Une nouvelle formule d’induction canonique pour modules de p-permutation

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Group theory

A new canonical induction formula for p-permutation modules

Une nouvelle formule d’induction canonique pour modules de p-permutation

Laurence Barker, Hatice Mutlu

DepartmentofMathematics,BilkentUniversity,06800Bilkent,Ankara,Turkey

a r t i c l e i n f o a b s t r a c t

Articlehistory:

Received1October2018

Acceptedafterrevision9April2019 Availableonlinexxxx

PresentedbytheEditorialBoard

Applying RobertBoltje’s theoryof canonical induction, wegive arestriction-preserving formulaexpressinganyp-permutationmoduleasaZ[¹/p]^-linearcombinationofmodules induced and inflated from projective modules associated with subquotient groups. The underlyingconstructionsinclude,foranygivenfinitegroup,aringwithaZ-basisindexed by conjugacyclasses oftriples(U,K,E) where U is a subgroup, K isa p^-residue-free normalsubgroupofU ,andE isanindecomposableprojectivemoduleofthegroupalgebra ofU/K .

©^{2019Académiedessciences.PublishedbyElsevierMassonSAS.Allrightsreserved.}

r é s u m é

Enapplication de lathéoriedel’induction canonique deRobert Boltje,nous présentons uneformule stable par restrictionau moyendelaquelle tout modulede p-permutation estexprimésousformedecombinaisonZ[¹/p]^{-linéairedesinductionsdesinflationsdes} modulesprojectifsassociésàdesgroupesdesous-quotients.Lesconstructionsconcernées comprennent,pourtoutgroupefini,unanneauquiauneZ-baseindexéeparlesclassesde conjugaisondestriplets(U,K,E)avecU unsous-groupe,O^p(K)=^K^{U etE unmodule} projectifindécomposabledel’algèbredegroupedeU/K .

©^{2019Académiedessciences.PublishedbyElsevierMassonSAS.Allrightsreserved.}

1. Introduction

WeshallbeapplyingBoltje’stheoryofcanonicalinduction[2] totheringof p-permutation modules.Ofcourse, p isa prime.Weshallbeconsidering p-permutationmodulesforfinitegroupsoveranalgebraicallyclosedfield

F

ofcharacteris- tic p.A reviewofthetheoryofp-permutationmodulescanbefoundinBouc–Thévenaz[6,Section2].

E-mailaddresses:barker@fen.bilkent.edu.tr(L. Barker),hatice.mutlu@bilkent.edu.tr(H. Mutlu).

https://doi.org/10.1016/j.crma.2019.04.004

1631-073X/©^{2019Académiedessciences.PublishedbyElsevierMassonSAS.Allrightsreserved.}

Acanonicalinductionformulaforp-permutationmoduleswasgivenbyBoltje[3,Section4] andshowntobe

Z

-integral.

Itexpressesany p-permutationmodule,uptoisomorphism,asa

Z

-linearcombinationofmodulesinducedfromaspecial kindofp-permutationmodule,namely,the1-dimensionalmodules.

We shallbe inducingfromanotherspecial kindof p-permutationmodule. LetG bea finitegroup. Weunderstandall

F

G-modulestobefinite-dimensional.Anindecomposable

F

G-moduleM issaidtobe exprojective providedthefollowing equivalent conditions hold up to isomorphism: there exists a normal subgroup K



^{G such that M is inflated from a} projective

F

G

/

K -module;thereexistsK



^{G suchthatM isadirectsummandofthe}permutation

F

G-module

F

G

/

K ;every vertexof M acts triviallyon M;some vertexofM actstrivially on M. Generally,an

F

G-module X iscalled exprojective providedeveryindecomposabledirectsummandof X isexprojective.

Theexprojectivemodulesdoalreadyplayaspecialroleinthetheoryofp-permutationmodules.Indeed,theparametriza- tionoftheindecomposable p-permutationmodules,recalledinSection2,characterizesanyindecomposablep-permutation moduleasaparticulardirectsummandofamoduleinducedfromanexprojectivemodule.

We shallgive a

Z [

¹

/

p

]

^{-integral canonicalinductionformula,expressingany} p-permutation

F

G-module,uptoisomor- phism,asa

Z [

¹

/

p

]

^-linearcombinationofmodulesinducedfromexprojectivemodules.Moreprecisely,weshallbeworking withtheGrothendieckringforp-permutation modulesT

(

G

)

andweshallbeintroducinganothercommutativering

T (

^G

)

which, roughly speaking,has a free

Z

-basis consistingoflifts ofinduced modules of indecomposableexprojective mod- ules. We shallconsider a ringepimorphism linG

: T (

^G

) →

^T

(

G

)

and its

Q

-linear extension linG

: QT (

^G

) → Q

^T

(

G

)

. The latter is split by a

Q

-linear map canG

: Q

^T

(

G

) → QT (

^G

)

which, as we shall show, restricts to a

Z [

¹

/

p

]

^{-linear map} canG

: Z[

¹

/

p

]

^T

(

G

) → Z[

¹

/

p

]T (

^G

)

.

Let

K

beafieldofcharacteristiczerothatissufficientlylargeforourpurposes.Tomotivatefurtherstudyofthealgebras

Z [

1

/

p

]T (

G

)

and

K T (

G

)

,wementionthat,notwithstandingtheformulasfortheprimitiveidempotentsof

K

T

(

G

)

inBoltje [4,3.6],Bouc–Thévenaz[6,4.12] and[1],therelationshipbetweenthoseidempotentsandthebasis

{[

^MG_P,E

: (

^P

,

E

) ∈

G

P(

^E

)}

remains mysterious. In Section 4,we shallprove that

K T (

^G

)

is

K

-semisimple aswell as commutative,in other words, the primitive idempotents of

K T (

^G

)

comprise a basis for

K T (

^G

)

. We shall also describe how, via lin_G, each primitive idempotentof

K

T

(

G

)

liftstoaprimitiveidempotentof

K T (

^G

)

.

2. Exprojectivemodules

Weshallestablishsomegeneralpropertiesofexprojectivemodules.

Given H

≤

^{G, we write} GInd_H and _HRes_G to denotethe inductionand restrictionfunctors between

F

G-modules and

F

H -modules.WhenH



^G,wewriteGInf_G/H todenotetheinflationfunctorto

F

G-modulesfrom

F

G

/

H -modules.Givena finitegroup L andanunderstoodisomorphismL

→

^G,wewrite LIsoG todenotetheisogationfunctorto

F

L-modulesfrom

F

G-modules,wemeantosay,LIsoG

(

X

)

isthe

F

L-moduleobtainedfroman

F

G-module X bytransportofstructureviathe understoodisomorphism.

Letusclassifytheexprojective

F

G-modulesuptoisomorphism.Wesaythat G is p^-residue-free providedG

=

^Op

(

G

)

, equivalently, G is generated by the Sylow p-subgroups of G. Let

Q(

^G

)

denote the set of pairs

(

K

,

F

)

, where K is a p^-residue-free normalsubgroup of G and F is an indecomposableprojective

F

G

/

K -module,two such pairs

(

K

,

F

)

and

(

K^

,

F^

)

being deemed the same provided K

=

^{K and F}

∼ =

^{F. We define an} indecomposable exprojective

F

G-module M^K_G^,F

=

GInf_G/K

(

F

)

.Byconsideringvertices,weobtainthefollowingresult.

Proposition2.1.TheconditionM

∼ =

^MG^K,Fcharacterizesabijectivecorrespondencebetween:

(a) theisomorphismclassesofindecomposableexprojective

F

G-modulesM, (b) theelements

(

K

,

F

)

of

Q(

^G

)

.

In particular, for a p-subgroup P of G, the condition E

∼ =

^NG(P)InfN_G(P)/P

(

E

)

characterizes a bijective correspondence between, uptoisomorphism,theindecomposableexprojective

F

NG

(

P

)

-modules E withvertex P andtheindecomposable projective

F

NG

(

P

)/

P -modulesE.Itfollowsthatthewell-knownclassificationoftheisomorphismclassesofindecomposable p-permutation

F

G-modules, asinBouc–Thévenaz [6, 2.9] for instance,can be expressedas inthe next result. Let

P(

^G

)

denotethesetofpairs

(

P

,

E

)

where P isa p-subgroupofG and E isanexprojective

F

NG

(

P

)

-modulewithvertex P ,two suchpairs

(

P

,

E

)

and

(

P^

,

E^

)

beingdeemedthesameprovided P

=

^PandE

∼ =

^E.Wemake

P(

G

)

becomea G-setviathe actions onthecoordinates.Wedefine M^G_P,E tobe theindecomposable p-permutation

F

G-modulewithvertex P inGreen correspondencewithE.

Theorem2.2.TheconditionM

∼ =

^MGP,Echaracterizesabijectivecorrespondencebetween:

(a) theisomorphismclassesofindecomposablep-permutation

F

G-modulesM, (b) theG-conjugacyclassesofelements

(

P

,

E

) ∈ P(

^G

)

.

WenowgiveanecessaryandsufficientconditionforM^G_P,E tobeexprojective.

Proposition2.3.Let

(

P

,

E

) ∈ P(

G

)

.LetK bethenormalclosureofP inG.ThenM^G_P,EisexprojectiveifandonlyifNK

(

P

)

actstrivially onE.Inthatcase,K isp^-residue-free,P isaSylowp-subgroupofK ,wehaveG

=

^NG

(

P

)

K ,theinclusionNG

(

P

)  →

^{G inducesan} isomorphismNG

(

P

)/

NK

(

P

) ∼ =

^G

/

K ,andM^G_P,E

∼ =

^MG^K,F,whereF istheindecomposableprojective

F

G

/

K -moduledetermined,upto isomorphism,bytheconditionE

∼ =

NG(P)Inf_NG(P)/NK(P)Iso_G/K

(

F

)

.

Proof. WriteM

=

^MG_P,E^{.IfM is}exprojectivethen K actstriviallyonM and,perforce,N_K

(

P

)

actstriviallyon E.

Conversely,supposeNK

(

P

)

actstriviallyonE.ThenP ,beingavertexofE,mustbeaSylowp-subgroupofNK

(

P

)

.Hence, P isaSylow p-subgroupofK .ByaFrattiniargument,G

=

^NG

(

P

)

K andwehaveanisomorphismN_G

(

P

)/

N_K

(

P

) ∼ =

^G

/

K as specified.Let X

=

GIndN_G(P)

(

E

)

.Theassumptionon E impliesthat X haswell-defined

F

-submodules

Y

= 

kk

⊗

NG(P)x

:

^x

∈

^E



,

Y^

= 

kk

⊗

NG(P)x_k

:

^xk

∈

^E

, 

kx_k

=

⁰



summed over a left transversalkNK

(

P

) ⊆

^{K . Making use of the}well-definedness, an easy manipulation shows that the actionof NG

(

P

)

on X stabilizes Y andY^.Similarly, K stabilizes Y andY^.So Y andY^ are

F

G-submodulesof X .Since

|

^K

:

^NK

(

P

)|

^{is coprime to p, we have Y}

∩

^Y

=

^{0. Since}

|

^K

:

^NK

(

P

)| = |

^G

:

^NG

(

P

)|

^{, a} consideration ofdimensions yields X

=

^Y

⊕

^Y.

Fixalefttransversal

L

^forNK

(

P

)

inK .Forg

∈

^NG

(

P

)

and

 ∈ L

^{,wecanwrite g}

 = 

ghg with



g

∈ L

^andhg

∈

^NK

(

P

)

. Bytheassumptionon E again,h_gx

=

^{x forallx}

∈

^E.So

g





 ⊗

^x

= 



g

 ⊗

^gx

= 





g

⊗

^gx

= 



 ⊗

^gx

summedover

 ∈ L

.Wehaveshownthat NG(P)ResG

(

Y

) ∼ =

^{E.Asimilarargumentinvolvingasumover}

L

showsthat K acts trivially onY .Therefore,Y

∼ =

^MKG^,F.Ontheotherhand,Y is indecomposablewithvertex P and,bytheGreencorrespon- dence,Y

∼ =

^MGP,E.

2

Weshallbemakinguseofthefollowingclosureproperty.

Proposition2.4.Givenexprojective

F

G-modulesX andY ,thenthe

F

G-moduleX

⊗

FY isexprojective.

Proof. Wemayassumethat X andY areindecomposable.Then X andY are,respectively,directsummandsofpermutation

F

G-moduleshavingtheform

F

G

/

K and

F

G

/

L whereK



^G



^L.ByMackeydecompositionandtheKrull–SchmidtTheorem, everyindecomposabledirectsummandof X

⊗

^{Y isadirectsummandof}

F

G

/(

K

∩

^L

)

.

2

3. Acanonicalinductionformula

Throughout,welet

K

beaclassoffinitegroupsthatisclosedundertakingsubgroups.WeshallunderstandthatG

∈ K

^. Weshallabusenotation,neglectingtousedistinct expressionstodistinguishbetweena linearmap andits extensiontoa largercoefficientring.

Specializingsome generaltheory in Boltje[2], we shallintroduce a commutative ring

T (

^G

)

anda ring epimorphism linG

: T (

^G

) →

^T

(

G

)

.Weshallshowthatthe

Z [

¹

/

p

]

^{-linearextensionlin}G

: Z[

¹

/

p

]T (

^G

) → Z[

¹

/

p

]

^T

(

G

)

hasasplittingcanG

: Z [

¹

/

p

]

^T

(

G

) → Z[

¹

/

p

]T (

G

)

.Asweshallsee,canG istheuniquesplittingthatcommuteswithrestrictionandisogation.

To be clear about the definition of T

(

G

)

, the Grothendieck ring of the category of p-permutation

F

G-modules, we mentionthatthesplitshortexactsequencesarethedistinguishedsequencesdeterminingtherelationson T

(

G

)

.Themul- tiplicationon T

(

G

)

isgivenbytensorproductover

F

.Givena p-permutation

F

G-module X ,we write

[

^X

]

^todenotethe isomorphismclassof X .Weunderstandthat

[

^X

] ∈

^T

(

G

)

.ByTheorem2.2,

T

(

G

) = 

(P,E)∈GP(^G)

Z [

^MGP,E

]

as a direct sum ofregular

Z

-modules, the notation indicating that the index runs over representatives of G-orbits.Let T^ex

(

G

)

denote the

Z

-submodule of T

(

G

)

spanned by the isomorphism classesof exprojective

F

G-modules. ByProposi- tion2.4,T^ex

(

G

)

isasubringofT

(

G

)

.ByProposition2.1,

T^ex

(

G

) = 

(K,F)∈GQ(^G)

Z[

^M_G^K,F

] .

For H

≤

^{G, theinductionand}restriction functors _GInd_H and_HRes_G give rise toinductionandrestriction maps_Gind_H and HresG betweenT

(

H

)

andT

(

G

)

.Similarly, given L

∈ K

^andan isomorphism

θ :

^L

→

^{G,we havean evident isogation} map _Liso^θ_G

:

^T

(

L

) ←

^T

(

G

)

.Inparticular,giveng

∈

^{G,we haveanevident}conjugationmap gHcon^g_H.Boltjenotedthat,when

K

isthe set ofsubgroups of agiven fixed finitegroup, T is aGreen functor in thesense of [2, 1.1c].Forarbitrary

K

,a classofadmitted isogationsmustbe understood,andthe isogationsandinclusionsbetweengroupsin

K

mustsatisfythe

axiomsofacategory. Grantedthat,then T is stillaGreenfunctorinan evidentsense wherebytheconjugationsreplaced byisogations.

Followingaconstruction in[2,2.2],adaptationtothecaseofarbitrary

K

beingstraightforward,weformtheG-cofixed quotient

Z

-module

T (

G

) =  

U≤G

T^ex

(

U

) 

G

whereG actsonthedirectsumviatheconjugationmapsgUcon^g_U.HarnessingtheGreenfunctorstructureofT ,therestric- tionfunctorstructureofT^exandnotingthat T^ex

(

G

)

isasubringofT

(

G

)

,wemake

T

^{becomeaGreenfunctormuchasin} [2,2.2],withtheevident isogationmaps.Inparticular,

T (

^G

)

becomesa ring,commutativebecause T

(

G

)

iscommutative.

GivenxU

∈

^Tex

(

U

)

,wewrite

[

^U

,

xU

]

G todenotetheimageofxU in

T (

^G

)

.Anyx

∈ T (

^G

)

canbeexpressedintheform

x

= 

U≤GG

[

^U

,

x_U

]

G

where thenotationindicates that theindexrunsover representativesofthe G-conjugacyclassesofsubgroupsof G.Note that x determines

[

^U

,

xU

]

^{and x}G butnot, in general, xU. Let

R(

^G

)

be the G-set of pairs

(

U

,

K

,

F

)

where U

≤

^{G and}

(

K

,

F

) ∈ Q(

U

)

.Wehave

T (

G

) = 

U≤GG,(K,F)∈_NG(U)Q(^U)

Z [

^U

, [

^M_U^K,F

]] = 

(U,K,F)∈GR(^G)

Z [

^U

, [

^MK_U^,F

]] .

Wedefinea

Z

-linearmaplinG

: T (

^G

) →

^T

(

G

)

suchthat linG

[

^U

,

xU

] =

GindU

(

xU

)

.Asnotedin[2,3.1],thefamily

(

linG

:

G

∈ K)

^{isa morphismofGreenfunctors lin}

: T →

^{T .In}particular,themap lin_G

: T (

^G

) →

^T

(

G

)

isa ringhomomorphism.

Extendingtocoefficientsin

Q

,weobtainanalgebramap

lin_G

: Q T (

G

) → Q

^T

(

G

) .

Let

π

G

:

^T

(

G

) →

^Tex

(

G

)

bethe

Z

-linearepimorphismsuch that

π

G actsastheidentity onT^ex

(

G

)

and

π

G annihilates theisomorphismclassofeveryindecomposablenon-exprojectivep-permutation

F

G-module.By

Q

-linearextensionagain, weobtaina

Q

-linearepimorphism

π

G

: Q

^T

(

G

) → Q

^Tex

(

G

)

.After[2,5.3a,6.1a],wedefinea

Q

-linearmap

canG

: Q

T

(

G

) → Q T (

G

) , ξ →

¹

|

G

|



U,V≤^G

|

U

|

möb

(

U

,

V

)[

U

,

UresV

( π

V

(

VresG

(ξ )))]

G wheremöb

()

denotestheMöbiusfunctionontheposetofsubgroupsofG.

Theorem3.1.Considerthe

Q

-linearmapcanG. (1) Wehavelin_G◦can_G

=

^idQT(G).

(2) ForallH

≤

^G,wehaveHresG◦canG

=

^canH◦^HresG.

(3) ForallL

∈ K

^andisomorphisms

θ :

^L

←

^G,wehaveLiso^θ_G◦can_G

=

^canL◦Liso^θ_G. (4) canG

[

^X

] = [

^X

]

^forallexprojective

F

G-modulesX .

Thosefourproperties,takentogetherforallG

∈ K

^{,determinethemapscan}G.

Proof. By[2,6.4],part(1)willfollowwhenwehavecheckedthat,foreveryindecomposablenon-exprojectivep-permutation

F

G-module M, wehave

[

^M

] ∈ 

K<G GindK

(Q

T

(

K

))

.By[3,2.1,4.7],we mayassumethat G is p-hypoelementary.By[3, 1.3(b)], M isinducedfromN_G

(

P

)

whereP isavertexofM.ButM isnon-exprojective,so P isnotnormalinG.Thecheck iscomplete.Parts(2),(3),(4)followfromtheproofof[2,5.3a].

2

Parts(2)and(3)ofthetheoremcanbeinterpretedassayingthatcan_∗

:

^T

→ T

^{isamorphismof}restrictionfunctors.It is nothard tocheckthat, when

K

isclosedunderthe takingofquotientgroups, thefunctors T , T^ex,

T

^{can beequipped} withinflationmaps,andthemorphismslin_∗andcan_∗arecompatiblewithinflation.

Thelatesttheoremimmediatelyyieldsthefollowingcorollary.

Corollary3.2.Givenap-permutation

F

G-moduleX ,then

[

^X

] =

¹

|

^G

|



U,V≤G

|

^U

|

^möb

(

U

,

V

)

_Gind_Ures_V

( π

_V

(

_Vres_G

[

^X

])) .

Givenp-permutation

F

G-modules M and X ,withM indecomposable,wewritemG

(

M

,

X

)

todenotethemultiplicity of M asadirectsummandof X .Wewrite

π

G

(

X

)

todenotethedirectsummandof X , well-defineduptoisomorphism,such that

[ π

G

(

X

) ] = π

G

[

^X

]

^.

Lemma3.3.Let

p

beasetofprimes.Supposethat,forallV

∈ K

^,allp-permutation

F

V -modulesY ,allU

^{V suchthatV}

/

U isa cyclic

p

-group,andallV -fixedelements

(

K

,

F

) ∈ Q(

U

)

,wehave

mU

(

M_U^K,F

, π

U

(

UResV

(

Y

))) = 

(J,E)∈Q(V)

mU

(

M_U^K,F

,

UResV

(

M_V^J,E

))

mV

(

M_V^J,E

, π

V

(

Y

)) .

Then,forallG

∈ K

^,wehave

|

^G

|

p^canG

[

^Y

] ∈ T (

G

)

,where

|

^G

|

p^denotesthe

p

^-partof

|

^G

|

^. Proof. Thisisaspecialcaseof[2,9.4].

2

Wecannowprovethe

Z [

¹

/

p

]

-integralityofcanG.

Theorem3.4.The

Q

-linearmapcan_Grestrictstoa

Z [

¹

/

p

]

^-linearmap

Z [

¹

/

p

]

^T

(

G

) → Z[

¹

/

p

]T (

^G

)

.

Proof. Let

p

be the set of primes distinct from p. Let V , Y , U , K , F be as in the latest lemma. We must obtain the equalityinthelemma.WemayassumethatY isindecomposable.IfY isexprojective,then

π

U

(

UResV

(

Y

)) ∼ =

^UResV

(

Y

)

and

π

V

(

Y

) ∼ =

^{X , whencetherequiredequalityisclear.So wemayassumethat Y is} non-exprojective.Then

π

V

(

Y

)

isthe zero module.Itsuffices toshow that M_U^K,F isnot adirectsummand of_URes_V

(

Y

)

.Foracontradiction, supposeotherwise.The hypothesison

|

^V

:

^U

|

^{impliesthat U containstheverticesofY .So Y}

|

VIndU

(

X

)

forsome indecomposable p-permutation

F

U -module X . Bearing in mind that

(

K

,

F

)

is V -stable, a Mackey decomposition argument shows that M^K_U^,F

∼ =

^{X . The} V -stabilityof

(

K

,

F

)

alsoimpliesthat K

^{V .So}

Y

|

VInd_UInf_U/K

(

F

) ∼ =

VInf_V/KInd_U/K

(

F

) .

WededucethatY isexprojective.Thisisacontradiction,asrequired.

2

Proposition3.5.The

Z

-linear maplinG

: T (

^G

) →

^T

(

G

)

issurjective.However,the

Z [

¹

/

p

]

^{-linear mapcan}G

: Z[

¹

/

p

]

^T

(

G

) → Z [

¹

/

p

]T (

^G

)

neednotrestricttoa

Z

-linearmapT

(

G

) → T (

^G

)

.Indeed,puttingp

=

^{3 andG}

=

^SL2

(

3

)

,lettingY betheisomorphi- callyuniqueindecomposablenon-simplenon-projectivep-permutation

F

G-moduleandX theisomorphicallyunique2-dimensional simple

F

Q8-module,thenthecoefficientofthestandardbasiselement

[

^Q8

,

X

]

GincanG

([

^Y

])

^isequalto2

/

3.

Proof. Sinceevery 1-dimensional

F

G-moduleisexprojective,thesurjectivityofthe

Z

-linearmap lin_G followsfromBoltje [3,4.7].Routinetechniquesconfirmthecounter-example.

2

4. The

K

-semisimplicityofthecommutativealgebra

K

T (^G)

Let

I(

^G

)

betheG-setofpairs

(

P

,

s

)

where P isap-subgroupofG ands isap^-elementofNG

(

P

)/

P .Let

K

beafieldof characteristiczerosuchthat

K

hasrootsofunitywhose orderisthe p^-partoftheexponentofG.Choosingandfixingan arbitraryisomorphismbetweenasuitable torsionsubgroupof

K − {

⁰

}

^{andasuitabletorsionsubgroupof}

F − {

⁰

}

^,wecan understandBrauercharactersof

F

G-modulestohavevaluesin

K

.Forap^-elements

∈

^{G,wedefineaspecies}



₁^G_,sof

K

T

(

G

)

, we mean,an algebra map

K

T

(

G

) → K

^{, such that}



₁^G_,s

[

^M

]

^{is thevalue, at s, ofthe Brauer character ofa} p-permutation

F

G-moduleM.Generally,for

(

P

,

s

) ∈ I(

^G

)

,we defineaspecies



_P^G_,s of

K

T

(

G

)

suchthat



^G_P,s

[

^M

] = 

₁^N_,^G_s^(P)/P

_[

M

(

P

) ]

^,where M

(

P

)

denotesthe P -relativeBrauerquotientof M^P.Thenext result,well-known,canbefoundinBouc–Thévenaz[6,2.18, 2.19].

Theorem4.1.Given

(

P

,

s

), (

P^

,

s^

) ∈ I(

^G

)

,then



^G_P,s

= 

_P^G,s^ ifandonlyifwehaveG-conjugacy

(

P

,

s

) =

G

(

P^

,

s^

)

.Theset

{ 

^G_P,s

: (

P

,

s

) ∈

G

I(

G

) }

^{isthesetofspeciesof}

K

T

(

G

)

anditisalsoabasisforthedualspaceof

K

T

(

G

)

.Thedualbasis

{

^eGP,s

: (

^P

,

s

) ∈

G

I(

G

) }

isthesetofprimitiveidempotentsof

K

T

(

G

)

.Asadirectsumoftrivialalgebrasover

K

,wehave

K

T

(

G

) = 

(P,s)∈GI(^G)

K

e^G_P,s

.

Let

J (

^G

)

betheG-setofpairs

(

L

,

t

)

whereL isap^-residue-freenormalsubgroupofG andt isap^-elementofG

/

L.We defineaspecies



_G^L,t of

K

T^ex

(

G

)

suchthat,givenanindecomposableexprojective

F

G-moduleM,then



_G^L,t

[

^M

] =

^{0 unlessM}

istheinflationofan

F

G

/

L-module M,inwhichcase,



_G^L,t isthevalue,att,oftheBrauercharacterofM.Itiseasytoshow that,givena p-subgroup P

≤

^{G anda p-element s}

∈

^NG

(

P

)/

P ,then



_P^G_,s

[

^M

] = 

_G^L,t

[

^M

]

^forallexprojective

F

G-modulesM ifandonlyifL isthenormalclosureofP inG andt isconjugatetotheimage ofs inG

/

L.Hence,viathelatesttheorem, weobtainthefollowinglemma.

Lemma4.2.Given

(

L

,

t

), (

L^

,

t^

) ∈ J (

^G

)

,then



_G^L,t

= 

_G^L,tifandonlyifL

=

^Landt

=

G/Lt^,inotherwords,

(

L

,

t

) =

G

(

L^

,

t^

)

.Theset

{ 

_G^L,t

_{: (}

L

,

t

) ∈

G

J (

^G

) }

^{isthesetofspeciesof}

K

T^ex

(

G

)

anditisalsoabasisforthedualspaceof

K

T^ex

(

G

)

.

Let

K(

^G

)

betheG-setoftriples

(

V

,

L

,

t

)

whereV

≤

^{G and}

(

L

,

t

) ∈ J (

^V

)

.Given

(

L

,

t

) ∈ J (

^G

)

,wedefineaspecies



_G^G_,L,t

of

K T (

^G

)

suchthat,forx

∈ T (

^G

)

expressedasasumasinSection3,



_G^G_,L,t

(

x

) = 

_G^L,t

(

x_G

) .

Generally,for

(

V

,

L

,

t

) ∈ K(

G

)

,wedefineaspecies



^G_V,L,t of

K T (

G

)

suchthat



^G_V,L,t

(

x

) = 

^V_V,L,t

(

VresG

(

x

)) .

Using Lemma4.2,astraightforward adaptationofthe argumentin[6,2.18] gives thenext result. Thisresultalsofollows fromBoltje—Raggi-Cárdenas—Valero-Elizondo[5,7.5].

Theorem 4.3.Given

(

V

,

L

,

t

), (

V^

,

L^

,

t^

) ∈ K(

^G

)

,then



_V^G_,L,t

= 

^G_V,L^,t^ if andonly if

(

V

,

L

,

t

) =

G

(

V^

,

L^

,

t^

)

.Theset

{ 

_V^G_,L,t

: (

V

,

L

,

t

) ∈

G

K(

G

) }

^{isthesetofspeciesof}

K T (

G

)

anditisalsoabasisforthedualspaceof

K T (

G

)

.Thedualbasis

{

^eGV,L,t

: (

^V

,

L

,

t

) ∈

G

K(

^G

)}

^{isthesetofprimitive}idempotentsof

K T (

^G

)

.Asadirectsumoftrivialalgebrasover

K

,wehave

K T (

G

) = 

(V,L,t)∈GK(G)

K

e^G_V,L,t

.

Wehavethefollowingeasycorollaryonliftsoftheprimitiveidempotentse^G_P,s.

Corollary4.4.Given

(

P

,

s

) ∈ I(

^G

)

,thene_^G_P,s,P,sistheuniqueprimitiveidempotente of

K T (

^G

)

suchthatlinG

(

e

) =

^eG_P,s^.

References

[1] L.Barker,Aninversionformulafortheprimitiveidempotentsofthetrivialsourcealgebra,J.PureAppl.Math.(2019),https://doi.org/10.1016/j.jpaa.2019. 04.008,inpress.

[2]R.Boltje,Ageneraltheoryofcanonicalinductionformulae,J.Algebra206(1998)293–343.

[3]R.Boltje,Linearsourcemodulesandtrivialsourcemodules,Proc.Symp.PureMath.63(1998)7–30.

[4] R.Boltje,Representationringsoffinitegroups,theirspeciesandidempotentformulae,preprint.

[5]R.Boltje,G.Raggi-Cárdenas,L.Valero-Elizondo,The−+and−⁺constructionsforbisetfunctors,J.Algebra523(2019)241–273.

[6]S.Bouc,J.Thévenaz,Theprimitiveidempotentsofthep-permutationring,J.Algebra323(2010)2905–2915.