Blocks of Mackey categories Journal of Algebra

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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 0021-8693/© 2015ElsevierInc.All rights reserved.

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-functorisa^ΠRM^←_F-module.Wedefinea block ofRM^←_F tobeablockof^ΠRM^←_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 acat- 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) satisfyingFx_G= 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(x_i)_G =

0 or _G(x_i)_F = 0. In that case, we define the sum 

ix_i ∈ ^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) Fx_G.

Theunityelementof^ΠL isthesum

1_L= 

G∈Obj(L)

id_G.

Proofofthenextremarkisstraightforward.

Remark2.1. Anyelementz ofthecentreZ(^ΠL) canbeexpressed asasum

z = 

G∈Obj(L)

zG

where z_G ∈ L(G,G). Conversely, given elements z_G ∈ 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 1_Λ. Inthatcase,we saythatΛ has a finiteblockdecomposition. Wedefinea block ofL to beablockof^ΠL.

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)

blk(L(G, G)) .

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 andf_n= e andeachf_i−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 ^ΠL.Given F,G∈ Obj(L) andx∈ L(F,G), then

bFx = bFx1_L= 

d∈EF

d

x 

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 b1 = 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 asubalgebraof^ΠL,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 MΠ =

GM (G) where an element x ∈ L(F,G) acts on M_Π as M (x),annihilating M (G^) for allobjects G^ distinct from G.Byrestriction, we obtain an L-module M_⊕. Note that LMΠ = M_Π, in other words, LM⊕ = M_⊕. Given anotherL-functor M^, then eachnaturaltransformation M → M^ givesrise, in anevidentway,toa^ΠL-mapMΠ→ MΠ^ whichisalsoanL-mapM_⊕→ M_⊕^.Conversely,

theL-mapsM_⊕ → M_⊕^ coincidewiththe^ΠL-mapsMΠ→ MΠ^ and giverisetonatural transformationsM→ M^.Puttingtheconstructionsinreverse,givenanL-moduleM_⊕ suchthat LM⊕ = M_⊕, we canextendM_⊕ to a^ΠL-module MΠ and we can alsoform anL-functor M suchthat M (G)= idGM_⊕ = idGMΠ. Henceforth, we shall neglectto distinguishbetween M andM_Π andM_⊕.Thatis to say,we identifythecategory ofL functorswiththecategoryof^ΠL-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 ∼= 1_KT .

Proof. We have ^ΠK = 1K.^ΠL.1_K. 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 andS^besimpleK-functors.LetT and T^ betheisomorphicallyuniquesimple L-functorssuchthatS ∼= 1_KT andS^∼= 1_KT^.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 ^ΠK acts as theidentity onS, so ab= a.Similarly,a^b^ = 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 ∈ Obj(L). Let T and T^ be simple L-functors. Then T and T^ belong to the same block

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 ande^T^(G^)= 0.SinceebT (G)= eT (G), wehaveeb= 0.Similarly,e^b= 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 thecon- 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

Find^α_G = [(F× G)/{(α(g), g) : g ∈ G}] , Gres^α_F = [(G× F )/{(g, α(g)) : g ∈ G}]

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

Find^α_Gind^β_H=_Find^αβ_H , _Hres^β_Gres^α_F =_Hres^αβ_F .

Whenα isinjective,wecall_Find^α_G an ordinaryinduction andwecall_Gres^α_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

Fdef_G^α=_Find^α_G , _Ginf_F^α=_Gres^α_F

whichwecall deflation and inflation.Notethat,forarbitraryα,wehavefactorizations

Find^α_G=_Find_α(G)def_G^α, _Gres^α_F =_Ginf_α(G)^α res_F . Whenα isanisomorphism,wewrite

Fiso^α_G=_Find^α_G =_Fres^α_G⁻¹

whichwecall isogation.InC,theidentity morphismonG istheisogationisoG=_Giso¹_G. Given g ∈ G,we letc(g) denoteleft-conjugation byg.Let V,V^ ≤ G. Againusing the aboveformulaforcomposition,werecover thefamiliarMackeyrelation

Vres_Gind_V = 

V gV^⊆G

Vind_V∩gV^iso^c(g)_Vg∩V^res_V .

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_Vres_G =_Find_{α(V )}def_V^αres_G=

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_Vres_Gind^β_Wres_H = 

V gβ(W )⊆G

Find^α_β−1^gc(g)β_(Vg^g)res_H

where α_g: F ← V ∩^gβ(W ) andβ_g: V^g∩ β(W )← β⁻¹(V^g) 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(V^g)) . Hence_Vres_Gind^β_W = 

V gβ(W )

Vind^c(g)β_β−1(V^gg)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,theconjugationmap^gV ^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 transitivemorphisms_Find^α_Vres_G 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_Ures^β_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

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, then_Fiso^γ_G : F ← G isanisomorphism inM^←_F.Sothefirstconditionimpliesthesecond.Trivially,thesecondconditionimplies thethird.Assumethethirdcondition.Letθ : F ← G andφ: G← F bemutuallyinverse isomorphismsinRM^←_F.Writingθ =

iλ_iθ_iandφ=

jμ_jφ_jaslinearcombinationsof 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) whereInn(G) denotesthegroupof innerautomorphismsofG.

Remark3.5.LetF beaMackeysystemonK.GivenF,G∈ K andα,α^∈ F(F,G),then thefollowingthreeconditionsareequivalent:that_Find^α_G=_Find^α_G^;that_Gres^α_F =_Gres^α_F^; thatα = α^.

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

Let PF,G^F denote the set of pairs (α,V ) where V ≤ G and α ∈ F(F,V ). We allow F × G toactonPF,G^F suchthat

(f,g)(α, V ) = (_(f,g)α,^gV )

forf ∈ F andg∈ G.LetP^FF,Gdenote thesetofF×G-orbits inPF,G^F .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 ]∈P^FF,G

R ._Find^α_Vres_G.

Proof. For V,V^ ≤ G and α ∈ F(F,V ) and α^ ∈ F(F,V^), we have _Find^α_Vres_G =

Find^α_V^res_G 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∈ OutF(G) actsas_Giso^η_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 S^RM_G,V determined up to iso- morphism by the condition that G is aminimal group for S_G,V^RM and S_G,V^RM(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 (G,V )↔ S provided S ∼= S_G,V^RM.

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,thenthefollowingtwoconditionsareequiv- alent:

(a) ForeveryfulllinearsubcategoryL0 ofL withonlyfinitelymanyobjects,thequiver algebraL0is semisimple.

(b) ThealgebraiLi issemisimpleforeveryidempotenti ofthequiveralgebraL.

Proof. If each iLi is semisimple then, given L0, we have L0 = 1_L0.L.1_L0, which is semisimple.Conversely,supposethateachL0issemisimple.Giveni,letL0beasubcat- egoryofL suchthatObj(L0) isfiniteandi hastheformi=

F,G∈Obj(L0) Fi_Gwitheach

Fi_G∈ L(F,G).Then1_L0i= i= i1_L0.SincethealgebraL0= 1_L0.L.1_L0 issemisimple, thealgebraiLi= i1_L0.L.1_L0i 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_H^β whereβ runsovertheelementsofG(G,H).Itistobeunderstoodthat_Gind_H^β =

Gind_H^β ifandonlyifβ = β^.Thus

L(G, H) = 

β∈G(G,H)

ZGind_H^β .

WedefineaZ-module

L = 

G,H∈K

L(G, H) .

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

π(_Gind_W^βres_H) =

⎧⎨

⎩

Gind_W^β 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(_Find^α_Vres_G)_Gind_H^β = 

V gβ(H)⊆G : V ≥^gβ(H)

Find_H^αc(g)β.

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

Z_Jiso^δ_K

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

I(H, H) ∼=ZOutG(H) .

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

τ (_Kiso^γ_J)_Gind_H^β =

⎧⎨

⎩

Gind_K^βγ−1 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) sendsthebasiselement_Gind_H^β tothebasiselement_Gind_H^βγ−1. Letusrecallthenotionofasuborbitmaponapermutationmodule.LetΓ beafinite group andΩ afinite Γ-set. Forω₁,ω₂ ∈ Ω,let(ω₁,ω₂) betheZ-linearendomorphism of ZΩ suchthat,givenω∈ Ω,then

(ω1, ω2)ω =



ω1 if ω = ω2, 0 if ω= ω2.

TheendomorphismringEnd_ZΓ(ZΩ) has aZ-basisconsistingof themaps

$(ω₁, ω₂) = 

(ω₁^,ω₂^)∈Ω×Ω : (ω^1,ω^₂)=Γ(ω1,ω2)

(ω^₁, ω^₂) .

We call $(ω₁,ω₂) a suborbit map on ZΩ. Since $(ω₁,ω₂) = $(ω₁^,ω^₂) if and only if (ω₁,ω₂)=_Γ(ω^₁,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_Find^α_Vres_G in N suchthat V ∼=_G H, (b) thesuborbitmaps$ on thepermutationOut_G(H)-module L(−,H).

Thecorrespondence_Find^α_Vres_G↔ $ ischaracterized bytheconditionthat _Find^α_Vres_G actsonL(–,H) asapositiveinteger multipleof $.

Proof. Fix F,G ∈ K. Two transitive morphisms _Find^α_Vres_G and _Find^α_V^res_G 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,_Gind^ν_H) and $(_Find^μ_H^,_Gind^ν_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=_Find^α_Vres_G,we defineasuborbitmap $= $(_Find_H^μ,_Gind_H^ν) asfollows. We chooseaG-isomorphismν₀: V ← H andextendν₀ toahomomorphismν : G← H by composingwiththeinclusionG← V .Wedefine μ= αν.Thesuborbitmap$ 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  = _Find^α_V^res_G. 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^ = α^ν^, thenν^ = c(g)ν and μ^ = c(f )μ.So$(_Find_H^μ,_Gind_H^ν)= $. Wehaveestablishedthat$ dependsonlyon.

Conversely,given asuborbit map $= $(_Find_H^μ,_Gind_H^ν),we define atransitive mor- phism  = _Find^α_Vres_G as follows. Let V = ν(H), letν₀ : V ← H be the isomorphism restrictedfrom ν and letα = μν₀⁻¹ : F ← V .We mustshow that dependsonly on$ and noton thechoiceof μ and ν.Supposethat $= $(_Find_H^μ,_Gind_H^ν). 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^= μ^ν₀^−1,then

α^= c(f )μγ⁻¹γν₀⁻¹c(g⁻¹) = c(f )αc(g⁻¹) . So_Find^α_V^res_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 =

$(_Find_H^μ1,_Gind_H^ν1) be a suborbit map with non-zero coefficient. We are to show that

$1 = $. Since σ()_Gind_H^ν1 = 0, Lemma 4.2 implies thatV = ^xν1(H) for some x∈ G.

Replacing ν1 withc(x)ν1 doesnotchange_Gind_H^ν1, so wemayassumethatV = ν1(H).

Then ν1 = νγ⁻¹ for some γ ∈ OutG(H). That is to say, _Gind_H^ν1 belongs to the same Out_G(H)-orbit as _Gind_H^ν. So we may assume that _Gind_H^ν1 = _Gind_H^ν. By Lemma 4.2 again, _Find_H^μ1 =_Find_H^αc(g)ν 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 Fκ_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

λ_j._Find^α_V^j

jres_G

as aZ-linear combination ofdistinct transitivemorphisms inN witheach λ_j = 0.Let V be maximalamong theV_j.Replacing someoftheV_j withG-conjugatesifnecessary, we can choosetheenumeration suchthatVj = V forj ≤ m andVj G V for j > m.

Invoking Proposition 4.3, let $j be the suborbit map corresponding to _Find^α_V^jres_G for j ≤ m.Notethatthe$_jaremutuallydistinct.ByLemma 4.2,σ(_Find^α_Vj^jres_G) annihilates L(–,V ) forj > m. So,byProposition 4.3, thereexist non-zerointegersz₁,. . . ,z_m such thattherestrictionofσ(κ) toL(–,V ) ism

j=1λjzj$j.Perforce,σ(κ)= 0. 2

Proposition4.5. IfK isfinitethentherepresentation σ :KN → EndKI(KL) isbijective.