The functor of complex characters of finite groups
aa r X i v : . [ m a t h . R T ] D ec The functor of complex characters of finite groups
Mehmet Arslan
MEF University, Faculty of Economics, Maslak, Istanbul, Turkey
Olcay Co¸skun ∗ Bogazici University, Department of Mathematics, Bebek, Istanbul, Turkey&Bogazici University Feza Gursey Center for Physics and Mathematics, Kandilli, Istanbul, Turkey
Abstract
We determine the structure of the fibered biset functor sending a finite group G to the complexvector space of complex valued class functions of G . Previously, it is studied as a biset functorby Bouc and as a C × -fibered biset functor by Boltje and the second author. In this paper, wecomplete the study by considering the other choices of the fiber group. As a corollary, we obtain a(previously known) set of composition factors of the biset functor of p -permutation modules. Keywords: fibered biset functors, character rings, primitive characters, p -permutation modules
1. Introduction
One of the fundamental constructions in representation theory of finite groups is the ring ofcharacters of a group. Over a field of characteristic zero, it determines all representations. Thefunctorial structure of character rings is studied in several ways, including Thvenaz and Webb’sdescription [8] of its Mackey functor structure, and Bouc’s description [2] of its biset functor struc-ture. Recently, it is investigated by Romero [7] as a Green biset functor and by Boltje and thesecond author [1] as a C × -fibered biset functor. In all these cases, it turns out to be a semi-simplefunctor. In the present paper, we are aiming to determine its structure as a fibered biset functorfor some other choices of fiber groups A < C × .Fibered biset functors are introduced by Boltje and the second author in [1] as a general frame-work for structures having actions of monomial bimodules. It originates from Bouc’s theory of bisetfunctors [2] where the basic tools are bisets and permutation bimodules. Basic theory of fiberedbisets and fibered biset functors together with a parametrization of simple functors is described in[1]. One of the most natural examples of fibered biset functors is the functor R C of complex char-acters under tensor product by monomial bimodules. It is shown in [1] that C R C is a C × -fiberedbiset functor, and hence by restriction, it is an A -fibered biset functor for any subgroup A ≤ C × . ∗ Corresponding author
Email addresses: mehmet [email protected] (Mehmet Arslan), [email protected] (Olcay Co¸skun) Both authors are supported by Tubitak-1001-113F240.
Preprint submitted to Elsevier December 24, 2018 n order to determine the A -fibered structure of C R C , we first analyze the structure of A -fibered bisets in some special cases. In [2], Bouc proves that any transitive biset is a product offive basic bisets, one corresponding to each one of common operations from module theory, namely,induction, inflation, transport of structure, deflation and restriction. In [1], Boltje and the secondauthor show that for a large fiber group A , a similar decomposition, with new basic elements, ispossible for fibered bisets. Later in [4], the second author and Ylmaz show that for abelian groups,this decomposition is simpler. One of our main results, Theorem 3.4, is a new decompositiontheorem which works for A -fibered bisets over cyclic groups when A is a small fiber group.The problem of determining the fibered biset functor structure of C R C falls into three casesdepending on the fiber group. For the first case, we consider a large fiber group for which thedecomposition for fibered bisets given in [1] holds. More precisely, we fix a set π of primes and let A be the group of all p n -th roots of unity for all n ∈ Z + and for all p ∈ π . In this case, the functor C R C turns out to be semisimple. The simple summands are given by the following theorem. Theorem 1
Let A be as above. There is an isomorphism of A -fibered biset functors C R C ∼ = M ( m,ζ ) ∈ [Υ π ] S A Z /m Z , , , C ζ . where [Υ π ] denote the set of all pairs ( m, ζ ) such that m ∈ N is a π ′ -number and ζ is a primitivecharacter of Aut( Z /m Z ) . Also given ( m, ζ ) ∈ [Υ π ] , we denote by S A Z /m Z , , , C ζ the simple A -fiberedbiset functor with minimal group Z /m Z and evaluation C ζ at Z /m Z . As an immediate corollary of this theorem, we recover a set of composition factors for the bisetfunctor of p -permutation modules. See Corollary 6.4. A more general set is already determined byBaumann in [3].We also consider the case where A is a finite cyclic p -group. In this case, the functor C R C is stillsemisimple. However determination of simple summands is more involved since now there are newbasic A -fibered bisets which do not appear in the previous case. We defer the detailed notation toSection 6 and state the following theorem. Theorem 2
Let A be a finite cyclic p -group. There is an isomorphism of A -fibered biset functors C R C ∼ = M ( m,ζ ) ∈ [Υ p ] S A Z /m Z , , , C ζ ⊕ M ( m,ζ ) ∈ [Υ > | A | ] S A Z /m Z ,A,α, C αζ . where [Υ p ] is defined as above, and Υ > | A | denotes the set of all pairs ( m, ζ ) as above with m p > | A | and [Υ > | A | ] is a full set of representatives of A -equivalence classes described in Section 6. The third and more general case where the fiber group is finite cyclic group of composite order ora product of a finite group and an infinite cyclic group as in the first case can be treated similarly.The paper is organized as follows. In Section 2, we include basic notations and results on fiberedbiset functors. Section 3 contains our main results on decomposition of fibered bisets over cyclicgroups when the fiber group is small. Using these results, we interpret the parametrization of simplefibered biset functors with cyclic minimal group in Section 4. We study actions of fibered bisets oncharacters in Section 5. The proofs of our main theorems are contained in the final section, Section6. 2e now give some notations that are valid throughout the paper. We let G and H be finitegroups and A be a multiplicatively written (not necessarily finite) abelian group. We also set G A := Hom( G, A ) and G ∗ := Hom( G, C )and view them as abelian groups with point-wise multiplication.
2. Preliminaries
In this section we collect necessary definitions and results from [1]. A set X is called an A -fibered G -set if X is an A × G -set such that the action of A is free with finitely many orbits. The categoryof A -fibered G -sets together with ( A, G )-equivariant functions is denoted by G set A . Disjoint unionof sets induces a coproduct on the category G set A . The Grothendieck group B A ( G ) of this categoryis called the A -fibered Burnside group of G .An A -fibered G -set is called transitive if the G -action on the set of A -orbits is transitive. It iseasy to show that there is a bijective correspondence between isomorphism classes of transitive A -fibered G -sets X and G -conjugacy classes of pairs ( U, φ ) where U is a subgroup of G and φ : U → A is a group homomorphism. The bijection is given by associating X to ( U, φ ) if U is the stabilizer ofsome A -orbit in X and U acts on this A -free orbit via φ . We call the pair ( U, φ ) corresponding to X the stabilizing pair of X . We denote by M G ( A ) the set of all such pairs ( U, φ ), and write [
U, φ ] G for the isomorphism class of the A -fibered G -set with the stabilizing pair ( U, φ ). The group G actson M G ( A ) via conjugation. We write M GG ( A ) or M GG (whenever A is clear from the content) forthe set of G -fixed points in M G ( A ).The set M G ( A ) has a poset structure given by the ordering ( K, κ ) ≤ ( L, λ ) if K ≤ L and κ = λ | K . Together with the above G -action, it becomes a G -poset.We further write G set AH for the category of A -fibered G × H -sets. By the usual convention,we regard any object in this category as an A -fibered ( G, H )-biset. We write (cid:2) G × HU,φ (cid:3) instead of[
U, φ ] G × H wherever we regard it as a fibered biset. With this notation, any ordinary biset (cid:2) G × HU (cid:3) is regarded as a fibered biset as (cid:2) G × HU, (cid:3) . Here 1 denotes the trivial homomorphism from U to A .Let K be another finite group, X be an A -fibered ( G, H )-biset and Y be an A -fibered ( H, K )-biset. We write AH = A × H . The usual amalgamated product of X and Y over AH is the set X × AH Y of AH -orbits in X × Y . Here AH acts on X × Y via( a, h ) · ( x, y ) = ( x · ( a − , h − ) , ( a, h ) · h )for any ( a, h ) ∈ A × H and ( x, y ) ∈ X × Y . Given ( x, y ) ∈ X × Y , we write ( x, AH y ) for its imagein X × AH Y . Given ( a, g, k ) ∈ A × G × K , we define( a, g ) · ( x, AH y ) · k = ( agx, AH yk )and with this action X × AH Y becomes an ( AG, K )-biset. In general, this is not an A -free set. Wedefine the tensor product X ⊗ AH Y of X and Y as the subset of X × AH Y consisting of A -freeorbits which is an A -fibered ( G, K )-biset. We denote a free orbit in X × AH Y by x ⊗ y . Note thatthe tensor product of A -fibered bisets is linear in both coordinates and induces a bilinear map B A ( G, H ) × B A ( H, K ) → B A ( G, K ) . A -fibered bisets. By Goursat’s Theorem,there is a bijective correspondence between subgroups U of G × H and quintuples ( P, K, η, L, Q )given by P = p ( U ) and Q = p ( U ), the first and the second projections of U . Also K = k ( U ) = p ( U ∩ ( G × L = k ( U ) = p ( U ∩ (1 × H )). We clearly have that K E P and L E Q and the subgroup U ≤ G × H induces an isomorphism η : Q/L → P/K given by η ( hL ) = gK if ( g, h ) ∈ U . We call this the Goursat correspondence and U the Goursat correspondent of( p ( U ) , k ( U ) , η, k ( U ) , p ( U )) and vice versa.Given a pair ( U, φ ) ∈ M G × H ( A ), we also write φ | K × L = φ × φ − . We call the triple l ( U, φ ) =(
P, K, φ ) (resp. r ( U, φ ) = (
Q, L, φ )) the left invariants (resp. right invariants) of ( U, φ ). Wesometimes shorten the invariants and write l ( U, φ ) = (
K, φ ) and r ( U, φ ) = (
L, φ ).With this notation, given transitive elements (cid:2) G × HU,φ (cid:3) and (cid:2) H × KV,ψ (cid:3) , by [1, Corollary 2.5] we have h G × HU, φ i ⊗ AH h H × KV, ψ i = X x ∈ p ( U ) \ H/p ( V ) φ | Hx = x ψ | Hx h G × KU ∗ ( x, V, φ ∗ ( x, ψ i where H x = k ( U ) ∩ x k ( V ), and the subgroup U ∗ V is the composition U ∗ V = { ( g, k ) ∈ G × K | ( g, h ) ∈ U, ( h, k ) ∈ V for some h ∈ H } and the homomorphism φ ∗ ψ : U ∗ V → A is defined by( φ ∗ ψ )( g, k ) = φ ( g, h ) · ψ ( h, k )for some choice of h ∈ H such that ( g, h ) ∈ U and ( h, k ) ∈ V . Note that the homomorphism φ ∗ ψ is independent of the choice of h ∈ H . This product is called the Mackey product of fibered bisets.Now we define fibered biset functors. Let A be an abelian group and R be a commutative ringwith unity. Let C := C AR be the category where(i) the objects of C are all finite groups.(ii) Given two finite groups G and H ,Hom C ( G, H ) := R ⊗ B A ( H, G ) = RB A ( H, G ) . (iii) The composition is the R -linear extension of the tensor product of A -fibered bisets introducedabove.An A -fibered biset functor over R is an R -linear functor C → R mod. The class of all A -fiberedbiset functors together with natural transformations between them forms a category, denoted by F := F AR . Since R mod is an abelian category, the category F is also abelian. Classification ofsimple fibered biset functors is done in [1, Section 9]. We review the parametrization in a specialcase in Section 4.
3. Fibered bisets for cyclic groups
By [2, Corollary 7.3.5], one parameter for the classification of simple summands of the bisetfunctor of character rings is finite cyclic groups. It turns out that, also in the case of fibered biset4unctors, one of the parameters runs over a set of finite cyclic groups. In this section, we specializesome results from [1] to the case of cyclic groups.First we introduce notation for basic fibered bisets which is used throughout the paper. Let H be a subgroup of G and N be a normal subgroup of G . Also let G ′ be a finite group and λ : G ′ → G be an isomorphism.We define induction from H to G and restriction from G to H as the transitive bisets given,respectively, by Ind GH := G G H and Res GH := H G G where we regard the set G as a ( G, H )-biset (resp. as an (
H, G )-biset) in the usual way, via leftand right multiplication by the corresponding group. We also define deflation from G to G/N and inflation from
G/N to G as the transitive bisets given, respectively, byDef GG/N := G/N ( G/N ) G and Inf GG/N := G ( G/N ) G/N . As above, we regard the set
G/N as a (
G/N, G )-biset (and as a (
G, G/N )-biset) in the usual way.Finally, we define the transport of structure from G ′ to G through λ as the transitive biset given byc λG,G ′ = Iso λG,G ′ := G G G ′ where the G action is the left multiplication and the G ′ -action is multiplication through λ . In allthese cases, the A -action is trivial.There are three other basic elements that we need in this paper. The first is twist biset definedas follows. Let φ ∈ G A be a homomorphism from G to A . Then the twist by φ at G is the A -fibered( G, G )-biset Tw φG = (cid:16) G × G ∆( G ) , ∆ φ (cid:17) where ∆( G ) is the diagonal subgroup of G × G and ∆ φ ( g, g ) = φ ( g ) for any g ∈ G .The other two basic elements are both idempotents in B A ( G, G ). Let (
K, κ ) ∈ M GG ( A ). Wewrite ∆ K ( G ) = { ( g, h ) ∈ G × G : gK = hK } and define φ κ : ∆ K ( G ) → A by φ κ ( g, h ) = κ ( gh − ) = κ ( h − g ). With this notation, following [1], we define E G ( K,κ ) = (cid:16) G × G ∆ K ( G ) , φ κ (cid:17) . For simplicity, we write e G ( K,κ ) for the isomorphism class of E G ( K,κ ) . Whenever there is no risk ofconfusion we drop the letter G and write e ( K,κ ) . As shown in [1, Section 4.3], for all ( K, κ ) , ( L, λ ) ∈M GG ( A ), we have e ( K,κ ) · e ( L,λ ) = ( e ( KL,κλ ) if κ | K ∩ L = λ | K ∩ L . The following construction introduces orthogonal idempotents from e ( K,κ ) . Let µ ⊳K,L denote theMobius coefficient with respect to the poset of normal subgroups of G . Given ( K, κ ) ∈ M GG , wedefine f ( K,κ ) = X ( K,κ ) ≤ ( L,λ ) ∈ M GG µ ⊳K,L e ( L,λ ) .
5y [1, Proposition 4.4], the idempotents f ( K,κ ) form a set of mutually orthogonal idempotents in B A ( G, G ) summing up to the identity element e (1 , . For further properties of these elements, werefer to Section 4 of [1].In [2], Bouc showed that any transitive ( G, H )-biset ( G × H ) /U is equal to a Mackey productof the above five basic bisets. More precisely, we have (cid:18) G × HU (cid:19) = Ind GP Inf
PP/K
Iso ηP/K,Q/L
Def
QQ/L
Res HQ where ( P, K, η, L, Q ) is the Goursat correspondent of U . A similar, but partial, decomposition fora transitive A -fibered ( G, H )-biset is given in [1, Proposition 2.8]. Particularly if φ : U → A , thenwe have (cid:18) G × HU, φ (cid:19) = Ind GP Inf
PP/ ˆ K ⊗ AP/ ˆ K Y ⊗ AQ/ ˆ L Def
QQ/ ˆ L Res HQ (1)where ˆ K and ˆ L are kernels of φ and φ , respectively. Here Y is a transitive A -fibered ( P/ ˆ K, Q/ ˆ L )-biset with full left and right projections and faithful left and right restrictions of the induced fiberhomomorphism φ ′ . A further decomposition of Y is given in [1, Section 10] when A satisfies acondition. We recall the condition. There is a (unique) set π of primes such that for every n ∈ N , the n -torsionpart of A is cyclic of order n π , where n π denotes the π -part of n .In [4], a simplified decomposition for transitive A -fibered ( G, H )-bisets is given when A satisfiesthe above hypothesis and G and H are abelian groups. We recall this decomposition. Let ( U, φ ) ∈M G × H ( A ). We write ( P, K, η, Q, L ) for the Goursat correspondent of U . We also write ˜ φ = ˜ φ × ˜ φ for an extension of φ to P × Q which exists since the group P × Q is abelian and A is divisible (bythe hypothesis). Then by [4, Theorem 1], we have (cid:16) G × HU, φ (cid:17) ∼ = Ind GP Tw ˜ φ P Inf
PP/K
Iso ηP/K,Q/L
Def
QQ/L Tw ˜ φ Q Res HQ . (2)Note that, this decomposition is valid whenever | A | p ≥ max( | G | p , | H | p ) for any prime number p since the extension ˜ φ would still exist under this last condition. For the rest of this section, wedetermine the decomposition of A -fibered ( G, H )-bisets when
G, H and A are all finite cyclic groups.By the next lemma, we reduce this problem to p -groups. This is the fibered version of [2,Proposition 2.5.14]. Let X be an A -fibered G -set and Y be an A -fibered H -set. Then the Cartesianproduct X × Y is an A -fibered G × H -set via ( g, h, a ) · ( x, y ) = ( a · g · x, a · h · y ) for all ( g, h, a ) ∈ G × H × A and ( x, y ) ∈ X × Y . Moreover the correspondence ( X, Y ) X × Y induces a bilinear map from B A ( G ) × B A ( H ) to B A ( G × H ) and hence a group homomorphism B A ( G ) ⊗ Z B A ( H ) → B A ( G × H ) . This is a unital injective ring homomorphism which becomes an isomorphism when G and H areof coprime orders. X × Y is free as an A -set underthe diagonal action a · ( x, y ) = ( a · x, a · y ) which clearly holds. We skip the details of the proof.Now given an abelian group G , we decompose it as G ∼ = Y p G p where the product is over the set of distinct prime divisors of | G | and for a prime divisor p , wewrite G p for the Sylow p -subgroup of G . Therefore, by the above lemma, given any element X ∈ B A ( G, G ), we can decompose it as X ∼ = Y p X p where X p ∈ B A ( G p , G p ) for each prime divisor p of | G | . Also it is easy to show that if H is abelianof order less than | G | and X ∼ = Y ⊗ AH Z , then we also have X ∼ = Y p Y p ⊗ AH p Z p where H p is the Sylow p -subgroup of H and Y p and Z p are chosen as above. Therefore the decom-position can be done by considering one prime at a time. Notice that if A has trivial p -torsion then B A ( G p , G p ) ∼ = B ( G p , G p ) and hence in this case, any A -fibered ( G p , G p )-biset may be regardedas an ordinary ( G p , G p )-biset and hence can be decomposed using Bouc’s result. If | A | p ≥ | G | p ,then we can use Equation 2 to decompose X p . Hence, for abelian groups G and H , to determine afactorization of an A -fibered ( G, H )-biset X , we only need to determine the case where the groups G, H and A are p -groups for a fixed prime p and | A | < | G | ≥ | H | . Then it turns out that thedecomposition depends on the order of | A | relative to | H | .More precisely, we fix l ∈ N and let A = h a i be a cyclic group of order p l , written multiplicatively.Also let G and H be cyclic groups of orders p m and p n , respectively. We assume that l < m ≥ n .Note that the case m ≤ n can be done similarly or by taking opposites.Let ( U, φ ) ∈ M G × H ( A ). By [1, Proposition 2.8], we may also assume that p ( U ) = G, p ( U ) = H, ker( φ ) = 1 , ker( φ ) = 1 . Since A is finite, the above assumptions put some conditions on ( U, φ ). Indeed, to have a faithfulhomomorphism φ : k ( U ) → A , we should have | k ( U ) | ≤ | A | . Similarly, for k ( U ). Thus, if( U, φ ) ∈ M G × H ( A ) satisfies the above conditions, we also have | k ( U ) | ≤ | A | ≥ | k ( U ) | . Now given a pair (
U, φ ) as above, we let (
G, K, η, L, H ) be its Goursat correspondent. We fix agenerator h of H and let g be a generator of G such that η ( hL ) = gK . This is possible since p ( U ) = G . Note that with this choice, we get ( g, h ) ∈ U , and hence ∆ η ( G ) := h ( g, h ) i ≤ U . Noticealso that | ∆ η ( G ) ∩ (1 × L ) | = 1 and | ∆ η ( G ) · (1 × L ) | = | G | · | L | = | U | . Hence we have U = ∆ η ( G ) · (1 × L ) and G × H = ∆ η ( G ) · (1 × H ) .
7e further write φ = φ η · φ L where φ η (resp. φ L ) is the restriction of φ to ∆ η ( G ) (resp. to 1 × L ). Note that, φ L = 1 × φ andhence it is faithful by our assumption.Next suppose | A | ≥ | H | , then φ L extends to a homomorphism φ H : 1 × H = H → A . Given( x, y ) ∈ G × H , we write ( x, y ) = ( x, ˜ x ) · (1 , (˜ x ) − y ) as an element of ∆ η ( G ) · (1 × H ). Define ψ : G × H → A by ψ ( x, y ) = φ η ( x, ˜ x ) φ H ((˜ x ) − y ). This is a homomorphism since both φ η and φ H are homomorphisms and e xt = ˜ x ˜ t for any x, t ∈ G . Also ψ is an extension of φ . Indeed for any x ∈ G , we have ( x, ˜ x ) ∈ U by definition, and hence if ( x, y ) is also in U , then (1 , (˜ x ) − y ) ∈ U . Inparticular, we can apply Equation 2 and get (cid:16) G × HU, φ (cid:17) ∼ = Tw ψ G G Inf
GG/K
Iso ηG/K,H/L
Def
HH/L Tw φ H H . Here, we write ψ G for the restriction of ψ to G = G ×
1. Note that the term Tw ψ G G may bedecomposed further, but for our aims, this version is sufficient.For the second case where | A | < | H | , by [1, Proposition 4.2.(b)], there is an isomorphism (cid:16) G × HU, φ (cid:17) = (cid:16) G × H ∆ η ( G ) · (1 × L ) , φ η · φ L (cid:17) ∼ = (cid:16) G × H ∆ η ( G ) , φ η (cid:17) ⊗ AH E H ( L,φ L ) . (3)of A -fibered ( G, H )-bisets. We consider the factorization of the terms in the above isomorphismseparately. First consider the pair (∆ η ( G ) , φ η ) ∈ M G × H . We clearly have p (∆ η ( G )) = G, p (∆ η ( G )) = H and k (∆ η ( G )) = 1 . In general, k (∆ η ( G )) = K , and we let K η = k (∆ η ( G )), and˜ η : H → G/K η be the canonical isomorphism determined by ∆ η ( G ). Also let ˜ φ : G → A be the character given by˜ φ ( g ) = φ η ( g, h ). With this notation, we have (cid:16) G × H ∆ η ( G ) , φ η (cid:17) = Tw ˜ φG Inf
GG/K η Iso ˜ ηG/K η ,H . Indeed, to prove the equality, write (
V, ψ ) for the stabilizing pair of the right hand side. Then V = ∆( G ) ∗ { ( x, xK η ) : x ∈ G } ∗ { (˜ η ( y ) , y ) : y ∈ H } (4)= { ( x, y ) ∈ G × H : xK η = ˜ η ( y ) } = ∆ η ( G ) . On the other hand, since both the inflation and the isomorphism bisets have trivial fiber homomor-phism, we get ψ = ˜ φ , proving the equality.Once more the term Tw ˜ φG may be decomposed further but we do not consider it. For the otherterm in Equation 3, we prove a more general criterion for an idempotent e G ( K,κ ) to be reduced. Let G and A be finite cyclic p -groups. Then i) If | A | ≥ | G | , then e ( K,κ ) / ∈ I G if and only if ( K, κ ) = (1 , .(ii) If | A | < | G | , then e ( K,κ ) / ∈ I G if and only if κ is faithful. Proof If | A | ≥ | G | , then Equation 2 is applicable and hence any A -fibered ( G, G )-biset can bedecomposed into a product of basic fibered bisets. It is clear from the decomposition that the onlyreduced elements are products of twist and isogation. It is also clear that the only element e ( K,κ ) which can be written in this form is e (1 , .For the second case note that by [1, Proposition 8.6], if e ( K,κ ) / ∈ I G , then κ is faithful. Converselysuppose κ is faithful. Assume, for a contradiction, that E G ( K,κ ) is not reduced, and write E G ( K,κ ) = (cid:16) G × HU, φ (cid:17) ⊗ AH (cid:16) H × GV, ψ (cid:17) for a group H of order smaller than | G | and for pairs ( U, φ ) ∈ M G × H ( A ) and ( V, ψ ) ∈ M H × G ( A ).We further let g be a generator of the cyclic group G . Then since ( g, g ) ∈ ∆ K ( G ), there exists h ∈ H such that ( g, h ) ∈ U and ( h, g ) ∈ V . Notice that since | H | < | G | , the order of h is less thanthe order of g . Now consider the subgroup U ′ of U generated by ( g, h ). Clearly, U ′ is isomorphicwith G and k ( U ′ ) is isomorphic to the cyclic group of order | G | / | H | . Also k ( U ′ ) ≤ k ( U ) ≤ k ( U ∗ V ) = K. Here the second inequality follows from [1, Proposition 2.6 (a)]. Also by [1, Proposition 2.6 (b)],the restriction of φ to the subgroup k ( U ′ ) × κ ×
1. In particular,since κ is faithful, the homomorphism φ | k ( U ′ ) × is faithful. But clearly φ | U ′ extends φ | k ( U ′ ) × andsince U ′ is cyclic, the extension must be faithful. This is a contradiction because we assumed thatthe order of G is strictly greater than | A | . Hence E G ( K,κ ) must be reduced, as required. (cid:3) Note that when | A | < | G | and κ ∈ K A is not faithful, we have a decomposition e ( K,κ ) = Inf GG/ ˆ K e ( K/ ˆ K, ˆ κ ) Def
GG/ ˆ K (5)where ˆ K is the kernel of κ and ˆ κ is the character induced by κ . Moreover e ( K/ ˆ K, ˆ κ ) / ∈ I G/ ˆ K .With the above result, we see that the second term in the factorization (3) is reduced, andhence it does not factor through a group of smaller order. This completes the proof of the followingtheorem. Let p be a prime number, let G, H and A be finite cyclic p -groups with | G | ≥ | H | and let ( U, φ ) ∈ M G × H such that p ( U ) = G, p ( U ) = H, ker( φ ) = 1 , ker( φ ) = 1 . Put K = k ( U ) and L = k ( U ) . Then(i) Suppose | A | ≥ | H | . Then φ extends to φ H : H → A , and let ψ G : G → A be as above. Thenthe A -fibered ( G, H ) -biset ( G × HU,φ ) factors as (cid:16) G × HU, φ (cid:17) ∼ = Tw ψ G G Inf
GG/K
Iso ηG/K,H/L
Def
HH/L Tw φ H H . ii) Suppose | A | < | H | . Let H = h h i , G = h g i such that ( g, h ) ∈ U and η : H/k ( U ) → G/k ( U ) bethe canonical isomorphism induced by U . Set ∆ η ( G ) = h ( g, h ) i ≤ G × H, K η = k (∆ η ( G )) .Let ˜ η : H → G/K η be the canonical isomorphism induced by ∆ η ( G ) . Finally let ˜ φ : G → A, ˜ φ ( g ) = φ ( g, h ) , and φ L = φ | × L . Then the A -fibered ( G, H ) -biset ( G × HU,φ ) factors as (cid:16) G × HU, φ (cid:17) ∼ = Tw ˜ φG Inf
GG/K η Iso ˜ ηG/K η ,H E HL,φ L .
4. Simple fibered biset functors with cyclic minimal groups
A classification of simple fibered biset functors is given in [1, Section 9]. In this paper we onlyneed a special case of this result when the minimal group is cyclic. In this section we specialize tothis case.Let S be a simple A -fibered biset functor and G be a group such that S ( G ) = 0 but S ( H ) = 0for any H with | H | < | G | . In this case, we call G a minimal group for S . As explained in [2, Section4], for any finite group K , the evaluation S ( K ) is either zero or a simple module over the algebra E K = End C ( K ). Moreover, for a minimal group G , the module S ( G ) is a simple module over theessential algebra ¯ E G which is defined as follows. Let I G be the ideal generated by all elements in E G which factor through a group of smaller order. Then ¯ E G = E G /I G . A detailed structure of theessential algebra is described in [1].Conversely given a simple module V over the essential algebra ¯ E G , there is a unique simple A -fibered biset functor S with minimal group G and S ( G ) ∼ = V . However, for the simple functor S , theminimal group G is not always unique up to isomorphism. By [1, Theorem 9.2], the minimal groupsfor a given simple fibered biset functor are parameterized by linkage classes of certain triplets.Therefore to get a parametrization of simple fibered biset functors with cyclic minimal groups,we need to determine simple modules over the essential algebra and the linkage classes for cyclicgroups. The following result recalls the structure of the essential algebra for two particular cases. Let R = k be a field. Then(i) (Bouc, [2]) If | G | -torsion of A is trivial, then ¯ E G ∼ = k Out( G ) .(ii) (Co¸skun - Yılmaz, [4]) If G is abelian and A has a non-trivial element of order | G | , then ¯ E G ∼ = k [ G ∗ ⋉ Aut( G )] . Suppose G is cyclic. By Lemma 3.2, for any abelian group A , there is an isomorphism of algebras¯ E G ∼ = Y p i ¯ E P i . Here p i runs over all distinct prime divisors of | G | and for each i , we denote by P i the Sylow p i -subgroups of G . In particular if p i does not divide | A | , or if the p i -part of | A | is greater than thatof | G | , the structure of ¯ E P i is given by Theorem 4.1. Thus we need only to discuss the case where p i -part of | A | is non-trivial and less than that of | G | . Hence let p be such a prime and assume that G and A are finite p -groups with | A | < | G | . 10y [1, Corollary 8.5], simple modules over the algebra ¯ E G are parameterized by simple modulesover certain group algebras. To describe this parametrization, we introduce further notation. Let R G be the subset of M G consisting of reduced pairs, that is, R G = { ( K, κ ) ∈ M G : e ( K,κ ) / ∈ I G } . The triplets (
G, K, κ ) , ( G, L, λ ) are said to be linked , written (
G, K, κ ) ∼ ( G, L, λ ), if there is an A -fibered ( G, G )-biset ( G × GU,φ ) such that l ( U, φ ) = (
G, K, κ ) and r ( U, φ ) = (
G, L, λ ). In this casewe say that ( G × GU,φ ) links (
G, K, κ ) to (
G, L, λ ). This induces an equivalence relation on M G . Theequivalence class, called the linkage class, containing ( K, κ ) is denoted by { K, κ } . Note that if( G, K, κ ) ∼ ( G, L, λ ), then | K | = | L | as noted in [1, Definition 5.1]. Since G is cyclic, we furtherhave K = L . Finally note that if ( G, K, κ ) ∼ ( G, K, λ ), then (
K, κ ) ∈ R G if and only if ( K, λ ) ∈ R G ,by [1, Section 8.1].Given ( K, κ ) ∈ R G , we write Γ ( G,K,κ ) for the set of isomorphism classes of all A -fibered ( G, G )-bisets ( G × GU,φ ) such that l ( U, φ ) = (
G, K, κ ) = r ( U, φ ). By [1, Section 6.1], it becomes a group underthe tensor product, with the identity element e ( K,κ ) and inverses given by opposite fibered bisets.With this notation, Corollary 8.5 in [1] gives a bijective correspondence between(i) isomorphism classes of simple ¯ E G -modules and(ii) triples ( K, κ, [ V ]) where ( K, κ ) runs over R G , up to linkage, and [ V ] runs over isomorphismclasses of simple k Γ ( G,K,κ ) -modules.The correspondence is given by ( K, κ, [ V ]) ¯ E G ¯ f ( K,κ ) ⊗ k Γ ( G,K,κ ) V where ¯ f ( K,κ ) is the image of f ( K,κ ) under the canonical projection onto ¯ E G .It turns out that when G is cyclic, there is only one linkage class of reduced pairs once we fixthe subgroup K . More precisely we prove the following proposition. Let p be a prime number and G and A be finite cyclic p -groups with | A | < | G | .Given K ≤ G , the triplets ( G, K, κ ) and ( G, K, λ ) are linked for any faithful κ, λ ∈ K A . Proof
For simplicity, we write κ ∼ λ instead of ( G, K, κ ) ∼ ( G, K, λ ). Since κ and λ are faithful,the pairs ( K, κ ) and (
K, λ ) are both reduced by Theorem 3.3. Thus if κ ∼ λ , then by Theorem 3.4,any link ( G × GU,φ ) between κ and λ is of the formTw ψG Iso αG,G E G ( K,κ ) for some ψ ∈ G A and α ∈ Aut( G ). In particular, the linkage class of κ is determined by the leftinvariants of products of the above type. To determine these invariants, let κ ∈ K A , ψ ∈ G A and α ∈ Aut( G ). We consider the productTw ψG Iso αG,G E G ( K,κ ) = (cid:16) G × G ∆( G ) , ∆ ψ (cid:17)(cid:16) G × G α ∆( G ) , (cid:17)(cid:16) G × G ∆ K ( G ) , φ κ (cid:17) where α ∆( G ) := { ( α ( g ) , g ) | g ∈ G } . By the Mackey product formula for fibered bisets, the aboveproduct is a transitive fibered biset with stabilizing pair ( U, φ ) where U = ∆( G ) ∗ α ∆( G ) ∗ ∆ K ( G ) = { ( g , g ) : α ( g − ) K = g K } φ ( g , g ) = ψ ( g ) κ ( g − α ( g − )) for ( g , g ) ∈ U . In particular we have p ( U ) = G = p ( U ) , k ( U ) = K = k ( U ) . Here we note that since G is cyclic, we have α ( K ) = K . Moreover for g ∈ K , we have φ ( g,
1) = ψ ( g ) κ ( α ( g − )) = ( ψ · α − κ )( g ) and φ (1 , g ) = ψ (1) κ ( g − ) = κ − ( g ) . Therefore we obtain l ( U, φ ) = (
G, K, ψ · α − κ ) and r ( U, φ ) = (
G, K, κ ) . In particular we see that the pairs (
K, κ ) and (
K, λ ) are linked if and only if there exists φ ∈ G A and α ∈ Aut( G ) such that λ = φ · α κ . To see that κ ∼ λ for any faithful κ and λ , note that, sincethe characters κ and λ are faithful, their extensions χ and χ to G are also faithful. Hence thereexists an automorphism σ of G such that χ = σ χ and hence λ = σ κ . (cid:3) Therefore when G and A are finite cyclic p -groups with | A | < | G | , the set of linkage classes on R G is in one-to-one correspondence with the set of subgroups of G of order at most | A | . Hence inthis case, there is a bijective correspondence between(i) the isomorphism classes of simple ¯ E G -modules(ii) the pairs ( K, [ V ]) where K ≤ G runs over subgroups of order at most | A | and [ V ] runs overisomorphism classes of simple k Γ G,K,κ -modules for a fixed faithful character κ of K .Now let G be a cyclic p -group and A be abelian. Given a simple A -fibered biset functor S withminimal group G and S ( G ) = V . If | G | -torsion of A is trivial, then A -fibered ( G, G )-bisets can beidentified with (
G, G )-bisets and hence by [2, Theorem 4.3.10], G is the unique minimal group for S ,up to isomorphism and the only reduced pair for G is (1 , S = S G,V = S AG, , , [ V ] .Otherwise if A has an element of order | G | , then by Equation 2, G is unique up to isomorphism,again there is only one reduced pair for G . Thus we have S = S AG, , , [ V ] .Finally in the remaining case, suppose S ( G ) corresponds to the triple ( K, κ, [ V ]). By [1, Propo-sition 9.7], any other minimal group H for S must be abelian of order | G | and there must exist( L, λ ) ∈ M H such that ( G, K, κ ) is linked to (
H, L, λ ). As we shall see at the end of Section 6, forsimple subfunctors of the functor of complex characters, H must be isomorphic to G .
5. Actions on characters
We denote by Irr( G ) the set of irreducible complex characters of the group G and by R C ( G ) = M χ ∈ Irr ( G ) Z · χ the ring of complex characters of G . As usual, we identify it with the Grothendieck ring of thecategory of finite dimensional C G -modules, and identify the set Irr( G ) by a complete set of iso-morphism classes of simple C G -modules. Given a C G -module M , we denote its image in R C ( G )by χ M . The functor sending G to R C ( G ) is a C × -fibered biset functor as described in [1]. In the12ame way it has a structure of an A -fibered biset functor for any subgroup A of C × . We recall thisstructure.Let A be a subgroup of C × . As in the previous section, let B A denote the A -fibered bisetfunctor of Burnside groups, sending any group G to the Burnside group B A ( G ). Given a transitive A -fibered G -set X = [ U, φ ] G , we can construct the monomial C G -module C X with monomial basis X , that is, the C -vector space C X with basis the A -orbits X/A of X and the C G -action inheritedfrom the G -action on X/A . It is easy to show that C X ∼ = Ind GU C φ as C G -modules, where C φ denotes the 1-dimensional representation of U with character φ . Thenthe well-known linearization map Lin G : B A ( G ) → R C ( G )is defined as the linear extension of this correspondence. Similarly, if Y is an A -fibered ( H, G )-biset,then the linearization of Y can be regarded as a monomial ( C H, C G )-bimodule and hence inducesa group homomorphism R C ( Y ) : R C ( G ) → R C ( H )given by R C ( Y )( χ M ) = χ C Y ⊗ C G M . For simplicity, we denote this homomorphism by Y GH . It isshown in [1, Section 11.4] that with this induced action of fibered bisets, the functor R C becomesan A -fibered biset functor. Note that when A is trivial, the above definition gives a biset functorstructure on the functor R C which also becomes a Green biset functor. The following theoremsummarizes the results describing the biset functor structure of C R C . We refer to [8] for theMackey functor structure of it. (i) (Bouc, [2]) The biset functor C R C is semisimple and there is an isomorphismof biset functors C R C ∼ = M ( m,ζ ) S Z /m Z , C ζ where the sum is over all pairs ( m, ζ ) with m ∈ N and ζ runs over all primitive characters of( Z /m Z ) × .(ii) (Boltje - Co¸skun, [1]) The C × -fibered biset functor C R C is simple and there is an isomorphismof C × -fibered biset functors C R C ∼ = S , , , where the right hand side is the unique simple C × -fibered biset functor with minimal group1.(iii) (Romero, [7]) The Green biset functor C R C is simple.Note that Bouc’s decomposition can be thought as a decomposition of 1-fibered biset functors, andhence parts (i) and (ii) of the above theorem cover two extreme cases where the fiber group is thesmallest and the largest. For the rest of the paper, we want to determine the structure of C R C as an A -fibered biset functor for other subgroups of C × . For this aim, we need the following descriptionof action of fibered bisets on characters. Note that this is a special case of [2, Lemma 7.1.3].13 .2 Lemma Let M be a C G -module with character χ and Y be a transitive A -fibered ( H, G ) -biset with stabilizing pair ( V, ψ ) . Then for any h ∈ H , the value at h of the character Y GH χ of the C H -module C Y ⊗ C G M is given by ( Y GH χ )( h ) = 1 | V | X x ∈ H,g ∈ G ( h x ,g ) ∈ V ψ ( h x , g ) χ ( g ) . Proof
We apply Bouc’s formula [2, Lemma 7.1.3] for characters of tensor product of bimodules toobtain ( Y GH χ )( h ) = 1 | G | X g ∈ G θ ( h, g ) χ ( g )where θ is the character of the monomial ( C H, C G )-bimodule C Y . By definition it is the functionsending any ( h, g ) ∈ H × G to the trace of the endomorphism y ( h, g ) y of C Y . Since Y is an A -fibered ( H, G )-biset and A ≤ C × , any set [ Y /A ] of representatives of A -orbits Y /A of Y is a basisof C Y . Precisely θ ( h, g ) = X y ∈ [ Y/A ]( h,g )[ y ]=[ y ] ψ y ( h, g )where ψ y ( h, g ) ∈ A is determined by the equation ( h, g ) y = ψ y ( h, g ) y and we write [ y ] for the A -orbit containing y . On the other hand, since Y is transitive, for any [ y ] , [ y ′ ] ∈ Y /A , there existssome ( a, b ) V ∈ ( H × G ) /V such that ( a, b )[ y ] = [ y ′ ]. Also, if ( h, g ) stabilizes [ y ], then ( h, g ) ( a,b ) stabilizes [ y ′ ]. Thus,( Y GH χ )( h ) = 1 | G | X g ∈ G θ ( h, g ) χ ( g ) = 1 | G | X g ∈ G X ( a,b ) V ∈ ( H × G ) /V ( ha,gb ) ∈ V ψ ( h a , g b ) χ ( g ) . Replacing g with b g we get Y GH ( χ )( h ) = 1 | G | X b g ∈ G X ( a,b ) V ∈ ( H × G ) /V ( ha,g ) ∈ V ψ ( h a , g ) χ ( g ) . If ( a, b ) V ∈ ( H × G ) /V , then for each ( u, v ) ∈ V and for any [ y ] ∈ Y /A ,( h au , g v )[ y ] = ( h a , g ) v [ y ] = ( h a , g )[ y ] . Hence ( Y GH χ )( h ) = 1 | G || V | X b g ∈ G X ( a,b ) ∈ ( H × G )( ha,g ) ∈ V ψ ( h a , g ) χ ( g ) = 1 | G || V | X g ∈ G X ( a,b ) ∈ H × G ( ha,g ) ∈ V ψ ( h a , g ) χ ( g ) (6)= 1 | G || V | X g ∈ G X a ∈ H X b ∈ G :( ha,g ) ∈ V ψ ( h a , g ) χ ( g ) = 1 | V | X a ∈ H,g ∈ G ( ha,g ) ∈ V ψ ( h a , g ) χ ( g ) . This completes the proof of the lemma. (cid:3)
We also need explicit descriptions of actions of certain fibered bisets. Note that if Y is one ofinduction, restriction, inflation, deflation or isogation bisets, then the above formula becomes thecorresponding well-known map from character theory. We also have the following maps.14 .3 Corollary Let χ ∈ R C ( G ) . Then(i) For any φ ∈ G A , we have Tw φG χ = φ · χ. (ii) Suppose G is abelian and let ( K, κ ) ∈ M G . Then E G ( K,κ ) χ = X φ ∈ Irr( G ): φ | K = κ < χ, φ > φ. In other words, the action of E G ( K,κ ) on a C G -module M is given by projection onto thesubmodule of M generated by all extensions of κ to G . Proof
For the first part, recall that the stabilizing pair of Tw φG is (∆( G ) , ∆ φ ). Thus by Lemma5.2, for any g ∈ G , we getTw φG ( χ )( g ) = 1 | ∆( G ) | X x ∈ G,g ′∈ G ( gx,g ′ ) ∈ ∆( G ) ∆ φ (( g x , g ′ )) χ ( g ′ ) = 1 | G | X x ∈ G ∆ φ (( g x , g x )) χ ( g x ) (7)= 1 | G | X x ∈ G φ ( g ) χ ( g ) = φ ( g ) χ ( g ) = ( φχ )( g ) . For the second part, suppose G is abelian. The stabilizing pair of E G ( K,κ ) is (∆ K ( G ) , φ κ ) where φ κ ( g , g ) = κ ( g g − ). To simplify calculations, we suppose χ is an irreducible character. Thegeneral case follows from the linearity of the action.Once more, by Lemma 5.2, for any h ∈ G , we have( E G ( K,κ ) χ )( h ) = 1 | G || K | X x ∈ G,g ∈ G ( h x ,g ) ∈ ∆ K ( G ) φ κ ( h x , g ) χ ( g ) = 1 | K | X g ∈ G :( h,g ) ∈ ∆ K ( G ) κ ( hg − ) χ ( g ) (8)= 1 | K | X k ∈ K κ ( k − ) χ ( hk ) = χ ( h ) (cid:16) | K | X k ∈ K κ ( k − ) χ ( k ) (cid:17) . Here note that the last equality holds since G is abelian and hence χ is a homomorphism. Noticethat the second term in the last row above is the inner product of the characters κ and χ | K in thespace of class functions on K , and hence is equal to 1 if κ = χ | K and zero otherwise. Hence theresult follows. (cid:3) We also need action of the idempotent f G ( K,κ ) when G is cyclic. In this case, the description ofthe idempotent is simpler. Let p be a prime and G and A be finite cyclic p -groups with | A | < | G | . Also let ( K, κ ) ∈ R G . Then(i) If | K | = | A | , then f G ( K,κ ) = e G ( K,κ ) . In particular, f G ( K,κ ) annihilates all irreducible characters of G whose restriction to K doesnot coincide with κ . ii) If | K | < | A | , let L ≤ G be the unique subgroup of G of order p · | K | . Then f G ( K,κ ) = e G ( K,κ ) − X λ ∈ L A : λ | K = κ e G ( L,λ ) . Moreover, in this case, f G ( K,κ ) annihilates all characters of G . Proof
For both parts, the second claims follows form the above corollary once we prove the firstparts. Thus we only prove the first claims. Recall that, by definition, for any pair (
K, κ ) ∈ M G ,we have f G ( K,κ ) = X ( K,κ ) ≤ ( L,λ ) ∈M G µ ⊳K,L e G ( L,λ ) . Since G is cyclic, the Mobius function µ ⊳K,L coincides with the number theoretic Mobius function µ ( | L : K | ) and hence by [5, Proposition 10.1.10] we have µ ⊳K,L = K = L, − | L : K | = p, . Now given (
K, κ ) ∈ R G , there is a unique subgroup L ≤ G such that | L : K | = p . Also if( K, κ ) ≤ ( L, λ ), then λ is an extension of κ . Since κ is faithful, an extension exists if and only if | L | ≤ | A | . (cid:3) Finally we discuss actions of isogations and twists on primitive characters. Again, by Lemma5.7, it is sufficient to consider the case of p -groups. Let p be a prime, A be cyclic of order p l and m = p k for some k ∈ N with l < k . We write G = Z /m Z and identify A with the unique subgroup of G of order p l . Also let ν be a primitive character modulo m and write ˜ ν for the canonical extensionof ν to G .Recall that a primitive character ν modulo m is a multiplicative homomorphism ν : ( Z /m Z ) × → C × that is not induced from any character of smaller modulus. With this notation, ˜ ν : Z /m Z → C is given by ˜ ν ( x ) = ν ( x ) if x ∈ ( Z /m Z ) × and zero otherwise.The set G ∗ = Irr( G ) is also cyclic, isomorphic to G . We write χ for the generator of G ∗ forwhich χ (1) = e πim . With this choice, we get G ∗ = { , χ, χ , . . . , χ m − } and χ j (1) = χ ( j ) for any j = 0 , , . . . , m −
1. Hence by [5, Corollary 2.1.42], the generalizedcharacter ˜ ν can be expressed as ˜ ν = τ (˜ ν ) X ( j,p )=1 ˜ ν ( j ) χ j where the sum is over all indexes j = 0 , , . . . , m − j, p ) = 1 and where τ (˜ ν ) = < ˜ ν, χ > is the coefficient of the irreducible character χ in the generalized character ˜ ν .Recall that by [1, Section 4.3], the idempotents f { K,κ } sum up to the identity endomorphism of G as ( K, κ ) runs over the linkage classes. Furthermore, by Corollary 5.4, the idempotents f { K,κ } annihilates any character of G if ( K, κ ) ∈ R G and K < A . Moreover, if (
K, κ ) is not reduced,16hen each summand of f ( K,κ ) is also non-reduced, and by Equation 5, it annihilates all primitivecharacters. Finally, by Proposition 4.2, all faithful α ∈ A ∗ are linked. Therefore we get˜ ν = 1 G · ˜ ν = f { A,α } ˜ ν = X α ∈ A ∗ : α faithful f ( A,α ) ˜ ν. For simplicity, we put ˜ ν α = f ( A,α ) ˜ ν and get˜ ν = X α ∈ A ∗ : α faithful ˜ ν α . Notice that by Corollary 5.4, the summand ˜ ν α is the projection of ˜ ν to the summand whoserestriction to A is a multiple of α , that is,˜ ν α = τ (˜ ν ) X j : χ j | A = α ˜ ν ( j ) χ j where the sum is over all distinct extensions of α to G . This set can be described more explicitly. Let j α be the smallest natural number such that χ j α extends α . Then any other extension is a productof χ j α and an irreducible character of G inflated from the quotient group G/A . In particular, if χ j is another extension of α , then j ∼ = j α (mod | A | ). Hence we have { χ j ∈ G ∗ : χ j | A = α } = { χ j α + k ·| A | : k = 0 , , . . . , | G : A | − } . Therefore we have˜ ν α = τ (˜ ν ) | G : A |− X k =0 ˜ ν ( j α + k | A | ) χ j α + k | A | = χ j α · (cid:0) τ (˜ ν ) | G : A |− X k =0 ˜ ν ( j α + k | A | ) χ k | A | (cid:1) . (9)Now we determine the actions of fibered bisets on these summands. First, for any σ ∈ Aut( G ),the action of the isogation biset Iso σG on ˜ ν follows from [2, Theorem 7.3.4]. We identify the groups G × = Aut( G ) via σ ↔ σ (1). With the above notation, given σ ∈ Aut( G ) , we have Iso σG ˜ ν α = ˜ ν ( σ (1))˜ ν σ α where σ α ( x ) = α ( σ ( x )) for any x ∈ A . Next we determine the action of twist bisets Tw φG for φ ∈ G A on ˜ ν α for a fixed fatihful α ∈ A ∗ .Since G A < G ∗ , for any φ ∈ G A , we have φ = χ s for some s ∈ N such that | G : A | divides s . Let χ s ∈ G A be a character of G and let α ∈ A ∗ be faithful. Then Tw χ s G ˜ ν α = ν ( x )˜ ν β where β = χ s · α and x ∈ G × is given by j − β ( j β − s ) . roof By Corollary 5.3, we have Tw χ s G ˜ ν α = χ s ˜ ν α . Hence given χ j ∈ G ∗ , we have < Tw χ s G ˜ ν α , χ j > = < ˜ ν α , χ j − s > . In particular, the irreducible character χ j appears in Tw χ s G ˜ ν α with a non-zero coefficient if and onlyif χ j − s appears in ˜ ν α with a non-zero coefficient. This is equivalent to say that χ j − s | A = α . Hencewe have Tw χ s G ˜ ν α = τ (˜ ν ) X χ j | A = β ˜ ν ( j − s ) χ j where we put β = χ s | A · α . As above, we can rewrite this sum in the following way.Tw χ s G ˜ ν α = χ j β · (cid:0) τ (˜ ν ) | G : A |− X k =0 ˜ ν ( j β − s + k | A | ) χ k | A | (cid:1) . (10)Now since s is a multiple of | G : A | , both j β and j β − s are coprime to p . Therefore there is anautomorphism x ∈ G × such that xj β ≡ j β − s mod | G | . Then since s is a multiple of | G : A | , wealso get xj β + k | A | ≡ x ( j β + k | A | ) (mod | G | )for all k = 0 , , . . . , | G : A | −
1. With this congruence, and the fact that ˜ ν is periodic with period | G | , we can rewrite Equation 10 asTw χ s G ˜ ν α = χ j β · (cid:0) τ (˜ ν ) | G : A |− X k =0 ˜ ν ( x ( j β + k | A | )) χ k | A | (cid:1) . (11)Furthermore, since both x and j β + k | A | , k = 0 , , . . . , | G : A |− G × , we have ˜ ν ( x ( j β + k | A | )) = ν ( x ( j β + k | A | )) = ν ( x ) ν ( j β + k | A | ) and hence Equation 11 becomesTw χ s G ˜ ν α = ν ( x ) χ j β (cid:0) τ (˜ ν ) | G : A |− X k =0 ˜ ν ( j β + k | A | ) χ k | A | (cid:1) (12)= ν ( x )˜ ν β . (cid:3) The following result describes action of fibered bisets on characters of direct products of groupsof coprime orders. We left the straightforward verification to the reader. Recall that by Theorem10.33 of [6], given groups G and H , we have C R C ( G × H ) ∼ = C R C ( G ) ⊗ Z C R C ( H ) . In particular, if G = G × G and H = H × H , we can write C R C ( G × H ) ∼ = C R C ( G × H ) ⊗ Z C R C ( G × H ) . .7 Lemma Let G = G × G and H = H × H be finite groups such that ( | G || H | , | G || H | ) = 1 and let X be an A -fibered ( G, H ) -biset. Also let χ ∈ C R C ( H ) . Then X GH χ = Y G H ( χ ) × Z G H ( χ ) where X = Y × Z is factored according to Lemma 3.2 and χ = χ × χ is factored according to theabove isomorphism.
6. Functor of complex characters
In this section, we prove our main theorems. We distinguish between two cases. The first caseis where the fiber group is large enough so that Equation 1 can be applied. In the second case,we consider a finite fiber group. It turns out that, in both cases, the functor C R C is semisimple.We also determine its simple summands. We explain our approach briefly. Let A be a group asdescribed above. Also let F be an A -fibered biset subfunctor of C R C . Then by restriction, F isalso a biset functor and hence it is a direct sum of simple biset functors of the form S Z /m Z , C ζ forsome set of pairs ( m, ζ ) with m ∈ N and ζ is a primitive character of ( Z /m Z ) × . Thus to determinesimple A -fibered subfunctors of C R C , one needs to put an appropriate equivalence relation on theset Υ of all such pairs. Throughout this section, let π be a set of prime numbers and A = π ∞ be the group of all p -power roots of unity for all primes p ∈ π , that is, π ∞ = Y p ∈ π lim ←− n ∈ N ( Z /p n Z ) × . Let Υ be the set of all pairs ( m, ζ ) defined above. We define a relation on Υ as follows. Twopairs ( m, ζ ) , ( n, ν ) ∈ Υ are said to be π -equivalent , written ( m, ζ ) ≡ ( n, ν ) if the π ′ -parts m π ′ and n π ′ of m and n are equal and after identifying the groups Z /m π ′ Z ∼ = Z /n π ′ Z , we have C ζ π ′ ∼ = C ν π ′ . The last condition means that the π ′ -parts ζ π ′ and ν π ′ of ζ and ν afford the same one dimensionalrepresentation of Z /m π ′ Z ∼ = Z /n π ′ Z .Clearly, this is an equivalence relation on the set Υ. We denote the equivalence class containingthe pair ( m, ζ ) by [ m, ζ ] and write Υ π ∞ for the set of equivalence classes. It is also clear that eachequivalence class contains a unique pair ( n, ν ) where n is a π ′ -number. Hence the set [Υ π ∞ ] of allpairs ( m, ζ ) with m π = 1 is a full set of representatives of equivalence classes on Υ π ∞ .With this notation, we first determine the π ∞ -fibered biset subfunctor of C R C generated by asimple biset subfunctor of it. Let ( m, ζ ) ∈ Υ and let T Z /m Z , C ζ be the π ∞ -fibered biset subfunctor of C R C gener-ated by the image of S Z /m Z , C ζ in C R C . Then there is an isomorphism of biset functors T Z /m Z , C ζ ∼ = M ( n,ν ) ∈ [ m,ζ ] S Z /n Z , C ν . roof To simplify the notation, we put Z /n Z = Z n for any n ∈ N throughout the proof. Let S [ m,ζ ] = M ( n,ν ) ∈ [ m,ζ ] S Z n , C ν be the right hand side of the above isomorphism. By its definition, S [ m,ζ ] is a biset functor. Wehave to prove that S [ m,ζ ] ∼ = T Z m , C ζ as biset functors.We first prove that S [ m,ζ ] ⊆ T Z m , C ζ . Let ( n, ν ) ∈ [ m, ζ ]. We shall show that the simple bisetsubfunctor S Z n , C ν of C R C is contained in T Z m , C ζ . Since m is a π ′ -number, there is a π -number k such that n = mk . We write ˜ ν = ˜ ν π × ˜ ν π ′ where ν π ∈ C R C ( Z k ) and ν π ′ ∈ C R C ( Z m ). By the definition of the equivalence relation on Υ, wemay assume that the π ′ -part ν π ′ of ν coincides with ζ . On the other hand, the π -part ˜ ν π of ˜ ν is avirtual character of Z k , and hence it is a complex linear combination of the irreducible charactersof Z k , say ˜ ν π = X χ ∈ Irr ( Z k ) c χ χ for some complex numbers c χ . Moreover, since k is a π -number, the group Z k embeds in A andhence each irreducible character χ of Z k induces a twist biset Tw χ Z k . Thus, by Corollary 5.3, puttingTw ν π = X χ c χ Tw χ Z k , we obtain ˜ ν π = Tw ν π · Z k . Now by Lemma 5.7, we have˜ ν = ˜ ν π × ˜ ν π ′ = (Tw ν π · × ˜ ζ = (Tw ν π × id) · (1 × ˜ ζ ) = (cid:0) (Tw ν π × id)Inf Z n Z m (cid:1) ˜ ζ. In particular, ˜ ν is contained in the A -fibered biset subfunctor generated by ˜ ζ . We already knowthat the simple biset functor S Z n , C ν is generated by ˜ ν . Thus S Z n , C ν ⊆ T Z m , C ζ for all ( n, ν ) ∈ [ m, ζ ] and hence S [ m,ζ ] ⊆ T Z m , C ζ as required.To prove the reverse inclusion, it is sufficient to show that any simple biset subfunctor of T Z m , C ζ is parameterized by a pair equivalent to ( m, ζ ). Indeed, since C R C is semisimple as a biset functor,the subfunctor T Z m , C ζ is also semisimple as a biset functor and hence it is the sum of its simplesubfunctors. In particular, any simple biset subfunctor of T Z m , C ζ is of the form S Z n , C ν for some( n, ν ) ∈ Υ.Let S Z n , C ν ⊆ T Z m , C ζ . We have to show that ( n, ν ) is equivalent to ( m, ζ ). Since S Z n , C ν ( Z n ) ⊆ T Z m , C ζ ( Z n ) and the functor T Z m , C ζ is generated by ˜ ζ , we must have˜ ν = γ · ˜ ζ γ ∈ B A ( Z n , Z m ). Then by Equation 2, we deduce that ˜ ν is a C -linear combination ofelements of the form Ind Z n P Tw φ P Inf
PP/K
Iso ηP/K,Q/L
Def
QQ/L Tw φ Q Res Z m Q ˜ ζ for some L ≤ Q ≤ Z m and K ≤ P ≤ Z n such that η : Q/L → P/K is an isomorphism and φ ∈ P A and φ ∈ Q A . But Z m is a minimal group for the functor T Z m , C ζ . Thus the maps factoring througha group of smaller order annihilates ˜ ζ and hence any transitive summand of γ must be of the formInd Z n Z s Tw φ Z s Inf Z s Z m Iso η Z m , Z m ˜ ζ where η ∈ Aut( Z /m Z ) and m | s, s | n and φ : Z s → A is a group homomorphism. Also, since Z n isa minimal group for the biset functor S Z n , C ν , the element ˜ ν is not in the image of induction mapsand hence there must exist terms where s = n , that is, terms of the formTw φ Z n Inf Z n Z m Iso η Z m , Z m ˜ ζ. (13)Let n = n π n ′ m ′ where n π is the π -part of n , and the π ′ -part of n satisfies n π ′ = n ′ m ′ such that( m, n ′ ) = 1. With this notation, we write Z n = Z n π × Z n ′ × Z m ′ and hence by Lemma 3.2, we getInf Z n Z m Iso η Z m , Z m = Inf Z nπ × Z n ′ × Z m ′ × × Z m Iso η Z m , Z m = Inf Z nπ × Inf Z n ′ × (Inf Z m ′ Z m Iso η Z m , Z m )and Tw φ Z n = Tw φ π Z nπ × id × idwhere φ π is the restriction of φ to the π -part of the group Z n . Also we can regard ˜ ζ as an elementof C R C (1 × × Z m ) in the obvious way. Then (13) becomesTw φ Z n Inf Z n Z m Iso η Z m , Z m ˜ ζ = (cid:0) (Tw φ π Z nπ × id × id)(Inf Z nπ × Inf Z n ′ × (Inf Z m ′ Z m Iso η Z m , Z m )) (cid:1) (˜ ζ ) (14)= (cid:0) (Tw φ π Z nπ Inf Z nπ ) × (Inf Z n ′ ) × (Inf Z m ′ Z m Iso η Z m , Z m ) (cid:1) (˜ ζ ) (15)= φ π × × (Inf Z m ′ Z m Iso η Z m , Z m ˜ ζ ) = Inf Z n Z nπ × × Z m ( φ π × × Iso η Z m , Z m ˜ ζ ) (16)= ζ ( η )Inf Z n Z nπ × × Z m ( φ π × × ˜ ζ ) . Here the last equality holds since the action of Z × m on ˜ ζ is by ζ . In particular the element givenin (13) is in the image of inflation maps unless n ′ m ′ = m . Hence γ must have a summand with m = n ′ m ′ . But this is possible only if m is the π ′ -part of n and hence all the summands of γ mustbe of the form Tw φ Z n Inf Z n Z m Iso η Z m , Z m ˜ ζ = (Tw φ Z nπ × Iso η Z m , Z m ) (1 × ˜ ζ ) . (17)In particular, we obtain that the π ′ -parts of n and m are equal. Moreover, we obtain˜ ν = γ · ˜ ζ = X (Tw φ Z nπ × Iso η Z m , Z m ) (1 × ˜ ζ )which clearly shows that the π ′ -parts of ν and ζ induces the same representation of Z m . Hence wehave proved that T Z m , C ζ ⊆ S [ m,ζ ] . (cid:3) As remarked above, each equivalence class [ m, ζ ] is represented by its unique member for which m is a π ′ -number. In particular, any other member is of the form ( n, ν ) with n = m · k where k isa π -number and hence the group Z /m Z is minimal for the functor T Z /m Z , C ζ and we have T Z /m Z , C ζ ( Z /m Z ) ∼ = C ζ where C ζ is the C Aut( Z /m Z )-module with character ζ . We further have the following identification. The π ∞ -fibered biset functor T Z /m Z , C ζ is simple and isomorphic to the functor S π ∞ Z /m Z , , , C ζ . Proof
Let 0 ⊂ T ⊆ T Z /m Z , C ζ be a π ∞ -fibered biset subfunctor of T Z /m Z , C ζ . Then since T Z /m Z , C ζ is semisimple as a biset functor, there is a pair ( n, ν ) equivalent to ( m, ζ ) such that S Z /n Z , C ν ⊆ T .Then by the proof of the previous theorem, we have˜ ν = ˜ ν π × ˜ ζ. Now since Tw φ Tw φ − = id for any φ ∈ G π ∞ and Def HH/N
Inf
HH/N = id for any groups N E H , weget ˜ ζ = Def Z /n ZZ /m Z (Tw ν − π × id)˜ ν. In particular, ˜ ζ is in T ( Z /m Z ) and since T Z /m Z , C ζ is generated by ˜ ζ , we get T = T Z /m Z , C ζ . Hence T Z /m Z , C ζ is simple. The second part follows from the remark above and the parametrization ofsimple A -fibered biset functors. (cid:3) With this proof we have also completed the proof of Theorem 1 as the π ∞ -fibered biset functor C R C is a sum of its simple subfunctors.As an immediate corollary we obtain the following result describing the restriction of the simple π ∞ -fibered biset functors that appear above to the category of biset functors. Let A be as above and ( m, ζ ) ∈ [Υ π ∞ ] . There is an isomorphism of biset functors S π ∞ Z /m Z , , ,ζ ∼ = M ( n,ν ) ∈ [ m,ζ ] S Z /n Z , C ν . Another immediate corollary is the case where π contains all prime numbers. Then we canreplace π ∞ by the unit group C × of complex numbers. By the definition of π -equivalence it isclear that there is only one equivalence class which can be represented by (1 , C R C isisomorphic to the simple C × -fibered biset functor S , , , , recovering the first part of [1, Theorem11.3].As another special case let p be a prime number and p ′ be the set of all primes q = p . Also let F be an algebraically closed field of characteristic p . Then the group A containing all q n -th roots ofunity for all n ∈ N and q ∈ p ′ can be identified with the torsion subgroup of F × . Thus ( p ′ ) ∞ -fiberedbiset functors are equivalent to F × -fibered biset functors. In particular, the above theorem gives adecomposition of C R C as an F × -fibered biset functor.We denote by T the biset functor of p -permutation modules, see [3] for details. We also write C T = C ⊗ T . By [1, Section 11D], T is is an F × -fibered biset functor, and it has a unique simplequotient isomorphic to S F × , , , . Together with Corollary 6.3, we obtain the following result. Notethat a more general result appears in [3, Corollary 44].22 .4 Corollary (Baumann) For any m ∈ N with ( m, p ) = 1 and any primitive character ζ mod-ulo m , the simple biset functor S Z /m Z , C ζ is a composition functor of the biset functor C T of p -permutation modules. Next we consider the case where A is a finite cyclic p -group for a fixed prime number p . Generalcase can be treated similarly. We follow the same steps as in the previous section and determinean equivalence relation on the set Υ so that the equivalence classes corresponds to A -fibered bisetsubfunctors of C R C . This case turns out to be more involved since we have a new distinguishedfibered biset, namely f ( K,κ ) for ( K, κ ) ∈ R G .Throughout this section, fix a prime number p , a natural number l and let A be the group of p l -th roots of unity in C . We still write Υ for the set of all pairs ( m, ζ ) defined previously. Wedefine a relation ∼ = A on Υ as follows. Let ( m, ζ ) , ( n, ν ) ∈ Υ. Then ( m, ζ ) ∼ = A ( n, ν ) if1. n p ′ = m p ′ and C ν p ′ ∼ = C ζ p ′ and2. either n p , m p ≤ | A | or n p = m p and for each faithful α ∈ A ∗ , there is a complex number c α such that f ( A,α ) · ˜ ν p = c α f ( A,α ) · ˜ ζ p . (18)It is straightforward to check that this relation is an equivalence relation on Υ. Also we shall seelater that if Equation 18 holds for some α , then actually it holds for all faithful characters in A ∗ .We again write [ m, ζ ] for the equivalence class containing the pair ( m, ζ ). In contrast with theprevious case, there are two types of equivalence classes in Υ / ∼ = A . Given m ∈ N , if m p ≤ | A | , then[ m, ζ ] = { ( p k m p ′ , ν ) : k ≤ l, C ν p ′ ∼ = C ζ p ′ } and otherwise if m p > | A | , then[ m, ζ ] = { ( m, ν ) : C ν p ′ ∼ = C ζ p ′ and f ( A,α ) · ˜ ν p = c α f ( A,α ) · ˜ ζ p for some faithful α ∈ A ∗ } . We write [Υ p ] for a set of representatives of equivalence classes of the first type. As in theprevious case, it can be identified with the set of all pairs ( m, ζ ) where m is a p ′ -number. We alsowrite [Υ > | A | ] for a set of representatives of equivalence classes of the second type.As in the previous section, for a given pair ( m, ζ ) ∈ Υ, we write T A Z /m Z , C ζ for the A -fibered bisetsubfunctor of C R C generated by its biset subfunctor S Z /m Z , C ζ . With this notation, we have For any ( m, ζ ) ∈ Υ , there is an isomorphism of biset functors T A Z /m Z , C ζ ∼ = M ( n,ν ) ∈ [ m,ζ ] S Z /n Z , C ν . Proof
Given ( m, ζ ) ∈ Υ. We let S [ m,ζ ] be the right hand side of the above isomorphism. As inthe previous case, we first show that S [ m,ζ ] ⊆ T A Z /m Z , C ζ .If m p ≤ | A | , then the proof is almost identical to the proof of the corresponding part of Theorem6.1. We skip the details.For the second case, suppose m p > | A | and let ( m, ν ) ∼ = A ( m, ζ ) so that C ν ′ p ∼ = C ζ ′ p and f ( A,α ) · ˜ ν p = c α f ( A,α ) · ˜ ζ p α ∈ A ∗ where c α ∈ C × . We have to prove that ˜ ν can be written as a linearcombination of elements obtained from ˜ ζ by actions of fibered bisets.Fix a faithful character α ∈ A ∗ . Since the linkage class of ( A, α ) consists of all pairs (
A, β ) with β ∈ A ∗ faithful, the second equality above implies that f { A,α } · ˜ ν p = X β ∈ A ∗ :faithful c β f ( A,β ) · ˜ ζ p . Also since sum of the idempotents f ( K,κ ) is 1 and by Corollary 5.4, f ( K,κ ) annihilates all primitivecharacters if ( K, κ ) is not reduced or if
K < A , we obtain˜ ν p = f { A,α } · ˜ ν p = X β ∈ A ∗ :faithful c β f ( A,β ) · ˜ ζ p . In particular, ˜ ν p is a linear combination of elements obtained from ˜ ζ by actions of fibered bisets, asrequired. Thus we have proved that S [ m,ζ ] ⊆ T A Z /m Z , C ζ .Conversely, let S Z /n Z , C ν be a biset subfunctor of T A Z /m Z , C ζ . We have to prove that ( n, ν ) ∼ = A ( m, ζ ). By the choice of ( n, ν ), there is an element X ∈ B A ( Z /n Z , Z /m Z ) such that˜ ν = X · ˜ ζ. We have two cases. First suppose that m p ≤ | A | . Then we can represent the equivalence class[ m, ζ ] by the pair ( m, ζ ) where ( m, p ) = 1.By Equation 1, any transitive summand of X can be written asInd Z /n Z P Inf
PP/K Y Def
QQ/L
Res Z /m Z Q for some L ≤ Q ≤ Z /m Z and K ≤ P ≤ Z /n Z . Here Y is a transitive A -fibered ( Q/L, P/K )- bisetwith stabilizing pair (
U, φ ) such that p ( U ) = P/K, p ( U ) = Q/L and restrictions of φ to k ( U ) and k ( U ) are both faithful. As in the previous case of Theorem 6.1, since both ζ and ν are primitive,any summand of X where | P/K | < n annihilates ˜ ζ and, also ˜ ν is not in the image of any transitivebiset of the above form if | Q/L | < m . Hence Y can be chosen as a linear combination of transitive A -fibered ( Z /n Z , Z /m Z )-bisets with stabilizing pairs ( U, φ ) such that p ( U ) = Z /n Z , p ( U ) = Z /m Z and restrictions of φ to k ( U ) and k ( U ) are both faithful.Given such a summand Y , by Lemma 3.2, we write Y = Y p ′ × Y p where Y p ∈ B A ( Z /n p Z ,
1) and Y p ′ ∈ B A ( Z /n p ′ Z , Z /m Z ). Furthermore, since A is a p -group, the group B A ( Z /n p ′ Z , Z /m p ′ Z ) canbe identified by the Burnside group of ( Z /n p ′ Z , Z /m Z )-bisets. In particular, Y p ′ is a ( Z /n p ′ Z , Z /m Z )-biset which must be an isogation since restrictions of φ to the corresponding subgroups must befaithful. Hence we get n p ′ = m p ′ and C ν p ′ ∼ = C ζ p ′ after identifying Z /m Z = Z /n p ′ Z .On the other hand, the above conditions imply that the stabilizing pair for Y p is of the form( U p , φ p ) such that p ( U p ) = Z /n p Z , p ( U p ) = Z /m p Z and restrictions of φ to k ( U p ) and k ( U p ) areboth faithful. Since ( m, p ) = 1, then we necessarily have ( U p , φ p ) = (( Z /n p Z ) × , φ ×
1) for somefaithful ϕ : Z /n p Z → A . Hence, in this case, we must have n p ≤ | A | and Y p = Tw ϕ Z /n p Z Inf Z /n p Z .Thence we have proved that the p ′ -parts of n and m are equal and n p , m p ≤ | A | , as required.For the second case, if m p > | A | , then we also have n p > | A | . The condition on p ′ -parts canbe obtained in the same way as the previous case. Hence, by Lemma 5.7, we can drop the indexes24nd write m = m p and n = n p and Y = Y p . Also let ( U, φ ) be a stabilizing pair for Y . Then byTheorem 3.4, we can factor Y as[ Y ] = Tw ˜ φ Z /n Z Inf Z /n Z ( Z /n Z ) /K Iso ˜ η ( Z /n Z ) /K, Z /m Z e ( L,φ L ) where the notation is chosen according to Theorem 3.4. Now suppose K = 1. Then since n > | A | ,the homomorphism ˜ φ : Z /n Z → A has a non-trivial kernel, say K ′ . If K ′ > K , then ˜ φ can beregarded as a homomorphism ( Z /n Z ) /K → A and hence we have[ Y ] = Inf Z /n Z ( Z /n Z ) /K Tw ˜ φ ( Z /n Z ) /K Iso ˜ η ( Z /n Z ) /K, Z /m Z e ( L,φ L ) . On the other hand, if K ′ ≤ K , then ( Z /n Z ) /K ∼ = ( Z /n Z ) /K ′ K/K ′ , hence we have[ Y ] = Inf Z /n Z ( Z /n Z ) /K ′ Tw ˜ φ ( Z /n Z ) /K ′ Inf ( Z /n Z ) /K ′ ( Z /n Z ) /K Iso ˜ η ( Z /n Z ) /K, Z /m Z e ( L,φ L ) . In both cases, Y · ˜ ζ is in the image of inflation maps. But by its choice ˜ ν is not in this image. Hencethere is at least one transitive term Y such that K = 1. But then since Z /m Z ∼ = ( Z /n Z ) /K , wemust have n = m and hence any summand Y must be of the form[ Y ] = Tw ψ Z /m Z Iso σ Z /m Z , Z /m Z e ( L,λ ) for some ψ ∈ G A , σ ∈ Aut( Z /m Z ) and λ ∈ L A . Therefore, we get˜ ν = X ψ,σ,λ c ψ,σ,λ Tw ψ Z /m Z Iso σ Z /m Z , Z /m Z e ( L,λ ) ˜ ζ where c ψ,σ,λ ∈ C and the sum is over all ψ ∈ G A , σ ∈ Aut( Z /m Z ) , λ ∈ L A .Furthermore, by Corollary 5.4, e ( L,λ ) · ˜ ζ = 0 unless L = A . When L = A , then e ( A,λ ) = f ( A,λ ) and we put ˜ ζ α = f A,λ · ˜ ζ . Also by Corollary 5.5 and by Lemma 5.6, we obtain˜ ν = X ψ,σ,λ c ψ,σ,λ ˜ ζ ψ · σ α Finally since f ( A,α ) · ˜ ζ β = ( f ( A,α ) · f ( A,β ) ) · ˜ ζ β = 0 unless α = β and f ( A,α ) · ˜ ζ α = ˜ ζ α , multiplying theabove equality by f ( A,α ) we get f ( A,α ) · ˜ ν = X ψ,σ,λ : α = φ · σ α c ψ,σ,λ f ( A,α ) · ˜ ζ = c α f ( A,α ) · ˜ ζ for some non-zero complex number c α . Note that the same argument shows that if β ∈ A ∗ isanother faithful character, then f A,β · ˜ ν = c β f A,β · ˜ ζ for some non-zero complex number c β .This completes the proof of the last claim. Hence we have shown that ( n, ν ) ∼ = A ( m, ζ ), andhence T A Z /m Z , C ζ = S [ m,ζ ] . (cid:3) As in the previous case, the functor T A Z /m Z , C ζ is simple. Proof is almost identical to the previouscase, we skip details. To identify it, first assume ( m, ζ ) ∈ [Υ p ]. Then as in the case of large enoughfiber group, we have an isomorphism T A Z /m Z , C ζ ∼ = S A Z /m Z , , , C ζ . m, ζ ) ∈ [Υ > | A | ], then, by its construction, we have T A Z /m Z , C ζ ( Z /m Z ) ∼ = M ν :( m,ν ) ∼ = A ( m,ζ ) C ν as C Aut( Z /m Z )-modules. Fix a faithful character α of A . In T A Z /m Z , C ζ ( Z /m Z ), the one-dimensionalsubspace generated by ˜ ζ α × ζ p ′ = ( f ( A,α ) × · ˜ ζ is Γ ( Z /m Z ,A,α ) -invariant. Indeed, any element ofthis group is of the form Tw φ Z /m Z Iso σ Z /m Z , Z /m Z e ( A,α ) × p ′ -part of Z /m Z , for some φ ∈ Z /m Z A and σ ∈ Aut( Z /m Z ) such that φ · σ α = α . Thus givensuch an element Tw φ Z /m Z Iso σ Z /m Z , Z /m Z e ( A,α ) in Γ ( Z /m Z ,A,α ) , we get, by Corollary 5.5 and Lemma5.6, that Tw φG Iso σG,G e ( A,α ) · ˜ ζ α = ρ ζ,α ( φ, σ )˜ ζ α . for some complex number ρ ζ,α ( φ, σ ). Let C αζ be this one-dimensional C Γ ( Z /m Z ,A,α ) -module. Thenclearly, C αζ = ( f ( A,α ) × · T A Z /m Z , C ζ ( Z /m Z ) and T A Z /m Z , C ζ ( Z /m Z ) = ¯ E Z /m Z ⊗ C Γ ( Z /m Z ,A,α ) C αζ . Therefore by [1, Theorem 9.2], the simple functor T A Z /m Z , C ζ is parameterized by the quadruple( Z /m Z , A, α, C αζ ). This completes the proof of the following theorem. Let p be a prime number and A be a finite cyclic p -group. Then(i) The A -fibered biset subfunctor T A Z /m Z , C ζ of C R C is simple.(ii) If ( m, ζ ) ∈ [Υ p ] , then T A Z /m Z , C ζ is isomorphic to the simple functor S A Z /m Z , , , C ζ .(iii) If ( m, ζ ) ∈ [Υ > | A | ] , then T A Z /m Z , C ζ is isomoriphic to the simple functor S A Z /m Z ,A,α, C αζ . With this theorem, we have completed the proof of Theorem 2. The general case where the fibregroup A is of composite order, or where is arbitrary subgroup of C × can be obtained easily bycombining results from Theorem 1 and Theorem 2.Again as a corollary, we obtain the following result on restriction of simple fibered biset functorsthat appear above. Let A be a finite cyclic p -group. Then1. For any ( m, ζ ) ∈ [Υ p ] , there is an isomorphism of biset functors S A Z /m Z , , , C ζ ∼ = M ( n,ν ) ∈ [ m,ζ ] S Z /n Z , C ν .
2. For any ( m, ζ ) ∈ [Υ > | A | ] , there is an isomorphism of biset functors S A Z /m Z ,A,α, C αζ ∼ = M ( n,ν ) ∈ [ m,ζ ] S Z n , C ν . eferences [1] Boltje, R. & Co¸skun, O., Fibered Biset Functors , Advances in Mathematics (to appear).[2] Bouc, S.,
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