A new canonical induction formula for p-permutation modules [Une nouvelle formule d'induction canonique pour modules de p-permutation]

(1)

Contents lists available atScienceDirect

C. R.

Acad.

Sci.

Paris,

Ser. I

www.sciencedirect.com

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

Department of Mathematics, Bilkent University, 06800 Bilkent, Ankara, Turkey

a

r

t

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c

l

e

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n

f

o

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t

Article history:

Received1October2018

Acceptedafterrevision9April2019 Availableonline24April2019 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 .

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,Op₍_K₎₌_K_{U et}_{E un}_module projectifindécomposabledel’algèbredegroupedeU/K .

1. Introduction

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

F

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

E-mail addresses:barker@fen.bilkent.edu.tr(L. Barker),hatice.mutlu@bilkent.edu.tr(H. Mutlu). https://doi.org/10.1016/j.crma.2019.04.004

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Acanonicalinductionformulaforp-permutationmoduleswasgivenbyBoltje[3,Section4] andshowntobe

Z-integral.

Itexpressesany p-permutationmodule,uptoisomorphism,asa

Z-linear

combinationofmodulesinducedfromaspecial 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 isadirectsummandofthepermutation

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,the parametriza-tionoftheindecomposable p-permutationmodules,recalledinSection2,characterizesanyindecomposablep-permutation moduleasaparticulardirectsummandofamoduleinducedfromanexprojectivemodule.

We shallgive a

Z

[

1

/

p

]

-integral canonicalinductionformula,expressingany p-permutation

F

G-module,upto isomor-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

: Q

T (

G

)

→ Q

T

(

G

)

. The latter is split by a

Q-linear

map canG

: Q

T

(

G

)

→ Q

T (

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

KT

(

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 linG, each primitive idempotentof

KT

(

G

)

liftstoaprimitiveidempotentof

K

T (

G

)

.

2. Exprojectivemodules

Weshallestablishsomegeneralpropertiesofexprojectivemodules.

Given H

≤

G, we write GIndH and HResG to denotethe inductionand restrictionfunctors between

FG-modules

and

F

H -modules.WhenH

G,wewrite

G

InfG/H todenotetheinﬂationfunctorto

F

G-modulesfrom

FG

/

H -modules.Givena ﬁnitegroup L andanunderstoodisomorphismL

→

G,wewrite LIsoG todenotetheisogationfunctorto

F

L-modulesfrom

F

G-modules,wemeantosay,

L

IsoG

(

X

)

isthe

F

L-moduleobtainedfroman

FG-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 MK_G,F

=

GInfG/K

(

F

)

.Byconsideringvertices,weobtainthefollowingresult.

Proposition2.1.TheconditionM

∼

₌

M_GK,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)InfNG(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

FG-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

FN

G

(

P

)

-modulewithvertex P ,two suchpairs

(

P

,

E

)

and

(

P

,

E

)

beingdeemedthesameprovided P

=

PandE

∼

=

E.Wemake

P(

G

)

becomea G-setviathe actions onthecoordinates.Wedeﬁne MG

P,E tobe theindecomposable p-permutation

F

G-modulewithvertex P inGreen correspondencewithE.

Theorem2.2.TheconditionM

∼

₌

MG

P,Echaracterizesabijectivecorrespondencebetween:

(a) theisomorphismclassesofindecomposablep-permutation

FG-modules

M,

(b) theG-conjugacyclassesofelements

(

P

,

E

)

∈

P(

G

)

.

(3)

Proposition2.3.Let

(

P

,

E

)

∈

P(

G

)

.LetK bethenormalclosureofP inG.ThenMG_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 ,andMGP,E

∼

=

M

K,F

G ,whereF istheindecomposableprojective

FG

/

K -moduledetermined,upto isomorphism,bytheconditionE

∼

₌

NG(P)InfNG(P)/NK(P)IsoG/K

(

F

)

.

Proof. WriteM

=

MG

P,E.IfM isexprojectivethen K actstriviallyonM and,perforce,NK

(

P

)

actstriviallyon E.

Conversely,supposeNK

(

P

)

actstriviallyonE.ThenP ,beingavertexofE,mustbeaSylowp-subgroupofNK

(

P

)

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

=

NG

(

P

)

K andwehaveanisomorphismNG

(

P

)/

NK

(

P

)

∼

=

G

/

K as speciﬁed.Let X

=

GIndNG(P)

(

E

)

.Theassumptionon E impliesthat X haswell-deﬁned

F-submodules

Y

=

kk

⊗N

G(P)x

:

x

∈

E

,

Y

=

kk

⊗N

G(P)xk

:

xk

∈

E

,

kxk

=

0

summed over a left transversalkNK

(

P

)

⊆

K . Making use of thewell-deﬁnedness, an easy manipulation shows that the actionof NG

(

P

)

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

FG-submodules

of 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,hgx

=

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

∼

₌

MK_G,F.Ontheotherhand,Y is indecomposablewithvertex P and,bytheGreen correspon-dence,Y

∼

₌

MG

P,E.

2

Weshallbemakinguseofthefollowingclosureproperty.

Proposition2.4.Givenexprojective

FG-modules

X andY ,thenthe

FG-module

X

⊗

FY isexprojective.

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

F

G-moduleshavingtheform

F

G

/

K and

FG

/

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

]

-linearextensionlinG

: 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

)

.The mul-tiplicationon T

(

G

)

isgivenbytensorproductover

F.

Givena p-permutation

FG-module

X ,we write

[

X

]

todenotethe isomorphismclassof X .Weunderstandthat

[

X

]

∈

T

(

G

)

.ByTheorem2.2,

T

(

G

)

=

(P,E)∈GP(G)

Z

[

MG_P_,_E

]

as a direct sum ofregular

Z-modules,

the notation indicating that the index runs over representatives of G-orbits.Let Tex

(

G

)

denote the

Z-submodule

of T

(

G

)

spanned by the isomorphism classesof exprojective

F

G-modules. By Proposi-tion2.4,Tex

(

G

)

isasubringofT

(

G

)

.ByProposition2.1,

Tex

(

G

)

=

(K,F)∈GQ(G)

Z[

M_GK,F

] .

For H

≤

G, theinductionandrestriction functors GIndH andHResG give rise toinductionandrestriction maps

G

indH and HresG betweenT

(

H

)

andT

(

G

)

.Similarly, given L

∈ K

andan isomorphism

θ

:

L

→

G,we havean evident isogation map _Lisoθ_G

:

T

(

L

)

←

T

(

G

)

.Inparticular,giveng

∈

G,we haveanevidentconjugationmap g_Hcon

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

(4)

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

Tex

(

U

)

_G

whereG actsonthedirectsumviatheconjugationmapsg_Ucong_U.HarnessingtheGreenfunctorstructureofT ,the

restric-tionfunctorstructureofTexandnotingthat Tex

(

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

,

xU

]G

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

[

U

,

xU

]

and xG 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_UK,F

]] =

(U,K,F)∈GR(G)

Z

[

U

,

[

MK_U,F

]] .

Wedeﬁnea

Z-linear

maplinG

:

T (

G

)

→

T

(

G

)

suchthat linG

[

U

,

xU

]

=

GindU

(

xU

)

.Asnotedin[2,3.1],thefamily

(

linG

:

G

∈ K)

isa morphismofGreenfunctors lin

:

T →

T .Inparticular,themap linG

:

T (

G

)

→

T

(

G

)

isa ringhomomorphism. Extendingtocoeﬃcientsin

Q,

weobtainanalgebramap

linG

: Q

T

(

G

)

→ Q

T

(

G

) .

Let

πG

:

T

(

G

)

→

Tex

₍

_G

₎

_be_the

_Z-linear

_epimorphism_such _that

_πG

_acts_as_the_identity _on_Tex

₍

_G

₎

_and

_πG

_annihilates theisomorphismclassofeveryindecomposablenon-exprojectivep-permutation

F

G-module.By

Q-linear

extensionagain, weobtaina

Q-linear

epimorphism

πG

: Q

T

(

G

)

→ Q

Tex

(

G

)

.After[2,5.3a,6.1a],wedeﬁnea

Q-linear

map

canG

: Q

T

(

G

)

→ Q

T

(

G

) , ξ

→

1

|

G

|

U,V≤G

|

U

|

möb

(

U

,

V

)[

U

,

UresV

(

π

V

(V

resG

(ξ )))]G

wheremöb

()

denotestheMöbiusfunctionontheposetofsubgroupsofG.

Theorem3.1.Considerthe

Q-linear

mapcanG.

(1) WehavelinG◦canG

=

idQT(G).

(2) ForallH

≤

G,wehave

H

resG◦canG

=

canH◦HresG.

(3) ForallL

∈ K

andisomorphisms

θ

:

L

←

G,wehave_Lisoθ_G_◦can_G

=

can_L_◦_Lisoθ_G.

(4) canG

[

X

]

= [

X

]

forallexprojective

F

G-modulesX .

Thosefourproperties,takentogetherforallG

∈ K

,determinethemapscanG.

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

F

G-module M, wehave

[

M

]

∈

_K_<_{G G}indK

(Q

T

(

K

))

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

(

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

isamorphismofrestrictionfunctors.It is nothard tocheckthat, when

K

isclosedunderthe takingofquotientgroups, thefunctors T , Tex_,

_T

_can _be_equipped withinﬂationmaps,andthemorphismslin_∗andcan_∗arecompatiblewithinﬂation.

Thelatesttheoremimmediatelyyieldsthefollowingcorollary.

Corollary3.2.Givenap-permutation

F

G-moduleX ,then

[

X

] =

1

|

G

|

U,V≤G

(5)

Givenp-permutation

FG-modules

M and X ,withM indecomposable,wewritemG

(

M

,

X

)

todenotethemultiplicity of M asadirectsummandof X .Wewrite

πG

(

X

)

todenotethedirectsummandof X , well-deﬁneduptoisomorphism,such that

[

πG

(

X

)

]

=

πG

[

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_UK,F

,

π

U(UResV

(

Y

)))

=

(J,E)∈Q(V)

mU(M_UK,F

,

UResV

(

M_VJ,E

))

mV

(

M_VJ,E

,

π

V

(

Y

)) .

Then,forallG

∈ K

,wehave

|

G

|

pcanG

[

Y

]

∈

T (

G

)

,where

|

G

|

pdenotesthe

p

-partof

|

G

|

.

Proof. Thisisaspecialcaseof[2,9.4].

2

Wecannowprovethe

Z

_[

1

/

p

]

-integralityofcanG.

Theorem3.4.The

Q-linear

mapcanGrestrictstoa

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

πU

(

UResV

(

Y

))

∼

=

UResV

(

Y

)

and

πV

(

Y

)

∼

=

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

πV

(

Y

)

isthe zero module.Itsuﬃces toshow that M_UK,F isnot adirectsummand ofUResV

(

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 MK_U,F

∼

=

X . The V -stabilityof

(

K

,

F

)

alsoimpliesthat K

V .So

Y

|

VIndUInfU/K

(

F

) ∼

=

VInfV/KIndU/K

(

F

) .

WededucethatY isexprojective.Thisisacontradiction,asrequired.

2

Proposition3.5.The

Z-linear

maplinG

:

T (

G

)

→

T

(

G

)

issurjective.However,the

Z

[

1

/

p

]

-linear mapcanG

: Z[

1

/

p

]

T

(

G

)

→

Z

_[

1

/

p

]

T (

G

)

neednotrestricttoa

Z-linear

mapT

(

G

)

→

T (

G

)

.Indeed,puttingp

=

3 andG

=

SL2

(

3

)

,lettingY bethe isomorphi-callyuniqueindecomposablenon-simplenon-projectivep-permutation

FG-module

andX theisomorphicallyunique2-dimensional simple

F

Q8-module,thenthecoeﬃcientofthestandardbasiselement

[

Q8

,

X

]

GincanG

(

[

Y

])

isequalto2

/

3.

Proof. Sinceevery 1-dimensional

F

G-moduleisexprojective,thesurjectivityofthe

Z-linear

map linG 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

G

1,sof

KT

(

G

)

, we mean,an algebra map

KT

(

G

)

→ K

, such that

G

1,s

[

M

]

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

F

G-moduleM.Generally,for

(

P

,

s

)

∈

I(

G

)

,we deﬁneaspecies

G

P,s of

KT

(

G

)

suchthat

GP,s

[

M

]

=

NG(P)/P

1,s

[

M

(

P

)

]

,where M

(

P

)

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

Theorem4.1.Given

(

P

,

s

),

(

P

,

s

)

∈

I(

G

)

,then

G

P,s

=

PG,s ifandonlyifwehaveG-conjugacy

(

P

,

s

)

=

G

(

P

,

s

)

.Theset

{

GP,s

:

(

P

,

s

)

∈

G

I(

G

)

}

isthesetofspeciesof

KT

(

G

)

anditisalsoabasisforthedualspaceof

KT

(

G

)

.Thedualbasis

{

eGP,s

: (

P

,

s

)

∈

G

I(

G

)

}

isthesetofprimitiveidempotentsof

KT

(

G

)

.Asadirectsumoftrivialalgebrasover

K,

wehave

K

T

(

G

)

=

(P,s)∈GI(G)

K

eG_P_,_s

.

Let

J (

G

)

betheG-setofpairs

(

L

,

t

)

whereL isap-residue-freenormalsubgroupofG andt isap-elementofG

/

L.We deﬁneaspecies

_GL,t of

KT

ex

₍

_G

₎

_such_that,_given_an_{indecomposable}_exprojective

_F

_G-module_M,_then

L,t

(6)

istheinﬂationofan

F

G

/

L-module M,inwhichcase,

L,t

G isthevalue,att,oftheBrauercharacterofM.Itiseasytoshow that,givena p-subgroup P

≤

G anda p-element s

∈

NG

(

P

)/

P ,then

PG,s

[

M

]

=

L,t

G

[

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

_GL,t

=

_GL,tifandonlyifL

=

Landt

=

G/Lt,inotherwords,

(

L

,

t

)

=

G

(

L

,

t

)

.Theset

{

_GL,t

: (

L

,

t

)

∈

G

J (

G

)

}

KT

ex

(

G

)

KT

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,

_GG_,_L_,_t

(

x

)

=

_GL,t

(

xG

) .

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

(V

resG

(

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

G

V,L,t

=

GV,L,t if andonly if

(

V

,

L

,

t

)

=

G

(

V

,

L

,

t

)

.Theset

{

VG,L,t

:

(

V

,

L

,

t

)

∈

G

K(

G

)

}

K

T (

G

)

K

T (

G

)

.Thedualbasis

{

eGV,L,t

: (

V

,

L

,

t

)

∈

G

K(

G

)

}

isthesetofprimitiveidempotentsof

K

T (

G

)

.Asadirectsumoftrivialalgebrasover

K,

wehave

K

T

(

G

)

=

(V,L,t)∈GK(G)

K

eG_V_,_L_,_t

.

WehavethefollowingeasycorollaryonliftsoftheprimitiveidempotentseG P,s.

Corollary4.4.Given

(

P

,

s

)

∈

I(

G

)

,theneG_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−+_{constructions}_for_biset_functors,_J._Algebra₅₂₃₍₂₀₁₉₎_241–273. [6]S.Bouc,J.Thévenaz,Theprimitiveidempotentsofthep-permutationring,J.Algebra323(2010)2905–2915.