fibred permutation sets and the idempotents and units of monomial burnside rings

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Journal of Algebra 281 (2004) 535–566

www.elsevier.com/locate/jalgebr

Fibred permutation sets and the idempotentsand units of monomial Burnside rings

Laurence Barker

Department of Mathematics, Bilkent University, 06533 Bilkent, Ankara, Turkey

Received 3 March 2003

Available online 11 September 2004

Communicated by Michel Broué

Abstract

We study the units of monomial Burnside rings and the idempotents of monomial Burnsidebras. Introducing a tenduction map, we realise the unit group and the torsion unit group as a Mfunctor. 2004 Elsevier Inc. All rights reserved.

Keywords:Monomial Burnside rings; Units of Burnside rings; Möbius inversion; Tenduction; Tensor induction

1. Introduction

Our main purpose is to realise unit groups of monomial Burnside rings as a Mfunctors overZ. The two given ingredients are a finite groupG and a cyclic (or, moregenerally, supercyclic) groupC. Before studying the unit group of the monomial Burnsring B(C,G) (defined below), it will be necessary to examine the primitive idempotenmonomial Burnside algebrasRB(C,G) with coefficients in a suitable ringR. In the casewhereC is cyclic of odd prime orderp, andG is of odd order, the unique subgroupindex 2 in the torsion-unit group ofB(C,G) will be realized as a Mackey functor over thfield Fp of orderp.

E-mail address:[email protected].

0021-8693/$ – see front matter 2004 Elsevier Inc. All rights reserved.doi:10.1016/j.jalgebra.2004.07.022

536 L. Barker / Journal of Algebra 281 (2004) 535–566

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More important than proving any theorems about them, we must indicate whyis good reason to believe that monomial Burnside rings are worthy of attention. Weargue that there are motives for seeking to extend the ordinary Burnside ringB(G) andits unit groupB(G)∗. That requires, to begin with, some comments onB(G) andB(G)∗,themselves. First, let us note that they are much more than isomorphism invariantsfinite groupG. They are Mackey functors, and so they are susceptible to local metThe use of the ringB(G) in representation theory, in connection with induction theoreis very well known—see, for instance, Benson [1, Chapter 5]—but let us say a few won the rather more enigmatic role ofB(G)∗ in representation theory.

As an isomorphism invariant,B(G)∗ imposes very little constraint on the groupG,since it is an elementary abelian 2-group, hence its only parameter is its rank. As a Mfunctor over the Galois fieldF2, there is much richer scope for study. The unit groupB(G)∗plays a role in the theory ofG-spheres. Matsuda [20, Corollary 2.6] showed that the grof homotopy equivalences on the unit sphereS(M) of anRG-moduleM maps injectivelyto B(G)∗, and bijectively whenM is, in a suitable sense, sufficiently large. Writing squbrackets to denote isomorphism classes, and writingΛ to indicate a reduced Lefscheinvariant, tom Dieck [12, Proposition 5.5.9] observed that the assignment[M] �→ Λ(S(M))

determines a linear map expG from the real representation ring to the unit groupB(G)∗.Yoshida [27, Theorem A] extended expG to a map from the real valued virtual charactto B(G)∗. Further extension to complex virtual characters would surely be desirablcalculation with concrete examples (say, with the cyclic group of order 4) will reaconvince those who try it that any such extension of the domain also requires an extof the codomainB(G)∗.

At various levels of generality, monomial Burnside rings, explicitly or implicitly, hbeen studied in contexts involving or closely related to induction theorems. See, fstance, Boltje [2–6], Boltje–Külshammer [7,8], Conlon [11], Dress [14]. In such contextapplications rely on the fact that, if the supercyclic groupC is suitable and, in particular, sufficiently large, thenB(C,G) maps surjectively to the representation ringFG-modules, whereF is a splitting field forG. In effect, C is taken to be the grouof torsion-units ofF (we shall explain this substitution in Section 2). One of the motfor the level of generality we have selected is that (for reasons which will transpirecase where|C| is prime seems to be of special interest.

Following Dress [14], let us introduce the notion of a monomial Burnside ring iabstract manner that is entirely detached fromits applications in linear representation thory. The theory of permutation sets for a finite group extends quite easily to a thefibred permutation sets where the role previously played by the points of the permuset is now played by fibres, which are copies of a fixed abelian groupA called thefibregroup. We define anA-fibred G-set to be anA-free A × G-set with only finitely manyorbits. The category ofA-fibred G-sets admits a product and a coproduct given by,spectively, tensor product overA and set-wise coproduct (disjoint union). See Sectiofor details. Hence, we obtain a Grothendieck ring, denoted byB(A,G), called themono-mial Burnside ring forG with fibre groupA. In the special case whereA is trivial, werecover the ordinary Burnside ringB(1,G) = B(G). For a commutative ringΘ, the alge-braΘB(A,G) = Θ ⊗Z B(A,G) is called amonomial Burnside algebra with coefficiering Θ.

L. Barker / Journal of Algebra 281 (2004) 535–566 537

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In view of (all) the applications (that we know of), our interest is in the case wherfibre group is isomorphic to a subgroup of the torsion-unit group of an integral domSuch groups are said to besupercyclic(an equivalent and abstract definition of the termgiven in Section 3). Much of the material in this paper is a study of the idempotentsmonomial Burnside ringKB(C,G), whereK is a field of characteristic zero. In the speccase whereK has enough roots of unity andC is the unit group of an algebraically closefield, this has already been done by Boltje [6]. Weshall present this as an application omonomial version of Möbius inversion. Regarded as a preliminary to a study of unittreatment of idempotents is not maximally efficient, because we shall prove someon idempotents that will not be needed when we turn to units. Furthermore, the appthrough Möbius inversion is not the most direct way of deriving the idempotent formHowever, we consider idempotents and monomial Möbius inversion to be of intertheir own right (and the latter provides a meaningful rationale for the idempotent forin Section 5).

Still, the main section in this paper is the final one, where we introduce a tenduction (sor induction) functor onA-fibredG-sets (as always,A arbitrary abelian) and a tenductiomap on the Burnside ringB(C,−) (as always,C arbitrary supercyclic). The tenductiomap serves as the transfer map for the unit groupB(C,−)∗ as a Mackey functor.

Let us mention another possible avenue of investigation. Out of a need for terminlet us say that a ringΛ is asplit extensionof a subringΓ providedΛ = Γ ⊕ I , whereI

is a two-sided ideal. Thus,Γ is a quotient ring ofΛ, as well as a subring. In Section 2, wshall observe thatB(C,G) is a split extension ofB(G); hardly remarkable, since we wealready regardingB(C,G) as a generalization ofB(G). But, in Section 7, we shall observthat, if the exponent ofG divides the order ofC, thenB(C,G) is also a split extension othe group algebra of the abelianization of (the dual group of)G. If we now also assume thaG is abelian, we conclude that the unit groupB(C,G)∗ is a split extension (in the ususense) of the unit group(ZG)∗. In Section 8, we shall be making use of (a fairly trivresult in) the theory of unit groups of group rings. These observations seem to suggpossibility of a Mackey functor approach to the study of unit groups of commutative grorings.

Some conventions: Any ring is understood to have a unity element, and any subunderstood to have the same unity element. The unit group of a ringΘ is denoted byΘ∗and the torsion-unit group, byΘω. We write exp(G) to denote the exponent ofG. To avoidconflicts of terminology, we shall use the termspecializeto refer to restriction of functions

2. Monomial Burnside rings in general

We make some preliminary comments on monomial Burnside rings and monBurnside algebras in the general case where the fibre group is the arbitrary abelian gA.The material was introduced long ago by Dress [14], and he considered the even moeral case whereG acts onA. Our purpose is just to set up our preferred notationperspective.

We write A × G = AG = {ag: a ∈ A, g ∈ G}. An A-freeAG-set with finitely manyA-orbits is called anA-fibred G-set. An A-orbit of an A-fibred G-set is called afibre.


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Whenever an expression of the formAX denotes anA-fibredG-set, it is to be understoothatX is a set of representatives of the fibres. Note thatX is finite. Of course,G need notstabilizeX. Any element ofAX can be written uniquely in the formax wherea ∈ A andx ∈ X.

Let AX andAY beA-fibredG-sets. Thecoproductof AX andAY is defined to be theicoproduct as sets (their disjoint union)

AX � AY = A(X � Y )

regarded in an evident way as anA-fibredG-set. With respect to the action ofA onAX ×AY given bya(ξ, η) = (aξ, a−1η), we letξ ⊗ η denote theA-orbit of an element(ξ, η) ofAX ×AY , and we letAX ⊗AY denote the set ofA-orbits. Loosely, we callAX ⊗AY thetensor productof AX andAY overA. We letAG act onAX ⊗ AY by

ag(ξ ⊗ η) = agξ ⊗ gη.

We writex ⊗y = xy andXY = {xy: x ∈ X, y ∈ Y }. Theproductof AX andAY is definedto be the tensor product

AX ⊗ AY = AXY

regarded as anA-fibredG-set.Having imposed a product and a coproduct on the category ofA-fibredG-sets, we can

now construct the Grothendieck ringB(A,G). Let [AX] denote the isomorphism claof a A-fibredG-setAX. As an abelian group,B(A,G) is generated by the isomorphisclasses ofA-fibredG-sets. The relations are given by the condition that

[AX] + [AY ] = [AX � AY ] = [A(X � Y )

].

The multiplication is such that

[AX][AY ] = [AX ⊗ AY ] = [AXY ].

Evidently,B(A,G) is a commutative ring.To analyse the ringB(A,G), we need some more notation and terminology. LetA\AX

denote the set of fibres ofAX. Obviously,AX is transitive as anAG-set if and only ifA\AX is transitive as aG-set. In that case,AX is said to betransitive as anA-fibredG-set. As an abelian group,B(A,G) is freely generated by the isomorphism classetransitiveA-fibredG-sets.

We define anA-characterof G to be a group homomorphismG → A. We define anA-subcharacterof G to be a pair(V , ν) whereV � G andν is anA-character ofV . TheA-subcharacters ofG admit an action ofG by conjugation:g(V, ν) = (gV,g ν). Let AνG/V

denote a transitiveA-fibredG-set such thatV is the stabilizer of a fibreAx, andvx = ν(v)x

for all v ∈ V . Whenν is the trivialA-character ofV , we writeAνG/V = AG/V .Proofs of the following three remarks are left as easy exercises.


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Remark 2.1.GivenA-subcharacters(V , ν) and(W,ω) of G, thenAνG/V is isomorphicto AωG/W if and only if (V , ν) is G-conjugate to(W,ω). Every transitiveA-fibredG-setis isomorphic to anA-fibredG-set of the formAνG/V .

Remark 2.2.As an abelian group

B(A,G) =⊕(V ,ν)

Z[AνG/V ],

where(V , ν) runs over a set of representatives of theG-classes ofA-subcharacters ofG.

In particular, the abelian groupB(A,G) has finite rank if and only if the set of grouhomomorphisms Hom(V ,A) is finite for allV � G.

For subgroupsV � G � W , an A-characterν of V and anA-characterω of W , letus writeν.ω for theA-character ofV ∩ W such thatu �→ ν(u)ω(u) for u ∈ V ∩ W . Themultiplication onB(A,G) is given by the following Mackey formula.

Remark 2.3.GivenA-subcharacters(V , ν) and(W,ω) of G, we have

[AνG/V ][AωG/W ] =∑

VgW⊆G

[Aν.gωG/V ∩ gW

],

where the notation indicates thatg runs over a set of representatives of the cosets ofV andW in G.

Any mapα :A1 → A2 of abelian groups induces a ring homomorphismB(A1,G) →B(A2,G) such that[A1X] �→ [A2 ⊗ A1X], the tensor product being overA1. In particu-lar, extendingA to an overgroupA′, then the embeddingA ↪→ A′ induces an embeddinB(A,G) ↪→ B(A′,G). If A has a complementary subgroup inA′, then (in the termi-nology of Section 1),B(A′,G) is a split extension ofB(A,G). In particular,B(A,G)

is a split extension ofB(G). Another consequence of these observations (equally tritbut, again, tremendous) is as follows. Consider a commutative ringΘ, its group ofunits Θ∗ and its group of torsion-unitsΘω. The embeddingΘω ↪→ Θ∗ induces an embeddingB(Θω,G) ↪→ B(Θ∗,G). But G is finite, so we can make the identificatioB(Θω,G) = B(Θ∗,G).

The category ofA-fibredG-sets, denotedA-G-SET, is full subcategory of the categoof AG-sets. For a subgroupF � G, induction and restriction ofAF -sets andAG-setsspecialize to an induction functor

AIndGF :A-F -SET→ A-G-SET

and a restriction functor

AResG :A-G-SET→ A-F -SET.
F


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ond

the

ssume

e,,

Forg ∈ G, there is also an evident conjugation functor

ACongF :A-F -SET→ A-gF -SET.

The functorsAIndGF , AResGF , ACong

F induce the following linear maps on monomial Burside rings:

• an induction map

AindGF :B(A,F) → B(A,G),

• a restriction map

AresGF :B(A,G) → B(A,F),

• a conjugation map

AcongF :B(A,F) → B(A,gF

).

Thus, for instance,AresGF [AX] = [AResGF (AX)]. Sometimes, we omit the subscriptA.Given a commutative ringΛ, we call

ΛB(A,G) = Λ ⊗Z B(A,G)

a monomial Burnside algebra overΛ. The induction, restriction and conjugation mapsmonomial Burnside ringsB(A,−) extend uniquely toΛ-linear induction, restriction anconjugation maps on monomial Burnside algebrasΛB(A,−).

Recall that aGreen functoroverΛ is a Mackey functorM such that, for each groupGin the domain,M(G) is aΛ-algebra, and a couple of extra axioms hold in addition toaxioms of a Mackey functor. See Thévenaz [23] for details. It is easy to see thatB(A,−)

is a Green functor overZ and, more generally,ΛB(A,−) is a Green functor overΛ.Let us make a few comments on linearization. For the rest of this section, we a

thatA is a subgroup of the unit groupΘ∗ of a commutative ringΘ. Let R(ΘG) denotethe representation ring (also called theGreen ring) associated with the categoryΘG-MODof (finitely generatedΘ-free)ΘG-modules. For aΘG-moduleM, we write [M] for theisomorphism class ofM, regarded as an element ofR(ΘG). By linear extension, as abovthe induction, restriction, and conjugation functors forΘG-modules give rise to inductionrestriction, and conjugation maps between representation rings. ThusR(Θ−) becomes aGreen functor. Thelinearization functor

LinG = Θ ⊗A − :A-G-SET→ ΘG-MOD

gives rise to a ring homomorphism, thelinearization map

linG :B(A,G) → R(ΘG).


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Thus linG[AX] = [LinG(AX)]. Evidently, induction, restriction, and conjugation commwith linearization; linG is a morphism of Green functors.

Our reason for setting down all these obvious functorial formalities is that, in theof a supercyclic fibre groupC, we shall be introducing a tenduction map to realize thegroupB(C,−)∗ as a Mackey functor (not as Green functor). It will still be true that lG

and extension of the fibre group are morphisms of Mackey functors. In fact, it willbe obvious. But the point is worth making because there is another apparently reasonadefinition of tenduction that is not compatiblewith linearization or with extension of thfibre group.

3. Subcharacters and subelements

We collect together some definitions and observations concerning subcharactesubelements. The purpose of the material will become apparent in later sections.

Given an elements of aG-setS, we write[s]G for theG-orbit of s, and we writeNG(s)

for the stabilizer ofs in G. Whenever we call theG-actionconjugation, we also call[s]GtheG-classof s, and we callNG(s) thenormalizerof s in G. If elementss, t ∈ S belongto the sameG-orbit, then we writes =G t . We letG\S denote the set ofG-orbits inS.

A supernatural number, recall, is a formal product∏

p pα(p), wherep runs over therational primes and eachα(p) ∈ N ∪ {∞}. The arithmetic of the supernatural numbersentirely multiplicative. With respect to the divisibility relation, the supernatural numcomprise a lattice. In fact, every non-empty set of supernatural numbers has a lowesmon multiple and a highest common factor.

A group is said to besupercyclicprovided every finitely generated subgroup is finand cyclic. Given a supercyclic groupC, then theorderof C, denoted|C|, is defined to bethe lowest common multiple of the orders of the elements ofC. In other words, the ordeof C is the exponent ofC, understood to be a supernaturalnumber. For each supernatunumbern, there is a unique subgroupCn of Q/Z having order|Cn| = n. In the case whern is finite, Cn is generated by the coset ofZ owning 1/n. Thus, a supercyclic groupdetermined up to isomorphism by its order, and the supernatural numbers are precisorders of the supercyclic groups.

Let C be a supercyclic group. LetO(G) denote the intersection of the kernels of tC-characters ofG. ThusO(G) is the minimal normal subgroup ofG such thatG/O(G)

is abelian with exponent dividing|C|. The group ofC-characters ofG, denoted

G = Hom(G,C)

may be regarded as the dual of the group

G = G/O(G).

Recall that theC-subcharacters ofG were defined (in a more general context) in Stion 2. TheG-set ofC-subcharacters ofG is written as

ch(C,G) = {(V , ν): V � G, ν ∈ V

}.


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We define aC-subelementof G to be a pair(H,hO(H)) whereh ∈ H � G. Sometimeswe abbreviate(H,hO(H)) as(H,h). Thus, twoC-subelements(H,h) and(I, i) of G areequal if and only ifH = I andhO(H) = iO(H). We letG act on theC-subelements oG by conjugation:g(H,h) = (gH, gh). TheG-set ofC-subelements ofG is denoted by

el(C,G) = {(H,hO(H)): H � G, hO(H) ∈ H

}.

Lemma 3.1.We have|el(C,G)| = |ch(C,G)|.

Proof. For each subgroupF � G, the number ofC-subelements with first coordinaF and the number ofC-subcharacters with first coordinateF are both equal to|F :O(F)|. �

The next lemma is well-known.

Lemma 3.2.Let B be a finite group acting as automorphisms on a finite abelian grouA

and on the dual groupA, the actions preserving the duality. Then|B\A| = |B\A|.

Proof. EmbeddingQ/Z in the unit group ofC, we can regard the elements of the groA = Hom(A,Q/Z) as functionsA → C. For α ∈ A, let α+ :A → C be the sum of theB-conjugates ofα. The set{α+: α ∈ A} is linearly independent and has size|B\A|. Buteachα+ is B-invariant, and can be regarded as a functionB\A → C. Therefore|B\A| �|B\A|. The reverse inequality follows by duality.�Lemma 3.3.We have|G\el(C,G)| = |G\ch(C,G)|.

Proof. Let F � G. By Lemma 3.2, the number ofG-classes ofC-subelements with firscoordinate conjugate toF is equal to the number ofG-classes ofC-subcharacters with firscoordinate conjugate toF . �

4. Monomial Möbius inversion

The incidence function and the Möbius function for a finite poset are, essentiallytually inverse matrices with rows and columns indexed by the elements of the posimportant particular case is that where the poset is sub(G), the poset of subgroups ofG,partially ordered by inclusion. We shall replace sub(G) with the two sets el(C,G) andch(C,G). These two sets are, in some sense, dual to each other: el(C,G) indexes the rowsof the monomial incidence function and the columns of the monomial Möbius funcch(C,G) indexes the columns of the monomial incidence function and the rows omonomial Möbius function. As a banal but convenient manoeuvre, we shall also coG-invariant versions of the various incidence and Möbius functions.


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Let us introduce a little deviceof notation that sometimes facilitates symbolic manilations. TheKronecker valueof a propositionS is defined to be the rational integer

S� ={

1, if S holds,0, if S fails.

Consider a finite posetP . The incidence functionof P is defined to be the functioζ :P × P → Z such thatζ(x, y) = x � y� for x, y ∈ P . For an integern � −2, we definea functioncn :P × P → Z such thatc−2(y, x) = y = x� andc−1(y, x) = y < x� and, ifn � 0, thencn(y, x) is the number of chains inP having the formy < z0 < · · · < zn < x.TheMöbius functionof P is defined to be the functionµ :P × P → Z given by

µ(y, x) =∞∑

n=−2

(−1)ncn(y, x).

SinceP is finite, only finitely many of the terms of the sum are non-zero. (Our numbeconvention reflects the fact that, ify < x, thenµ(y, x) is the reduced Euler characterisof the simplicial complex associated with the open interval bounded byy andx.)

Let A be an abelian group, and letθ andφ be functionsP → A. The equation

θ(y) =∑x∈P

φ(x)ζ(x, y)

is called thetotient equation. The equation

φ(x) =∑y∈P

θ(y)µ(y, x)

is called theinversion equation. The principle of Möbius inversion asserts that the totiequation holds for ally ∈ P if and only if the inversion equation holds for allx ∈ P .A proof can be found in Kerber [19, Section 2.2]. The principle can also be expressthe matrix equation∑

y∈P

ζ(x, y)µ(y, z) = x = z� =∑y∈P

µ(x, y)ζ(y, z),

which holds for allx, z ∈ P . In particular, ifx < z, then∑y∈P : x�y�z

µ(y, z) = 0 =∑

y∈P : x�y�z

µ(x, y).

The Möbius functionµ is determined by this identity, together with the conditions thaµ(x, x) = 1 andµ(y, x) = 0 wheny � x.


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ns

Suppose now that the finite posetP is aG-poset. We define theG-invariant incidencefunctionζG and theG-invariant Möbius functionµG to be the functionsP × P → Z suchthat

ζG(x, y) =∑

x ′∈[x]Gζ(x ′, y), µG(y, x) =

∑y ′∈[y]G

µ(y ′, x).

Note that, ifx =G x ′ andy =G y ′, thenζ(x, y) = ζ(x ′, y ′) andµ(y, x) = µ(y ′, x ′). SoµG andζG may be regarded as functionsG\P × G\P → Z. If the functionsθ andφ areG-invariant (withG acting trivially onA) then the totient equation can be rewritten as

θ(y) =∑

x∈GP

φ(x)ζG(x, y)

and the inversion equation can be rewritten as

φ(x) =∑

y∈GP

θ(y)µG(y, x).

Here, the notation indicates that the indices of the sums run over representativesG-orbits inP . The equivalence of the totient equation and the inversion equation cexpressed as the matrix equation∑

y∈GP

ζG(x, y)µG(y, z) = x =G z� =∑

y∈GP

µG(x, y)ζG(y, z).

Let K be a field of characteristic zero, and suppose that an embeddingC ↪→ Kω isgiven. For aC-subcharacterω of G and a subsetT ⊆ G, we write

ω(T ) =∑t∈T

ω(t)

as an element ofK. We writeω−1 to denote the inverse ofω in the groupG.Whenever we write an expression of the formζ(H,V ) or µ(V,H), whereH andV are

subgroups ofG, it is to be understood thatζ andµ are the incidence and Möbius functioof the G-poset sub(G). We now generalize those two functions. We define amonomialincidence function

ζ : el(C,G) × ch(C,G) → K, ζ(H,h;V,ν) = ν(h)ζ(H,V )

and amonomial Möbius function

µ : ch(C,G) × el(C,G) → K, µ(V, ν;H,h) = ν−1(V ∩ hO(H))µ(V,H)/|V |

where(H,h) ∈ el(C,G) and(V , ν) ∈ ch(C,G).


f

ces are

,

Let (V , ν) and (W,ω) be C-subcharacters ofG. Using orthonormality properties ocharacters,

∑(H,h)∈el(C,G)

µ(V, ν;H,h)ζ(H,h;W,ω) =∑

H :V �H�W

µ(V,H)

|V : O(V )|∑

vO(V )⊆V

ν−1(v)ω(v)

=∑

H :V �H�W

µ(V,W)⌊ν = ResWV (ω)

⌋= ⌊

(V , ν) = (W,ω)⌋.

By Remark 3.1, we can interpret the equality as an assertion that two square matrimutual inverses. Therefore, givenC-subelements(H,h) and(I, i) of G, we have∑

(V ,ν)∈ch(C,G)

ζ(H,h;V,ν)µ(V, ν; I, i) = ⌊(H,h) = (I, i)

⌋.

Now consider functionsθ : ch(C,G) → A andφ : el(C,A) → A. By the comments abovethe totient equation

θ(V, ν) =∑

(H,h)∈el(C,G)

φ(C,G)ζ(H,h;V,ν)

holds for all(V , ν) ∈ ch(C,G) if and only if the inversion equation

φ(H,h) =∑

(V ,ν)∈ch(C,G)

θ(V, ν)µ(V, ν;H,h)

holds for all(H,h) ∈ el(C,G).Much as before, we define aG-invariant monomial incidence functionζG and a

G-invariant monomial Möbius functionµG such that

ζG(H,h;V,ν) =∑

(H ′,h′)∈[H,h]Gζ(H ′h′;V,ν),

µG(V, ν;H,h) =∑

(V ′,ν ′)∈[V,ν]Gµ(V ′, ν′;H,h).

Theorem 4.1(Monomial Möbius inversion). GivenG-invariant functionsθ : ch(C,G) → A

andφ : el(C,G) → A, then the totient equation

θ(V, ν) =∑

φ(C,G)ζG(H,h;V,ν)

(H,h)∈Gel(C,G)


Oddlyof the

vec-totient

f theh-rant to,

oftion

y

hosee

holds for all(V , ν) ∈ ch(C,G) if and only if the inversion equation

φ(H,h) =∑

(V ,ν)∈Gch(C,G)

θ(V, ν)µG(V, ν;H,h)

holds for all(H,h) ∈ el(C,G).

Proof. This is clear from the preceding discussion.�In our proof of Theorem 4.1, we somehow managed to avoid using Lemma 3.3.

enough, the implication is the other way around: the theorem yields another prooflemma.

We mention that, by letting the matrices act on column vectors rather than on rowtors, all of the downwards sums can be replaced by upwards sums. For instance, theequation

θ(H,h) =∑

(V ,ν)∈Gch(C,G)

ζG(H,h;V,ν)φ(V, ν)

is equivalent to the inversion equation

φ(V, ν) =∑

(H,h)∈Gel(C,G)

µG(V, ν;H,h)θ(H,h).

5. An idempotent formula

In this section and the next one, we shall examine the primitive idempotents omonomial Burnside algebraKB(C,G), whereK is a field of characteristic zero. Througout the present section, we shall assume thatK has enough roots of unity for all oupurposes. We shall remind ourselves of this standing hypothesis whenever we wbecause it will be dropped in the next section. As in Section 4, we embedC in the group oftorsion-unitsKω. Of course, the idempotents ofKB(C,G) do not depend on the choicethe embeddingC ↪→ Kω, but we shall be making use of the embedding in our descripof the idempotents.

A formula for the primitive idempotents ofQB(G) was discovered independently bGluck [16] and Yoshida [26]. Similar formulas for the primitive idempotents ofCD(G) andCDp(G) were given by Boltje [6, Section 3]. In this section, we unify and generalize tresults. We give a formula the primitive idempotents ofKB(C,G). We also characterizthe induction and restriction maps in terms of the primitive idempotents.

Remark 2.2 tells us that, as a direct sum of 1-dimensionalK-vector spaces,

KB(C,G) =⊕

K[CνG/V ]. (1)
(V ,ν)∈Gch(C,G)


ateofatri-

s as

atrix

,

rion

The notation, here, indicates that(V , ν) runs over a set of representatives of theG-classesof C-subcharacters ofG. The primitive idempotents ofKB(C,G) will be indexed by theG-classes ofC-subelements ofG. The primitive idempotent indexed by the (G-class ofthe) C-subelement(H,h) will be denotedeG

H,h. We shall prove thatKB(C,G) is semi-simple and

KB(C,G) =⊕

(H,h)∈Gel(C,G)

KeGH,h, (2)

as a direct sum of algebras isomorphic toK. Let us call the elements ofKB(C,G) havingthe form[CνG/V ] the transitive elements. Equations (1) and (2) express the coordinsystems associated with, respectively, the basis of transitive elements and the basisprimitive idempotents. The aim of this section is to determine the transformation mces between the two coordinate systems. Let us express the transformation matrice

[CνG/V ] =∑

(H,h)∈Gel(C,G)

mG(H,h;V,ν)eGH,h, (3)

eGH,h =

∑(V ,ν)∈Gch(C,G)

m−1G (V, ν;H,h)[CνG/V ]. (4)

After establishing the decomposition in Eq. (2), we shall give formulas for the mentriesmG(H,h;V,ν) andm−1

G (V, ν;H,h).Our first step is to define species (algebra maps to the ground field)

sGH,h :KB(C,G) → K.

Consider aC-fibredG-setCX and aC-subelement(H,h) of G. Given a fibreCx in CX

stabilized byH , let us writeφx for the C-character ofH such thathx = φx(h)x for allh ∈ H . Note thatφx is independent of the choice of the elementx of the fibreCx. WedefinesG

H,h to be the linear map such that

sGH,h[CX] =

∑Cx

φx(h),

whereCx runs over the fibres inCX that are stabilized byH . Let us show thatsGH,h

is a species. Consider anotherC-fibred G-setCY . A fibre Cxy ⊆ CXY is stabilized byH if and only if the fibresCx ⊆ CX andCy ⊆ CY are stabilized byH . In that caseφxy = φxφy . ThereforesG

H,h([CX])sGH,h([CY ]) = sG

H,h([CXY ]), as required.The next result is immediate from Dress [14, Theorem 1′(c)]. We mention that, in ou

special case, his argument simplifies and the first and second halves of the conclusfollow easily from Lemmas 3.2 and 3.3, respectively.

Lemma 5.1(Dress). Recall thatK is sufficiently large. GivenC-subelements(H,h) and(I, i) of G, thensG

H,h = sGI,i if and only if (H,h) = G(I, i). Every species ofKB(C,G) is

of the formsG , and the species span the dual space ofKB(C,G).
H,h


). We

, we

x

e

it is

mpo-

ps are

By the lemma, there exists a unique elementeGH,h ∈ KB(C,G) such that

sGI,i

(eGH,h

) = ⌊(I, i) = G(H,h)

⌋.

In the proof of the lemma, we saw that the algebraKB(C,G) is isomorphic to a direct sumof copies ofK. So eacheG

H,h is a primitive idempotent andKeGH,h

∼= K. The decompositionin Eq. (2) is now established.

We can now turn to the problem of evaluating the matrix entries in Eqs. (3) and (4have

mG(H,h;V,ν) = sGH,h[CνG/V ] =

∑gV ⊆G

gν(h)⌊H � gV

⌋.

Comparing with the definition (in Section 4) of the appropriate incidence functionobtain

mG(H,h;V,ν) = |NG(H,h)||V | ζG(H,h;V,ν). (5)

Theorem 4.1 says that the matrix with entriesµG(V, ν;H,h) is the inverse of the matriwith entriesζG(H,h;V,ν). Therefore

m−1G (V, ν;H,h) = |V |

|NG(H,h)|µG(V, ν;H,h). (6)

We have proved the following result.

Theorem 5.2(Idempotent formula). Recall thatK is sufficiently large. There is a bijectivcorrespondenceeG

H,h ↔ [H,h]G between the primitive idempotentseGH,h of KB(C,G)

and theG-conjugacy classes[H,h]G of C-subelements(H,h) of G. We have∣∣NG(H,h)∣∣eG

H,h =∑

(V ,ν)∈Gch(C,G)

|V |µG(V, ν;H,h)[CνG/V ].

The following remark is immediate from the definitions that we have made, butworth emphasising, because we use it veryfrequently (often without mentioning it).

Remark 5.3.With respect to the basis of primitive idempotents, the coordinate decosition of an elementζ ∈ KB(C,G) is

ζ =∑

(H,h)∈Gel(C,G)

sG(H,h)(ζ )eG

(H,h).

With respect to that coordinate decomposition, the induction and restriction magiven by the matrix equations in the next two propositions.


f

gtheee, for

Proposition 5.4.GivenF � G and aC-subelement(J, j) of F , then

indGF

(eGJ,j

) = ∣∣NG(J, j) : NF (J, j)∣∣eG

J,j .

Proof. Consider aC-subelement(H,h) of G. The Mackey decomposition formula

resGH indGF =

∑HgF⊆G

indHH∩ gF res

gFH∩ gF congF

clearly holds for monomial Burnside algebras. Therefore

sGH,h

(indG

F

(eFJ,j

)) = sHH,h

(resGH

(indG

F

(eFJ,j

))) =∑

HgF⊆G

⌊(H,h) =F

g(J, j)⌋

= ∣∣{gF ⊆ G: (H,h) =Fg(J, j)

}∣∣= |NG(H,h)|

|NF (J, j)|⌊(H,h) =G (J, j)

⌋. �

Proposition 5.5.GivenF � G and aC-subelement(H,h) of G, then

resGF(eGH,h

) =∑(J,j)

eFJ,j ,

where(J, j) runs over representatives of theF -classes ofC-subelements ofF such that(J, j) is G-conjugate to(H,h).

Proof. For an arbitraryC-subelement(J, j) of F , we have

sFJ,j

(resGF

(eGH,h

)) = sGJ,j

(eGH,h

). �

Corollary 5.6. Recall thatK is sufficiently large. GivenF � G then, as a direct sum oideals,

KB(C,G) = Im(indG

F

) ⊕ Ker(resGF

).

Proof. Let (H,h) ∈ ch(C,G). By Proposition 5.4,eGH,h ∈ Im(indG

F ) if and only if

H � GF . By Proposition 5.5,eGH,h ∈ Ker(resGF ) if and only if H � GF . �

6. The primitive idempotents over a characteristic zero field

We now allowK to be any field of characteristic zero. Our technique for determininthe primitive idempotents ofKB(C,G) is based on Galois actions. It is a variant ofmeans by which Berman dealt with the analogous problem for group algebras. Sinstance, Karpilovsky [18, Section 8.9].


p

tsion.

n

prim-

nt.

of

uiv--

is

ps, the

Throughout, we letCK be the maximum subgroup ofC such thatKω has a subgrouisomorphic toCK . Thus,|CK | = gcd(|C|, |Kω|). We embedCK in K arbitrarily. Let usbegin by determining an upper bound (with respect to divisibility) on the number of rooof unity needed for the primitive idempotents to be as described in the previous sect

Proposition 6.1. Let r = gcd(|C|,exp(G)). Suppose thatK has primitiverth roots ofunity. Then the primitive idempotents ofKB(C,G) are precisely the idempotentseG

H,h.Furthermore

KB(C,G) =⊕

(H,h)∈Gel(C,G)

KeGH,h

as a direct sum of algebrasKeGH,h

∼= K.

Proof. The values of the Möbius function in Eq. (6) all belong toK. So the assertiofollows from Theorem 5.2. �

Before turning to the general case, let us recall some generalities concerning theitive idempotents of an artinian subringΦ of a commutative artinian ringΘ. The ringsΘandΦ each have only finitely many idempotents, and they sum to the (same) unity elemeEach primitive idempotent ofΦ is a sum of primitive idempotents ofΘ. Each primitiveidempotent ofΘ is a summand of a unique primitive idempotent ofΦ. When primitiveidempotentse andf of Θ are summands of the same primitive idempotent ofΦ, we saythate andf areequivalentwith respect toΦ.

Remark 6.2.With Φ andΘ as above, the primitive idempotentsε of Φ are in a bijectivecorrespondence with the equivalence classes[e] of the primitive idempotentse of Θ. Thecorrespondence is such thatε ↔ e providedε is the sum of the primitive idempotentsΘ that are equivalent toe.

Thus, if the primitive idempotents ofΘ have been determined, and if the above eqalence relation has been determined, then the primitive idempotents ofΦ have been determined too. We shall use this principle to specify the primitive idempotents ofKB(C,G)

in the absence of the hypothesis in Proposition 6.1. Our strategy is to extendK to afield K[C], defined below, and then to putΘ = K[C]B(C,G) andΦ = KB(C,G). Weshall see that the primitive idempotentseG

H,h of K[C]B(C,G) are permuted by the Galogroup Gal(K[C]/K). It will turn out that the equivalence classes[eG

H,h] are the orbits ofGal(K[C]/K).

Let K[C] be the minimal extension field ofK such thatK[C] has primitivemth rootsof unity for every natural numberm dividing the supernatural number|C|. ThenK[C]ωhas a subgroup isomorphic toC. Let us choose and fix an embeddingC ↪→ K[C]ω thatextends the embeddingCK ↪→ Kω.

A field extension having the formK[C]/K is called asupercyclotomic extension. Justas the theory of supercyclic groups is much the same as the theory of cyclic grou


exten-avee shall

is

r-

m

g

theory of supercyclotomic extensions is much the same as the theory of cyclotomicsions. Actually, for fixedG, we could work with a cyclotomic extension but, once we hmade some little observations on the Galois theory of supercyclotomic extensions, wactually find it simpler to work with a (profinite) Galois group that applies globally.

Given a subgroupD � C, then we can regardK[D] as the subfield ofK[C] generatedover K by D. If we let D run over the finite subgroups ofC, thenK[C] = ⋃

D K[D].In particular,K[C] is a union of (finite degree) Galois extension fields ofK. So K[C]is a normal separable extension field ofK. In particular, the (possibly infinite) Galogroup Gal(K[C]/K) has fixed fieldK. Let Aut(C/CK) denote the subgroup of Aut(C)

fixing CK . The following two remarks are very easy and (no doubt) well-known.

Remark 6.3.Let C′ be such thatCK � C′ � C. Then:

(1) The action of Gal(K[C]/K) as automorphisms ofK[C′] gives rise to a group epimophism Gal(K[C]/K) → Gal(K[C′]/K).

(2) The action of Aut(C/CK) as automorphisms ofC′ gives rise to a group epimorphisAut(C/CK) → Aut(C′/CK).

Remark 6.4.The action of Gal(K[C]/K) as automorphisms of the subgroupC of K[C]∗gives rise to a group isomorphism Gal(K[C]/K) ↔ Aut(C/CK).

Via the isomorphism in Remark 6.4, we identify Gal(K[C]/K) with Aut(C/CK). Thenthe two epimorphisms in Remark 6.3 coincide. Let us write

Γ = Gal(K[C]/K) = Aut(C/CK).

Given an elementγ ∈ Γ , and aC-subcharacter(V , ν) of G, we write γν = γ ◦ν andγ(V , ν) = (V , γν). Thus, ch(C,G) becomes aΓ -set. Consider aC-subelement(H,h)

of G. The groupsH andH are mutual duals, so the action ofΓ on H induces an actionon H . Explicitly, γ sends each elementhO(H) ∈ H to the unique elementγhO(H) ∈ H

such thatγφ(γh) = φ(h) for all φ ∈ H . We write γ(H,h) = (H, γh). Thus, el(C,G) be-comes aΓ -set. The actions ofΓ and G on ch(C,G) commute, likewise on el(C,G).We have realized ch(C,G) and el(C,G) as permutation sets of the direct productΓ G =Γ × G.

By Proposition 6.1, the primitive idempotents ofK[C]B(C,G) are the elements havinthe formeG

H,h. Let

eG,KH,h =

∑(I,i)

eGI,i ,

where(I, i) runs over representatives of theG-classes ofC-subelements ofG such that(H,h) and(I, i) areΓ G-conjugate.


to

ethe

f-

),].c-

tsoef-

Theorem 6.5.There is a bijective correspondenceeG,KH,h ↔ [H,h]Γ G between the primitive

idempotentseG,KH,h of KB(C,G) and theΓ G-classes[H,h]Γ G of C-subelements(H,h)

of G.

Proof. Put Θ = K[C]B(C,G) andΦ = KB(C,G). By Remark 6.2, we are requiredshow that two primitive idempotentseG

H,h andeGI,i of Θ are equivalent with respect toΦ

if and only if theC-subelements(H,h) and(I, i) areΓ G-conjugate. SinceeGH,h = eG

I,i ifand only if(H,h) and(I, i) areG-conjugate, we need only consider actions ofΓ . We letΓ act as ring automorphisms ofΘ such that

γ(λ[CνG/V ]) = (

γλ)[CνG/V ],

whereγ ∈ Γ andλ ∈ K[C]. Since theΓ -fixed subfield ofK[C] is K, theΓ -fixed subringof Θ is Φ. The primitive idempotents ofΘ are permuted byΓ , and the orbit sums are thprimitive idempotents ofΦ. From the definition of the monomial Möbius function anddefinition of the action ofΓ on theC-subelements, we have

γµ(V,ν;H,γh

) = µ(V, ν;H,h).

By Theorem 5.2,γeGH,γh = eG

H,h. SoeG,KH,h is theΓ -orbit sum ofeG

H,h. �Corollary 6.6. The numbers|Γ G\el(C,G)| and |Γ G\ch(C,G)| are both equal to thenumber of primitive idempotents ofKB(C,G).

Proof. By Theorem 6.5, there are precisely|Γ G\el(C,G)| primitive idempotents oKB(C,G). The equality|Γ G\el(C,G)| = |Γ G\ch(C,G)| holds by an argument similar to the proof of Lemma 3.3. �

It is worth working through what the theorem says in the caseK = Q, which is likely tobe the main case of interest. The analysis will involve cyclicC-sections (defined belowwhich have appeared before, in the case whereC has prime order, in work of Bouc [10One motive for considering coefficients inQ is that the tenduction map, defined in Setion 9, is a polynomial function with coefficients inQ. Although the tenduction map exisfor coefficients inZ, we shall see that the polynomial formula is not closed when the cficient ring is an arbitrary ring of cyclotomic integers.

We define acyclic C-sectionof G to be a pair(V ,U) such thatU � V � G andV/U

is cyclic with order dividing|C|. We define aC-subcycleof G to be a pair(H,Z) whereH � G andZ is a cyclic subgroup ofH . As an abuse of notation, givenh ∈ H , we abbre-viate(H, 〈hO(H)〉) as(H, 〈h〉).

Lemma 6.7.Suppose thatK = Q, whenceΓ = Aut(C).

(1) TwoC-subcharacters(V , ν) and(W,ω) areΓ -conjugate if and only if(V ,Ker(ν)) =(W,Ker(ω)); they areΓ G-conjugate if and only if(V ,Ker(ν)) =G (W,Ker(ω)).


th

mof

nt-

-

-

Thus, theΓ G-classes ofC-subcharacters ofG are in a bijective correspondence witheG-classes of cyclicC-sections ofG.

(2) Two C-subelements(H,h) and (I, i) are Γ -conjugate if and only if(H, 〈h〉) =(I, 〈i〉); they areΓ G-conjugate if and only if(H, 〈h〉) =G (I, 〈i〉). Thus, theΓ G-classes ofC-subelements ofG are in a bijective correspondence with theG-classes ofC-subcycles ofG.

Proof. If the supernatural number|C| is even, then|CQ| = 2, otherwise|CQ| = 1. Eitherway, every automorphism ofC fixesCQ. So

Γ = Gal(Q[C]/Q

) = Aut(C/CQ) = Aut(C).

Let r be as in Proposition 6.1, and letQr be the cyclotomic number field obtained froQ by adjoining primitiverth roots of unity. LetZ/r denote the ring of residue classesrational integers modulor. Making evident identifications, we write

Γr = Gal(Qr/Q) = Aut(Cr) = (Z/r)∗.

Let π be the group epimorphismΓ → Γr specified in Remark 6.3. Given an elemeγ ∈ Γ , we interpretπ(γ ) as a rational integer coprime tor and well-defined up to congruence modulor. Let (V , ν) ∈ ch(C,G) and(H,h) ∈ el(C,G). Theπ(γ )th powerνπ(γ ) iswell-defined, because the order of the group elementν ∈ V dividesr. A similar commentholds for the elementhO(H) ∈ H . SinceΓ acts on ch(C,G) and el(C,G) via π , we have

γ(V , ν) = (V,νπ(γ )

), γ(H,h) = (

H,hπ(γ )−1).

So theΓ -conjugates of(V , ν) are theC-subcharacters having the form(V , νm) wherem is coprime tor. Part (1) is now established. TheΓ -conjugates of(H,h) are theC-subelements having the form(H,hm) where, again,m is coprime tor. Hence, part (2). �

By Theorem 6.5 and part (2) of Lemma 6.7,

eG,QH,h =

∑(I,i)

eGI,i ,

where(I, i) runs over representatives of theG-classes ofC-subcharacters ofG such that(I, 〈i〉) =G (H, 〈h〉). We have proved the following corollary.

Corollary 6.8. There is a bijective correspondenceeG,QH,h ↔ [H, 〈h〉]G between the prim

itive idempotentseG,QH,h of QB(C,G) and theG-classes[H, 〈h〉] of C-subcycles(H, 〈h〉)

of G. In particular, the number of primitive idempotents ofQB(C,G) is equal to the number ofG-classes ofC-subcycles ofG, and it is also equal to the number ofG-classes ofcyclicC-sections inG.


vert-

,e-s

fringe

ll also

ct

r

7. The primitive idempotents over a ring of integers

Let R be an integral domain of characteristic zero such that no rational prime is inible in R. The monomial Burnside ringRB(C,G) is, of course, an extension ofB(C,G).As we observed in Section 2,B(C,G) is a split extension ofB(G). Thus, in particularwe are regardingB(G) as a subring ofRB(C,G). The main result in this section, Thorem 7.3, says that all the idempotents ofRB(C,G) belong toB(G). The theorem wadiscovered independently by Boltje (private communication).

Throughout this section, we takeK to be a sufficiently large field containingR. Asusual, we choose and fix an arbitrary embedding ofC in Kω. Since the idempotents ointerest are (will eventually turn out to be) idempotents of the ordinary BurnsideB(G), let us explain how the notation simplifies in that case. In an evident way, th1-subcharacters ofG can be identified with the subgroups ofG, and similarly for the1-subelements ofG. When calculating with elements ofB(G), we judiciously delete thefiner details of the notation in the equations of Section 5. Thus

RB(G) =⊕

V �GG

R[G/V ], KB(G) =⊕

H�GG

KeGH ,

|V |NG(H)

[G/V ] =∑

H�G

ζ(H,V )eGH ,

|NG(H)||V | eG

H =∑V �G

µ(V,H)[G/V ].

These equations are due to Gluck [16] and Yoshida [26].Before proving the theorem, it is convenient to abstract some technicalities that wi

be useful in the next section. Consider the mutually dual abelian groupsG andG. Givenelementsω ∈ G andg ∈ G, thenω(g) is an element ofC. We have embeddedC in Kω, sowe may regardω(g) as an element ofK. The group algebraKG decomposes as a diresum of algebras

KG =⊕g∈G

Keg,

where eacheg is a primitive idempotent and

ω =∑g∈G

ω(g)eg

for all ω ∈ G. More generally, the group algebrasKG andKG are mutually dual vectospaces overK. Thus, for eachη ∈ KG we have an elementη(g) ∈ K, and we can write

η =∑

η(g)eg.

g∈G


nto be

ise.

ate

verse

je.

For a commutative ringΘ, let φΘ be theΘ-algebra monomorphismΘG → ΘB(C,G)

given by

φΘ(ω) = [CωG/G].

Let θΘ be theΘ-algebra epimorphismRB(C,G) → ΘG given by

θΘ [CνG/V ] ={

ν, if V = G,

0, otherwise.

We have decorated the symbolsφΘ andθΘ with the subscriptΘ because, in this sectioand the next, we shall be varying the coefficient ring, and we shall sometimes needclear as to which coefficient ring is under consideration. However,φΘ andθΘ are just theΘ-linear extensions ofφZ andθZ. We may drop the subscript when no ambiguity can arOf course,φ andθ also depend onC andG but, in all our discussions involvingφ andθ ,the groupsC andG will be fixed. SinceθΘ is a left-inverse ofφΘ , we have

RB(C,G) = φR(RG) ⊕ Ker(θΘ). (7)

Thus (in the terminology of Section 1),φΘ andθΘ realizeRB(C,G) as a split extensionof θG.

Lemma 7.1.Givenζ ∈ KB(C,G) thenθ(ζ ) = ∑g∈G sG

G,g(ζ )eg .

Proof. In view of the coordinate decomposition in Remark 5.3, we need only evaluθon the primitive idempotents ofKB(C,G). Givenω ∈ G andg ∈ G, then

µG(G,ω;G,g) = µ(G,ω;G,g) = ω(g)−1/|G|

whereg denotes the image ofg in G. Theorem 5.2 yields

θ(eGG,g

) =∑ω∈G

ω(g)−1ω/|G| = eg

andθ(eGH,h) = 0 for H < G. �

Lemma 7.2.An elementζ ∈ KB(C,G) belongs toKB(G) if and only ifsGH,h(ζ ) = sG

H,1(ζ )

for all C-subelements(H,h) of G. In that case,sGH,h(ζ ) = sG

H (ζ ).

Proof. Given aG-setS, thensGH,h[CS] = sG

H [S], which is independent ofh. ThereforeKB(G) is contained in the space of vectors satisfying the specified criterion. The reinclusion holds by considering dimensions.�

As we noted above, the following theorem was discovered independently by Bolt


f

dd

of

a

in, via

-

Theorem 7.3.The idempotents ofRB(C,G) are precisely the idempotents ofB(G).

Proof. We must show that an idempotentζ of RB(C,G) satisfies the criterion inLemma 7.2. By considering restrictions and arguing by induction on|G|, we reduce tothe task of showing thatsG

G,g(ζ ) = sGG,1(ζ ) for all g ∈ G. But θR(ζ ) is an idempotent o

RG. Since no rational prime is invertible inR, the only idempotents ofRG are 0 and 1. ByLemma 7.1, ifθR(ζ ) = 0, thensG

G,g(ζ ) = 0 for all g, while if θR(ζ ) = 1 thensGG,g(ζ ) = 1

for all g ∈ G. �To finish the matter off, the primitive idempotents ofRB(C,G) ought to be describe

explicitly in terms of the primitive idempotents ofKB(C,G). For that, we shall neeanother lemma.

Lemma 7.4.For H � G, the primitive idempotenteGH ∈ KB(G) decomposes as a sum

primitive idempotents ofKB(C,G), thus

eGH =

∑(I,i)

eGI,i ,

where(I, i) runs over representatives of theG-classes ofC-subelements ofG such thatI =G H .

Proof. Using Lemma 7.2,sGI,i (e

GH ) = sG

I (eGH ) = I =G H �. �

For a perfect subgroupQ of G, let

εGQ =

∑H

eGH ,

whereH runs over representatives of theG-classes of subgroups ofG such that the in-finitely derived subgroup ofH is G-conjugate toQ. Dress [13] showed that there isbijective correspondenceεG

H ↔ [Q]G between the primitive idempotentsεGQ of RB(G)

and theG-classes[Q]G of perfect subgroupsQ of G. The result can also be foundthe books by Benson [1, Corollary 5.4.8] and tom Dieck [12, Section 1.4]. HenceLemma 7.4, we have the following result.

Proposition 7.5.The primitive idempotents ofRB(C,G) coincide with the primitive idempotents ofB(G). They are precisely the elements having the form

εGQ =

∑(I,i)

eGI,i,

whereQ is a perfect subgroup ofG, and(I, i) runs over representatives of theG-classesof C-subelements ofG such that the infinitely derived subgroup ofI is G-conjugate toQ.


ts ofrtibleeertible

-l

d

]alçınOrder

the

o-

,

e

g

What remains open is the problem of determining the primitive idempotenRB(C,G) when we drop the hypothesis that none of the rational primes are invein R. In view of the 2-local decomposition ofB(G)∗ in Yoshida [27], it would be desirablto solve the problem in the case where all except one of the rational primes are invin the (characteristic zero) coefficient ring.

8. Units

We turn now to a study of the unit groupB(C,G)∗, the torsion-unit groupB(C,G)ω,and some subgroups ofB(C,G)ω . Let us recall some features of the unit groupB(G)∗of the ordinary Burnside ring. It is well known thatB(G)∗ is an elementary abelian 2group; to see this, observe that the set of speciesQB(G) → Q is a basis for the duaspace ofQB(G). SoB(G)∗ can be regarded as a vector space over the Galois fielF2.Long ago, tom Dieck [12, Proposition 1.5.1] observed that, supposing|G| is odd then,without the Odd Order Theorem,G is solvable if and only if|B(G)∗| = 2. That led him[12, Problem 1.5.2] to propose that, in the study ofB(G)∗, the “2-primary structure ofGis relevant” and he also signalled interest in the case whereG is a 2-group. Yoshida [27later vindicated the prediction as to the “2-primary structure.” Tornehave [24] and Y[25] have shown that the 2-group case has rich special features. But, by the OddTheorem,B(G)∗ captures nothing at all when|G| is odd. For arbitraryG, and an oddprime p, there is scant reason to expect much of a connection betweenB(G)∗ and the“p-primary structure” ofG.

For monomial Burnside rings, it is quite a different story. By Proposition 8.1,abelian groupB(C,G)ω is finite. Let us writeB(C,G)even andB(C,G)odd for the Sylow2-subgroup and the Hall 2′-subgroup ofB(C,G)ω . Theorem 9.6 implies that the decompsition B(C,−)ω = B(C,−)even⊕ B(C,−)odd is a direct sum of Mackey functors overZ.Proposition 8.1 implies that, for an odd primep, the groupB(Cp,G)odd is an elementaryabelianp-group, in other words,B(Cp,G)odd is a Mackey functor overFp . FurthermoreProposition 8.2 implies that, if|G| is odd, thenB(Cp,G)odd has index 2 inB(Cp,G)ω. Itseems reasonable to propose the Mackey functorB(Cp,−)odd as an odd prime analoguof B(−)∗.

All we shall do for the whole unit groupB(C,G)∗, in this section, is to explain howB(C,G)∗ can be regarded as a split extension of(ZG)∗. As special cases of two rinhomomorphisms defined in the previous section, consider the ring monomorphism

φZ :ZG → B(C,G)

and the ring epimorphism

θZ :B(C,G) → ZG.

Specializing to the unit groups, we have a group monomorphism

φ∗ : (ZG)∗ → B(C,G)∗


,ach

l

si-on

ert

tion 7,

l

e tech-at the

and a group epimorphism

θ∗ :B(C,G)∗ → (ZG)∗.

Observing thatθ∗ is a left-inverse forφ∗, we see that

B(C,G)∗ = φ∗(ZG)∗ ⊕ Ker(θ∗).

Proposition 8.1.Let r = gcd(|C|,exp(G)) and ρ = lcm(2, r). The groupB(C,G)∗ hasexponent dividingρ and rank at most|G\el(C,G)| = |G\ch(C,G)|.

Proof. Let K be the cyclotomic field generated overQ by the primitiverth roots of unity.The torsion-unit groupKω is cyclic with orderρ. Let ζ ∈ B(C,G)ω . By Proposition 6.1ζ decomposes as a linear combination of primitive idempotents as in Remark 5.3 and ecoordinatesG

H,h(ζ ) is a root of unity inK. �Proposition 8.2.If |G| is odd, thenB(C,G)ω = {±1} × B(C,G)odd.

Proof. Given a torsion-unitζ in B(C,G), thensG1,1(ζ ) must be±1 because it is a rationa

integer and a unit. Assume thatsG1,1(ζ ) = 1. We are to prove thatζ has odd order. LetK be

the cyclotomic field as in the proof of the previous proposition. In view of the decompotion in Remark 5.3, we are to show thatsG

H,h(ζ ) has odd order. An inductive argument|G| deals with the caseH < G.

In Section 2, we noted thatB(C,G) is a split extension ofB(G). The projectionπ : B(C,G) → B(G) is given by[CX] �→ [1 ⊗C CX] = [C\CX]. We havesG

H (π(ζ )) =sGH,1(ζ ) for all H � G. As noted by tom Dieck [12, Proposition 1.5.1],B(G)∗ = {±1};

the result can be deduced quickly from Yoshida’s criterion [27, Proposition 6.5] togethwith the Odd Order Theorem. Therefore,sG

G(π(ζ )) = sG1 (π(ζ )). We have shown tha

sGG,1(ζ ) = 1.

Consider the ring epimorphismθK :KB(C,G) → KG and its specialization

θω :B(C,G)ω → (ZG)ω.

SinceG is abelian, a weak version of Higman’s theorem says that

(ZG)ω = {±1} × G.

See, for instance, Sehgal [21, Corollary 1.6] or Serre [22, Exercise 6.3]. From Secrecall that the elementsη of KG have the coordinate decompositionη = ∑

g η(g)eg . When

η ∈ (ZG)ω, we haveη(1) = ±1, with the positive value if and only ifη belongs to the Hal2′-subgroupG of (ZG)ω. SincesG

G,1(ζ ) = 1, we see from Lemma 7.1 thatθω(ζ ) belongs to

the Hall 2′ subgroupG and, for the same reason,sGG,g(ζ ) has odd order for allg ∈ G. �

Let us mention some combinatorial bounds that can be proved using the samniques. We only sketch the arguments. Using Proposition 7.5, it is easy to show th


nk of

rs in8.9].

.

wea

arize

rcyclicngfll

rank of B(C,G)even is at most the number ofG-classes of cyclicC-sections inG. Us-ing Remark 5.3 and Dirichlet’s unit theorem, it is not hard to show that the free-raB(C,G)∗ is zero or at most|G\el(C,G)|(φ(r)/2−1), wherer is as above, andφ denotesthe Euler totient function. In particular, if lcm(2, r) � 6, then every unit inB(C,G) is atorsion unit. A very similar (but more refined) use of Dirichlet’s unit theorem appeathe proof of a general version of Higman’s theorem given in Karpilovsky [17, SectionEvidently, the theory ofB(C,G)∗ is related to the theory of commutative group rings.

9. Tenduction

The purpose of this section is to realize the unit groupB(C,−)∗ as a Mackey functorIt will follow immediately thatB(C,G)ω andB(C,G)even andB(C,G)odd are Mackeysubfunctors. For a subgroupF of G, we shall define a product-preserving map

ZC tenG

F :B(C,F ) → B(C,G)

called thetenduction mapfor coefficient ringZ. Except where emphasis is needed,shall tend to omit the left decorations. Since tenG

F is product-preserving, it specializes togroup homomorphism

tenGF :B(C,F )∗ → B(C,G)∗.

We shall find thatB(C,−)∗, equipped with the tenduction, restriction and conjugationmaps, is a Mackey functor.

The construction of the tenduction map is not at all straightforward. Let us summthe steps we shall be taking. We shall introduce a functor

ATenGF :A-F -SET→ A-G-SET

called thetenduction functor. Immediately from the definition, it will be clear thatATenGF

preserves products: givenA-fibredF -setsAX andAY , then

TenGF (AX ⊗ AY) = TenGF (AX) ⊗ TenGF (AY).

Then, we shall confine our attention to the case where the fibre group is a supegroupC. Let K be a field with characteristic zero. Most of the work will be in findia formula for the coordinatessG

H,h[TenGF (CX)] in the case whereK has enough roots ounity. Extending that formula, and still assuming thatK has enough roots of unity, we shabe able to introduce a function

KC tenG

F :KB(C,F) → KB(C,G)

called thetenduction mapfor coefficient ringK. The map tenGF will be related to thetenduction functor TenGF by the condition

C tenG[CX] = [CTenG(CX)

]. (8)
F F


s toto

does

-on

ingd intorexten-

. Thedatede

of co-s

t us

f cer-tion of

From the defining formula for tenGF , we shall show that the tenduction map specializea mapZ

C tenGF from B(C,F ) to B(C,G). We shall then (and only then) be in a positiondefineK

C tenGF for arbitraryK.

This complicated zigzag manoeuvre—over sufficiently largeK, overZ, then over ar-bitrary K—is necessary. Let us issue a couple of warnings. First warning: Eq. (8)not determineZ

C tenGF as a product-preserving map. Even in the caseF = G = 1, there

are two distinct product-preserving maps satisfying Eq. (8). Second warning: ifK is thefield of fractions of the integral domainR, then the tenduction mapKC tenG

F need not specialize to a functionRB(C,F ) → RB(C,G) and it need not specialize to a functiRB(C,F )∗ → RB(C,G)∗ . This is so even in the case whereC is trivial andR is a ring ofcyclotomic integers. We shall give a counter-example below.

It may be illuminating to make a comparison with the induction mapAindGF , which is,

of course, a sum-preserving map onB(A,F) satisfying

AindGF [AX] = [

AIndGF (AX)

]. (9)

The induction mapAindGF on KB(A,F) is not the unique sum-preserving map satisfy

Eq. (9). It is the uniqueK-linear map satisfying Eq. (9). Linear extension can be usethat way because the induction functor IndG

F preserves coproducts. The tenduction funcTenGF , though, does not preserve coproducts. To construct a tenduction map, linearsion is not an option.

Tenduction for permutation sets was defined by tom Dieck [12, Section 5.13]difficulty discussed above was noticed by Yoshida [27, Section 3b], who consolithe definition using a technique introducedby Dress [15]. We shall be following thsame approach. Let us recall the key notion behind the technique. Let{p1, . . . , pm} and{q1, . . . , qn} be bases for abelian groupsP andQ, respectively. A functionθ :P → Q issaid to bepolynomialprovided there exist polynomials

θ1, . . . , θn ∈ Q[X1, . . . ,Xm]

such that, for allα1, . . . , αm ∈ Z, we have

θ

(m∑

i=1

αipi

)=

n∑j=1

θj (α1, . . . , αm)qj .

Since composites of polynomial functions are polynomial, and since linear changeordinates is polynomial, the defining condition is independent of the choices of base{p1, . . . , pm} and {q1, . . . , qn}. We shall be making use of a uniqueness principle: lesay that a subsetD of P is densein P provided, for every finite subsetD0 ⊆ D, thesubgroup generated by the complementD − D0 has finite index inP . Two polynomialfunctions onP that agree on a dense subset ofP are equal. We shall characterizeC tenG

F

as the unique polynomial functionB(C,F ) → B(C,G) satisfying Eq. (8).Yoshida [27, Section 3a] characterized tenduction for permutation sets in terms o

tain hom-sets. Using ideas in Bouc [9, Section 6.7], the referee found a generaliza


f thecon-

ersal

er-n

ise

this to fibred permutation sets. We shall be working with an explicit construction otenduction functorATenG

F , but we shall also present the referee’s more systematicstruction.

For the explicit construction, we begin by choosing an ordered left-transv{t1, . . . , tm} for F in G. Let Sm denote the symmetric group of degreem = |G : F |. Theelements of the wreath productSm �F are the tuples(s;f1, . . . , fm) wheres ∈ Sm and eachfj ∈ F . The group operation is given by(

s′;f ′1, . . . , f

′m

)(s;f1, . . . , fm) = (

s′s;f ′s1f1, . . . , f

′smfm

).

We embedG in Sm � F via the inclusiong �→ (s(g);f1(g), . . . , fm(g)) where

gtj = ts(g)jfj (g).

It is easy to check that, up to conjugacy inSm � F , the embeddingG ↪→ Sm � F is indepen-dent of the choice of the transversal{t1, . . . , tm}.

Let AX be aA-fibredF -set. The tensor product, overA, of m copies ofAX, denoted

ATenGF (AX) =m⊗

AX,

is anA-fibredSm � F -set such that, forx1, . . . , xm ∈ X, we have

(s;f1, . . . , fm)(x1 ⊗ · · · ⊗ xm) = fs−11xs−11 ⊗ · · · ⊗ fs−1mxs−1m.

Observing thata1x1 ⊗ · · ·amxm = (a1 . . . am)x1 ⊗ · · · ⊗ xm for a1, . . . , am ∈ A, we seethat the action ofSm � F on ATenGF (AX) is independent of the choice of the setX ofrepresentatives of the fibres ofAX. Via the inclusionG ↪→ Sm � F , we regardATenG

F (AX)

as anA-fibredG-set. We makeATenGF become a functor, operating on maps by

ATenGF (α)(x1 ⊗ · · · ⊗ xm) = α(x1) ⊗ · · · ⊗ α(xm),

whereα is a map with domainAX. As before, it is easy to see thatATenGF (α) is inde-pendent of the choice of the setX. Since the inclusionG ↪→ Sm � F is well-defined up toconjugacy, the functorATenG

F is well-defined up to equivalence.The referee’s alternative construction ofATenGF avoids the need to choose any transv

sal for F in G. Let GF denoteG regarded as anF -set, with (left) action such that aelementf ∈ F sends an elementk ∈ GF to the elementkf −1. Allowing F to act triviallyonA, we form the setA= Hom(GF ,A), which becomes an abelian group with pointwmultiplication. Let∆ be the kernel of the group epimorphism

A � α �→∏

gF⊆G

α(g) ∈ A.

Via the action ofG onGF by left multiplication, we embedG in the groupΣ = Aut(GF ).Consider the setH = HomF (GF ,AX). We regardH as aΣ-set such that(σφ)(σk) = φ(k)


-

,

t

ction, Sec-

p

[12,

whereσ ∈ Σ andφ ∈ H. We regardH as anA-set such that(αφ)(k) = α(k)φ(k) whereα ∈ A. These two actions commute, soΣ and the groupA/∆ ∼= A have commuting actions on the orbit space∆\H. Moreover, the action ofA on ∆\H is free, so∆\H is anA-fibredΣ-set. By specialization,∆\H is anA-fibredG-set.

To show that

ATenGF (AX) ∼= ∆\H

asA-fibred G-sets, let us return to the transversal{t1, . . . , tm}. As observed in Bouc [9Proposition 6.11], there is an isomorphismΣ ∼= Sm � F such thatσ ↔ (s;f1, . . . , fm)

whereσ(tj ) = tsj fj . The embeddings ofG in Σ and inSm � F evidently commute withthis isomorphism, so we may identifyΣ with Sm �F . We identifyA with the direct producof m copies ofA via the correspondenceα ↔ (α(t1), . . . , α(tm)). We identifyH with thedirect product ofm copies ofAX via the correspondenceφ ↔ (φ(t1), . . . , φ(tm)). It iseasy to check that the bijection

ATenGF (AX) � a1x1 ⊗ · · · ⊗ amxm ↔ ∆(a1x1, . . . , amxm) ∈ ∆\H

is an isomorphism ofA-fibredSm � F -sets and, perforce, an isomorphism ofA-fibredG-sets.

Let us note that tenduction of fibred permutation sets is compatible with tenduof modules. The latter form of tenduction is discussed in, for instance, Benson [1tion 3.15]. EmbeddingA in the unit group of a commutative ringΘ, it is clear thatATenG

F

commutes with tenductionΘF -MOD → ΘG-MOD via the linearization functors LinFand LinG (defined in Section 2).

Lemma 9.1(Mackey decomposition). GivenE � G, then

ResGE TenGF (AX) ∼=⊗

EgF⊆G

ResEE∩ gF ResgFE∩ gF Cong

F (AX).

Proof. Consider the action ofE on the cosets ofF in G. �We now replaceA with the supercyclic groupC. Recall that the transfer ma

tGF :G → F is the group homomorphism given by

tGF (g) = πF

(f1(g) · · ·fm(g)

),

whereπF is the canonical projection fromF to F . The kernel oftGF containsO(G), so wemay regardtGF as a homomorphismG → F .

In the special case for permutation sets, the following formula is due to tom DieckProposition 5.13.1], and it can also be found in Yoshida [27, Section 3b].


-

e

.

ld

Lemma 9.2.Given aC-fibredG-setCX and a subelement(H,h) of G, then

sGH,h

[TenGF (CX)

] =∏

FgH⊆G

sFF∩ gH,tg(h)[CX],

wheretg(h) = tgHF∩ gH (gh).

Proof. First suppose thatH = G. The conditionCx1 = Cx determines a bijective correspondence between the fibresCx1 ⊗ · · · ⊗ xm in TenG

F (CX) stabilized byG and the fibresCx in CX stabilized byF . Suppose thatCx1 ⊗ · · · ⊗ xm is stabilized byG. Let φ be theC-character ofF such thatf x1 = φ(f )x1. Then

g(x1 ⊗ · · · ⊗ xm) = φ(tGF (h)

)x1 ⊗ · · · ⊗ xm

for g ∈ G. We have shown that the assertion holds in the caseH = G, indeed,

sGG,h

[TenG

F (CX)] = sF

F,tGF (g)[CX].

Suppose now thatH < G. Fork ∈ G, writeL(k) = H ∩ kF . Inductively, we may assumthat the assertion holds whenG is replaced byL(k). By Lemma 9.1,

sGH,h

[TenG

F (CX)] =

∏HkF⊆G

sHH,h

[TenHL(k) Res

kFL(k)

(kCX

)]=

∏HkF⊆G

sL(k)

L(k),tHL(k)(h)

[Res

kFL(k)

(kCX

)]=

∏HkF⊆G

skF

L(k),tHL(k)

(h)

[kCX

].

The argument is completed by substitutingg = k−1 and conjugating byg. �Still assuming thatK has enough roots of unity, we define a function

KC tenG

F :KB(C,F) → KB(C,G)

such that, givenξ ∈ KB(C,F) and (H,h) ∈ el(C,G), then the(H,h)-coordinate ofKC tenGF (ξ) is

sGH,h

(KC tenG

F (ξ)) =

∏FgH⊆G

sGF∩ gH,(gh)F

(ξ).

At present, we have definedKC tenGF only in the case whereK has enough roots of unity

The proofs of the following two results are to be interpreted only for suchK. However, assoon as we have definedK

C tenGF for arbitraryK, it will be obvious that the two results hoin general.


s

n

ficients

is

Lemma 9.3.The functionKC tenGF preserves products and satisfies Eq.(8).

Proof. The first part is immediate from the definition ofKC tenG

F . The second part followfrom Lemma 9.2. �Theorem 9.4.The functionK

C tenGF restricts to a polynomial functionZC tenGF :B(C,F ) →B(C,G).

Proof. Let (I1, i1), . . . , (Iu, iu) be a set of representatives of theF -classes ofC-subelements ofF , and let(J1, j1), . . . , (Jv, jv) be a set of representatives of theG-classesof C-subelements ofG. Consider an elementξ ∈ KB(C,F), and write

ξ =u∑

a=1

α′ae

FIa,ia

, tenGF (ξ) =

v∑b=1

β ′be

FJb,jb

.

For any givenξ , it is to be understood that eachα′a ∈ K and eachβ ′

b ∈ K. But, if weallow the coordinatesα′

a to vary, then eachβ ′b becomes a functionKu → K, written

(α′1, . . . , α

′u) �→ β ′

b(α′1, . . . , α

′u). The explicit formula in Lemma 9.2 shows that eachβ ′

b isa polynomial function whose coefficients are all 0 or 1. Let(U1,µ1), . . . , (Uu,µu) be a setof representatives of theF -classes ofC-subcharacters ofF , and let(V1, ν1), . . . , (Vv, νv)

be a set of representatives of theG-classes ofC-subcharacters ofG. Write

ξ =u∑

a=1

αa[Cµa F/Ua], tenGF (ξ) =

v∑b=1

βb[CνvG/Vb].

The linear changes of coordinates fromαa to α′a and fromβ ′

b to βb are expressed iEqs. (3)–(6) in Section 5. Thence, each function(α1, . . . , αu) �→ βb(α1, . . . , αv) is a poly-nomial function whose coefficients belong to some cyclotomic extension ofQ. Sinceβb takes rational values whenever its arguments are natural numbers, the coefof βb are rational. Consider rational integersn, α1, . . . , αu with n sufficiently large forour purposes. Given any rational primep, thenpn − α1, . . . , p

n − αu are non-negativeand pn does not divide the denominators of any of the coefficients ofβb. The integerβb(p

n + α1, . . . , pn + αu) differs fromβb(α1, . . . , αu) by a rational whose denominator

coprime top. Therefore, the denominator ofβb(α1, . . . , αu) is coprime top. Sincep isarbitrary,βb(α1, . . . , αu) is a rational integer. �

In the proof of the theorem, we describedZC tenGF as the polynomial function

ZC tenGF

(u∑

a=1

αa[Cµa F/Ua])

=v∑

b=1

βb(α1, . . . , αu)[CνvG/Vb].

Furthermore, we proved that the coefficients of the polynomialsβb are rationals. Nowletting K be arbitrary, we extendZ tenG to a functionK tenG :KB(C,F) → KB(C,G)
C F C F


nld for

tt,

t ringample

t

uck and

ic

f

ndof

s andty,

i-ndg

defined by the same formula. It is clear that, whenK has enough roots of unity, the functioKC tenGF is the same as before. It is also clear that Lemma 9.3 and Theorem 9.4 hoarbitraryK.

We have completed the construction of the tenduction mapRC tenG

F where the coefficienring R is eitherZ or else an arbitrary field of characteristic zero. It is easy to see thaembeddingC in the torsion unit groupΘω of a commutative ringΘ, thenR

C tenGF commuteswith the tenduction mapRR(ΘF) → RR(ΘG) on the representation rings.

Let us show that the constructions do not work in the case where the coefficienR is an arbitrary integral domain of characteristic zero. We present the counter-expromised earlier in this section. Letp be a prime, letC = 1, letG = Cp2 and letF be theproper non-trivial subgroup ofG. Letω be a primitive 2pth root of unity, letR = Z[ω] andlet K = Q[ω]. The elementξ = ω[F/F ] is a torsion-unit ofRB(C,G). We shall show thathe elementζ = K

C tenGF (ξ) of KB(C,G) does not belong toRB(C,G). Using Lemma 9.2and the coordinate transforms discussed in Section 5 (the special cases due to GlYoshida),

tenGF(α1[F/1] + α2[F/F ]) = (pα1 + α2)

p − αp

2

p2 [G/1] + αp

2 − α2

p[G/F ] + α2[G/G].

In particular,pζ/ω = (ωp−1 − 1)[G/F ] + [G/G]. It suffices to show that the algebrainteger 1− ωp−1 is not divisible byp. The casep = 2 is trivial. Assuming thatp is odd,thenωp−1 is a primitivepth root of unity. The product of thep − 1 Galois conjugates o1− ωp−1 is p. So at least one of the Galois conjugates is not divisible byp. It follows thatnone of the Galois conjugates are divisible byp, as required.

Proposition 9.5(Mackey formula). LetE � G. As functionsB(C,F ) → B(C,E) or, moregenerally, as functionsKB(C,F) → KB(C,G), we have

resGE tenGF =

∏EgF⊆G

tenEE∩ gF res

gFE∩ gF congF .

Proof. The two specified functions onB(C,F ) are equal because they are polynomial abecause they agree on the dense subset ofB(C,F ) consisting of the isomorphism classesC-fibredF -sets. The general case holds because any polynomial overZ extends uniquelyto a polynomial overK. �

Similar arguments show that the tenduction, restriction and conjugation functormaps satisfy all the relations that characterize Mackey functors: idempotence, transitiviand compatibility with conjugation. Furthermore, tenduction, restriction, and conjugationall preserve products. Therefore, regarding the ringB(C,G) as a module over the semring N, with actionn : ξ �→ ξn, thenB(C,−), equipped with tenduction, restriction, aconjugation, is a Mackey functor overN. More to the point, we have proved the followintheorem.


ps,tor.

th.

01..

6,

1.–83.

us

5.

Theorem 9.6.The functorB(C,−)∗, defined on the category of inclusions of finite grouand equipped with the tenduction, restriction and conjugation maps, is a Mackey funcoverZ. Furthermore,B(C,−)ω andB(C,−)evenandB(C,−)odd are Mackey subfunctors

In particular, for an odd primep, we have completed the realization ofB(Cp,G)odd asa Mackey functor over the fieldFp .

References

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fibred permutation sets and the idempotents and units of monomial burnside rings

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