(EndKOutG(H)(KL(–, H))) =
F,G∈K nF,GH
wherenF,GH isthenumberofsuborbitmapsL(F,H)← L(G,H).ByProposition 4.3,the numberoftransitivemorphismsF ← G in N is
dimK(KN (F, G)) = H nF,GH . SodimK(KN ) = F,G∈K H nF,GH = H F,G∈K nF,GH = dimK(EndKI(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,EndKI(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∈K0FiG with FiG ∈ 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 1KS and 1KS are non-zero and belong to the sameblock ofK.LetK0bethesetofsubgroupsofelementsofObj(K).LetN0bethefullsubcategory
ofN withObj(N0)= K0.Proposition 2.5,appliedtothesubcategoryK ofKN0,tellsus
that1N0S 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 = MG.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,Gα,V : KM(F, G) → K
suchthat,givenanF –G-bisetX whoseisomorphismclass[X] belongstoM(F,G),then
ρF,Gα,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,Gα,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 onPF,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,Gα,V(x)(α, V )F,G= [α,V ]∈PFF,G ρF,Gα,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 ) ∈ PF,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-functorSG,VKM and thesimple KN -functorSG,VKN arerelated by
SG,VKN ∼=KNResKM(SG,VKM) , SG,VKM∼=KMInfKN(SG,VKN)
wheretheinflationisviathecanonicalalgebraepimorphismKM→ KN withkernelKJ .
Proof. By the latest proposition, the description of the inflation functor KMInfKN makes sense.The KM-functorS =KMInfKN(SG,VKN) issimpleand S(G)∼= SG,VKN(G) ∼=
V as FOutG(G)-modules. By Theorem 3.7, S ∼= SG,VKM. It follows that, SG,VKN ∼=
KNResKM(S). 2
Theorem 5.3. Every idempotent of Z(ΠKM) belongs to Z(ΠKN ). In particular, every
block of KM isacentralidempotent ofΠKN .
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
1MGe ofZ(
ΠKM
G) mustbelongtoZ(ΠKNG).The(G,G)-coordinateeG ofe coincides
with the(G,G)-coordinate of 1MGe.SoeG ∈ N (G,G).ByRemark 2.1,e=
G∈KeG,
hencee∈ΠKN .Bute iscentralinΠKM,soe iscentralinΠKN . 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 MK =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 classesofsimpleCMΔp-functors.ThecategoryCMΔp hasexactly p blocks.
Proof. The first part follows from Theorem 3.7. The second part then follows from Corollary 4.7. 2
AsasteptowardsfindingtheblocksofCMp,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 =1resC, 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},letAC(G) denote thecharacterring ofCG.Since G isabelian, thecharacteralgebraCAC(G) canbeidentifiedwiththeC-moduleoffunctionsG→ C.
Lete1 be theelement ofCAC(1) such thate1(1)= 1.Let eC0,. . . ,eCp−1 betheelements ofCAC(C) suchthateC
i (ci)= 1 andeCi vanishesoff{ci}.Then{e1} and{eC0,. . . ,eCp−1}
arebasesforCAC(1) andCAC(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)eCi
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 = spanC{e1,eC0} and SC,χ = spanC{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 theelementsofCMΔp thatactastheprojectionstothesimplesummands.By direct calculation, e1,1 actsas theprojection to S1,1, while eC,χ acts as theprojection
to SC,χ. 2
Proposition6.3.TheblocksofCMp aree1,1+eC,1andeC,χwithχ∈ Irr(COut(C))−{1}. The blocke1,1+ eC,1 ownsexactly2 isomorphismclassesofsimpleCMp-functors.Each of theotherp− 1 blocks ownsaunique isomorphismclassof simpleCMp-functors.
Proof. By Theorem 5.3, everycentralidempotent of the algebraCMp =ΠCMp is a
centralidempotentofthealgebraCMΔ
p =ΠCMΔp.WehaveCMp =CMΔp ⊕ Cδ where δ = 1defC.Sothecentralidempotents ofCMp 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. SotheblocksofCMp areasasserted.
ByTheorem 5.2and theproof ofLemma 6.2,there exist simpleCMp-functorsS1,1
and SC,χ thatrestrict to the simpleCMΔ
p functors S1,1 and SC,χ, respectively, where χ∈ Irr(COut(C)).Furthermore,everysimpleCMp-functorisisomorphictoS1,1 orone of the SC,χ . Since e1,1+ eC,1 actsas the identity onS1,1 and SC,1, the CMp-functors S1,1 andSC,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=MK.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-functorS1,1KM.ToproveTheorem 7.1, wemustshowthatb= 1M.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.
LetPG,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,Gα,V(KN (G,G))={0}.In particular,whenα isnon-injective,ρG,Gα,V(bG)= 0.Therefore,
ρ(bG) =
(α,V )∈PΔ
G,G
ρG,Gα,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,Gα,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,Gα,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,Fβ,W(bF)(c(g)π, F )G,F(β, W )F,F = ρF,F1,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,Gα,V(bG)(α, V )G,G(c(g)π, F )G,F = α∈Out(G), gZ(G)⊆G ρG,Gα,G(bG)(αc(g)π, F )G,F .
Comparingcoefficients,wededucethatρG,Gα,G(bG)= 0 whenα= 1.Lemma 7.3nowyields ρG,G1,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 GindWβ resG 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 SG,VKM. So b owns SG,VKM. By Theorem 3.7, b owns every simple KM-functor.Therefore b= 1M.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.
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