Representation Rings of Finite Groups, Their Species, and Idempotent

Representation rings of ﬁnite groups, their species, and idempotent formulae∗

Robert Boltje† Department of Mathematics University of California Santa Cruz, CA 95064 U.S.A. [email protected] August 22, 2001

Abstract We systematically investigate various representation rings of a ﬁnite group as functors to the category of rings together with various natural transformations between them. Moreover, their species and formulae for their primitive idempotents over a splitting ﬁeld are determined.

Introduction

In this article we investigate various representation rings of a finite group G: The Burnside ring B(G), the rings of monomial representations in characteristic zero, D(G), and in positive characteristic p, Dp(G), the trivial source ring T p(G) and linear source ring Lp(G) in (residual) characteristic p, and the classical character ring R(G) and Brauer character ring Rp(G) in characteristic p. The rings D(G) and Dp(G) were used for the canonical induction formulae of the character ring, the Brauer character ring, the linear source ring, and the trivial source ring (cf. [Bo98b]). We consider these representation rings as ring valued functors on the category of finite groups and use a network of natural transformations between them, cf. Diagram (1.1.a). In Section 1 we recall the definitions of these representation rings. In order to view them as functors one has to be particularly careful in the case of linear source and trivial source modules, since usually they are defined using a complete discrete valuation ring that has enough p-power roots of unity for the finite group G. But there is no complete discrete valuation ring with residual characteristic p containing all p-power roots of unity that can be used simultaneously

∗MR Subject Classiﬁcation 19A22, 20C20 †Research supported by the NSF, DMS-0070630

1 for all finite groups. We also recall and define various natural transformations between them. One of them is a natural ring homomorphism from the Brauer character ring to the trivial source ring that uses the canonical induction formula for the Brauer character ring. Section 2 is devoted to the species of the above representation rings. Those for D(G) and Dp(G) cannot be found in the literature yet. The others are already known. However, it is somewhat surprising that for all representation rings the same pattern occurs: The species are indexed over G-orbits of some G-set, even better, these G-sets come from functors between the category of finite groups and the category of finite sets and the G-set structure comes via functoriality from the inner automorphisms of G. Moreover, each of the natural transformations between the representation rings we introduced is induced by a natural transformation between these set valued functors. This functorial picture is worked out in detail. In Section 3 we give explicit formulae for the primitive idempotents of the representation rings (over a splitting field) introduced in Section 1. Using the functorial properties established in Section 2 we can derive formulae for all representation rings from the idempotent formula for the monomial representation ring D(G). The formula for D(G) is a consequence of a Möbiusinversion formula already used in the context of canonical induction formulae. The functorial approach allows us also to determine which idempotents vanish under restriction maps or the considered natural transformations.

Acknowledgement The author would like to thank B. K¨ulshammerfor his helpful comments on parts of the paper.

Notation Throughout this article G denotes a finite group and n := exp(G) its exponent. For g ∈ G and a set of primes π we write gπ for the π-part of g and Gπ for the set of all π-elements of G, i.e., elements whose order is only divisible by primes in π. Similarly, we write mπ for the π-part of m ∈ N. If π = {p} we will omit the curly brackets in the index. If π is the set of primes 0 different from p we use the index p . By H < G (resp. H 6 G) we indicate that H is a subgroup (resp. a proper subgroup) of G. For a prime p the largest normal p-subgroup of G is denoted by Op(G). For H 6 G and g ∈ G we set g −1 0 H := gHg . If X is a G-set and x, x ∈ X we write [x]G for the G-orbit of x 0 0 and x =G x if [x]G = [x ]G. The stabilizer of x ∈ G is denoted by NG(x). Throughout we denote by K ⊂ C the subfield obtained by adjoining all roots of unity to Q and by OK its ring of algebraic integers. Moreover, we fix a prime p and a maximal ideal pK of OK containing p, and we denote by O the localization of OK with respect to OK rpK . The maximal ideal of the local ring ∼ O is denoted by p. Then, F := O/p = OK /pK is an algebraic closure of Z/pZ via the natural map induced by the inclusion Z ⊂ OK ⊂ O. For m ∈ N we denote by ζm a primitive m-th root of unity and we set Km := Q(ζm), Om := Km ∩ O, pm := Om ∩ p, and Fm := Om/pm. Then, Om is a discrete valuation ring with maximal ideal pm, finite residue field Fm, and field of fractions Km such that S S S S Km = K, Om = O, pm = p, and Fm = F . m∈N m∈N m∈N m∈N

2 By Ri we denote the category of associative rings with identity together with ring homomorphisms preserving the identity. For R ∈ Ri we denote by Rmod the category of ﬁnitely generated left R-modules and by R× the group of invertible elements of R. For a commutative ring R we denote the category of R-algebras that are ﬁnitely generated as R-modules by Ralg. For two categories C and D and two functors F, G : C → D we denote by Nat(F, G) the set of natural transformations between F and G. Whenever this notation is used we assume implicitly that the categories C and D are small.

1 The representation rings

1.1 In this section we will give a common framework for various representation rings which we consider as contravariant functors from the category gr of ﬁnite groups to the category Ri of rings. We will obtain a commutative diagram

ι qqqqq qqqqq qqqqqqqqqq qqqqqqqqqq p,ab qqqqqqqq qqqqqqqq ab R qqqqqqqqqq qqqqqqqqqq R qqqqq qqqqq qqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqq ρ qqqqqqqqqqqqqqqqqqqq q q qqqqqqqq qqqqqqqq qqqqqqqqqqqqqqqqqqqq qqqqqqqq qqqqqqqq qqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqq qqqqqqqqqq qqqqqqqqqq qqqqqqqqqq η π η π

qqqqqqqq qqqqqqqq qqqqqqqq qqqqqqqq ι qqqqqqqqqqqqqqqqqqq ι qqqqqqqqqqqqqqqqqqq qqqqq qqqqq qqqqq qqqqq qqqqqqqqqq qqqqqqqqqq p qqqqqqqqqq qqqqqqqqqq qqqqqqqq qqqqqqqq qqqqqqqq qqqqqqqq B qqqqqqqqqq qqqqqqqqqq D qqqqqqqqqq qqqqqqqqqq D qqqqq qqqqq qqqqq qqqqq qqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqq π qqqqqqqqqqqqqqqq ρ qqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqq

β β (1.1.a)

q q q q qqqqqqqqqq qqqqqqqqqq qqqqqqqqqq qqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqq ι qqqqqqqqqqqqqqqqqqq qqqqq qqqqq p qqqqqqqqqq qqqqqqqqqq p qqqqqqqq qqqqqqqq T qqqqqqqqqq qqqqqqqqqq L qqqqq qqqqq qqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqq ρ qqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqq qqqqqqqqqq qqqqqqqqqq γ τ γ

q q q q qqqqqqqqqq qqqqqqqqqq q qqqqqqqqqq qqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqq qqqqqqq qqqqqqq λ qqqqqqqqqqqqqqqqqqq qqqqq qqqqq p qqqqqqqqqq qqqqqqqqqq qqqqqqqq qqqqqqqq R qqqqqqqqqq qqqqqqqqqq R qqqqq qqqqq qqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqq δ of such functors and natural transformations between them which will be introduced in the following subsections. An exponent p always indicates that the functor depends on the prime p. For each of these functors we denote the ring homomorphism associated to a group homomorphism f in gr by resf . If G f is the inclusion of a subgroup H of G into G, we usually write resH . For any pair of arrows in opposite directions the monomorphism is a section of the epimorphism, i.e.,

π ◦ ι = id , ρ ◦ ι = id , δ ◦ λ = id , π ◦ η = id , γ ◦ τ = id .

3 Moreover, all possible squares commute in the sense that the composition of two arrows equals the composition of the other two, i.e.,

ι ◦ η = η ◦ ι , ι ◦ π = π ◦ ι , π ◦ ρ = ρ ◦ π , ρ ◦ η = η ◦ ρ , β ◦ ι = ι ◦ β , β ◦ ρ = ρ ◦ β , γ ◦ ι = λ ◦ γ , γ ◦ ρ = δ ◦ γ , (1.1.b) and sometimes the composition of three maps in a square equals the fourth, namely in the cases

ρ ◦ η ◦ ι = η , π ◦ ρ ◦ η = ρ , π ◦ ι ◦ η = ι , ρ ◦ π ◦ ι = π , ρ ◦ β ◦ ι = β , δ ◦ γ ◦ ι = γ , γ ◦ ι ◦ τ = λ . (1.1.c)

The groups B(G), Dp(G), D(G), T p(G), Lp(G), Rp(G), and R(G) allow induction maps from subgroups and all maps between these groups commute with induction. All these statements are immediate consequences of the deﬁnitions or well-known properties of these maps. The representation rings and maps of the above diagram will be deﬁned next.

1.2 The Burnside ring B(G) is the representation ring of the category Gset of ﬁnite G-sets. For S ∈ Gset we write [S] for its class in B(G). Recall that if H runs through a set of representatives of conjugacy classes of subgroups of G, then the elements [G/H] form a Z-basis of B(G). The disjoint union and the cartesian product of two G-sets with diagonal action induces a ring structure on B(G). If f : G˜ → G is a group homomorphism one obtains an induced functor resf : Gset → G˜ set which in turn induces a ring homomorphism X −1 g resf : B(G) → B(G˜) , [G/H] 7→ [G/f˜ ( H)] , g∈f(G˜)\G/H that makes B into a contravariant functor B : gr → Ri.

1.3 We define Rab(G) as the group ring of the multiplicative group Gb := × p,ab 0 Hom(G, K ) and R (G) as the group ring of the p -part Gbp0 over Z. Obviously we have natural identifications Rab(G) =∼ ZHom(G, O×) =∼ × ∼ × p,ab ∼ × ∼ ZHom(G, Kn ) = ZHom(G, On ) and R (G) = ZHom(G, F ) = × ZHom(G, Fn ) using the various natural ring homomorphisms between O, F , p,ab ab On, Fn, K, and Kn. There is a natural inclusion ιG : R (G) → R (G) and ab p,ab 0 a left inverse ρG : R (G) → R (G) mapping ϕ ∈ Gb to its p -part, or if we interpret Rp,ab(G) as ZHom(G, F ×) by mapping ϕ to its reduction mod p. If f : G˜ → G is a group homomorphism between finite groups, we obtain a restriction map Hom(G, K×) → Hom(G,˜ K×) which induces ring homomorphisms ab ab p,ab p,ab ab resf : R (G) → R (G˜) and resf : R (G) → R (G˜) so that R : gr → Ri becomes a contravariant functor and Rp,ab a subfunctor. The notation Rab(G) and Rp,ab(G) will become more meaningful after the introduction of the character ring functor R and the Brauer character ring functor Rp. It is immediate that Rab(G) =∼ R(Gab) and Rp,ab =∼ Rp(Gab), where Gab denotes the commutator factor group of G.

4 1.4 Next we describe D(G) and Dp(G) (the letter D paying tribute to Dress who studied similar rings earlier, cf. [Dr71]). For a commutative ring R we set

× MR(G) := {(H, ϕ) | H 6 G, ϕ ∈ Hom(H,R )} .

× We will view Hom(H,R ) as multiplicative group. The set MR(G) is a G- poset, i.e., a partially ordered set (with (I, ψ) 6 (H, ϕ) if and only if I 6 g g g H and ψ = ϕ|I ) and a G-set under conjugation (H, ϕ) := ( H, ϕ) (with gϕ(x) := ϕ(g−1xg) for x ∈ gH) such that the G-action respects the poset p structure. Similarly, we define MR(G) as the G-subposet consisting of those 0 p (H, ϕ) ∈ MR(G) where ϕ has p -order. Now we define DR(G) (resp. DR(G)) as the free abelian group on the G-conjugacy classes [H, ϕ]G of elements in p p MR(G) (resp. MR(G)). Similar to the Burnside ring, DR(G) and DR(G) can be considered as representation rings of certain categories of finite G-equivariant line bundles (cf.[Bo97b]) but we do not need this interpretation here. DR(G) is a commutative ring under

X g g g g [H, ϕ]G · [I, ψ]G = H ∩ I, ϕ|H∩ I · ψ|H∩ I G , g∈H\G/I

p see [Bo97b, 5.3]. With this deﬁnition, DR((G) is a subring of DR(G). If R is any domain containing O , then M (G) ∼ M (G), Mp (G) ∼ Mp (G), n On = R On = R D (G) ∼ D (G), and Dp (G) ∼ Dp (G) in a canonical way, using the On = R On = R inclusion O× R×. We set M(G) := M (G), Mp(G) := Mp (G), n 6 On On D(G) := D (G), and Dp(G) := Dp (G), and use the above identiﬁcations On On p whenever R is a domain containing On. Note also that we can identify M (G) p with MF (G) or MFn (G). There is an obvious inclusion ιG : D (G) → D(G). If f : G˜ → G is a group homomorphism, then

X −1 g g ˜ g resf : D(G) → D(G) , [H, ϕ]G 7→ f ( H), ϕ ◦ f|f −1( H) G˜ g∈f(G˜)\G/H is a ring homomorphism (see [Bo97b, 5.4]) which maps Dp(G) to Dp(G˜) and makes D and Dp into contravariant functors from gr to Ri. Note that the embedding p ιG : B(G) → D (G) , [G/H] 7→ [H, 1]G , is a ring homomorphism which is natural in G. The injective ring homomor- p p phisms ιG : B(G) → D (G) and ιG : D (G) → D(G) have natural left inverses

p πG : D (G) → B(G) , [H, ϕ]G 7→ [G/H] , p ρG : D(G) → D (G) , [H, ϕ]G 7→ [H, ϕp0 ]G ,

0 in the category Ri. Recall that ϕp0 denotes the p -part of ϕ. p p Note that D (G) is canonically isomorphic to DF (G) = DF (G) or DFn (G) = p × × × D (G) via the isomorphism Hom(H, O ) 0 → Hom(H,F ) → Hom(H,F ) Fn n p n induced by the reduction homomorphism On → Fn and the inclusion Fn → F .

5 With these identiﬁcations one can interpret ρG also as reduction in the second argument of [H, ϕ]G. We have a ring homomorphism

ab ηG : R (G) → D(G) , ϕ 7→ [G, ϕ]G , which is natural in G and restricts to a natural transformation η : Rp,ab → Dp. Moreover, we have a left inverse of η, given by ( ab ϕ, if H = G, πG : D(G) → R (G) , [H, ϕ]G 7→ (1.4.a) 0, otherwise , which again restricts to a natural transformation π : Dp → Rp,ab. It is now easy to check that the ﬁrst four equations in (1.1.b) and (1.1.c) hold.

1.5 Let R be a commutative ring. A permutation RG-module is an RG-module which has a finite G-stable R-basis. A monomial RG-module is an RG-module M which has a finite R-basis such that G permutes the rank-one R-submodules spanned by these basis elements. This is the same as saying that M is isomorphic G to a direct sum of modules of the form IndH (Rϕ), where (H, ϕ) ∈ MR(G) and Rϕ denotes the RH-module R with H-action defined by hr := ϕ(h)r for h ∈ H and r ∈ R. We call an RG-module a trivial source (resp. linear source) RG- module if it is isomorphic to a direct summand of a permutation (resp. monomial) RG-module. If R is a complete discrete valuation ring, these modules can be characterized via the sources of their indecomposable direct summands, cf. [Bo98a, Proposition 1.1, 1.2]. This explains the terminology. For general R it might be more suitable to call these modules quasi permutation (resp. monomial) RG-modules. We denote by RGtriv (resp. RGlin) the categories of trivial source (resp. linear source) RG-modules. We assume that the Krull-Schmidt- Theorem holds in RGlin. Then we define LR(G) as the free abelian group on the set of isomorphism classes [M] of indecomposable modules M ∈ RGlin. If M ∈ RGlin and M = M1 ⊕ · · · ⊕ Mr is a decomposition into indecomposable modules, we set [M] = [M1] + ··· + [Mr] ∈ LR(G). The group LR(G) has a commutative ring structure induced by the tensor product M ⊗R N of two modules M,N ∈ RGlin. Similarly, we define TR(G) as the representation ring of RGtriv. This is a subring of LR(G). If f : G˜ → G is a group homomorphism ˜ then every M ∈ RGlin is also an object in RG˜ lin using the G-action given by restriction along f. In fact, the restriction preserves permutation modules and monomial modules. This induces a ring homomorphism resf : LR(G) → LR(G˜) such that resf (TR(G)) ⊆ TR(G˜), and LR and TR become functors form gr to Ri.

We are mainly interested in R = Om for certain multiples m of n. Note that by [CR81, Theorem 30.18(iii)] the Krull-Schmidt-Theorem holds in OmGlin whenever m is a multiple of n. Moreover, by [Fe82, Lemma I.18.7] there exists a multiple m of n such that for any multiple l of m the functor

Ol ⊗Om −: OmGlin → OlGlin preserves indecomposability and induces an isomorphism LOm (G) → LOl (G), since every indecomposable linear source OmG- G module must be a direct summand of some IndP (Om)ψ for some p-subgroup P

6 × of G and some ψ ∈ Hom(P, Om). In this case we call Km a splitting ﬁeld for the linear source modules of G. If Km and Km0 are two such splitting ﬁelds, then we obtain a canonical isomorphism LOm (G) → LOm0 (G) via LOl (G) with l = lcm(m, m0) or any common multiple l of m and m0.

Let m be a multiple of n such that Km is a splitting field for the linear source modules of G. In order to relate the ring LOm (G) to the linear source rings used in the literature let Obm denote a p-adic completion of Om and bpm the maximal ideal of Obm. It is well-known that the Krull-Schmidt-Theorem holds for lin. By [CR81, Theorem 30.18], the functor Obm ⊗Om −: OmGlin → lin Obm ObmG preserves indecomposability and induces an isomorphism LOm (G) → L (G). Obm From the preceding discussions it follows that again for any multiple l of m the functor Obl ⊗ −: lin → lin preserves indecomposability and induces Obm ObmG OblG an isomorphism L (G) → L (G). Let Rb denote the valuation ring in a Obm Obl maximal unramified extension E of the field of fractions of Obm. In other words, E is isomorphic to the field obtained by adjoining an mp-th root of unity and 0 all roots of unity of p -order to the field Qp of p-adic numbers. The ring Rb is then a complete discrete valuation ring whose residue field is an algebraic closure of the field of p elements. Since every indecomposable linear source RGb -module is already defined over Obl for some multiple l of m, we obtain that the functor Rb ⊗ −: lin → lin preserves indecomposability and Obm ObmG RGb induces an isomorphism L (G) → L (G). So, finally, the functor Rb ⊗Om Obm Rb −: O Glin → lin preserves indecomposability and induces an isomorphism m RGb LO (G) → L (G) so that we can use result about L (G) from the literature m Rb Rb also for LOm (G). Everything we established above for linear source modules holds in particular for trivial source modules. Moreover, with Rb as above it is well-known that reduction modulo the radical of R gives a functor triv → triv that b RG FG preserves indecomposability and induces an isomorphismb T (G) → T (G), R F cf. [Br85]. Together with the above results reinterpreted for trivialb source modules one obtains that also the functors Fm ⊗Om −: OmGtriv → FmGtriv and

F ⊗Fm −: Fm triv → F triv preserve indecomposability and induces isomorphisms

TOm (G) → TFm (G) and TFm (G) → TF (G). p p Now we define L (G) := LOm (G) and T (G) := TOm (G), where Km is a splitting field for the linear source modules of G. Moreover, whenever m0 ∈ N is such that also Km0 is a splitting field for the linear source modules of G, then we p p identify L (G) and T (G) with LOm0 (G) and TOm0 (G) as above. In particular, if p p f : G˜ → G is a group homomorphism then resf : L (G) → L (G˜) is well-defined and restricts to a ring homomorphism T p(G) → T p(G˜). Summarizing the above considerations we obtain functors T p,Lp : gr → Ri such that ι: T p ⊆ Lp is a natural inclusion. p p Sometimes we will identify T (G) with TF (G) or TFm (G) and L (G) with LO (G) or L (G) always using the above isomorphism. m Rb

7 The homomorphism

G βG : D(G) → L(G) , [H, ϕ]G 7→ IndH (Om)ϕ , defines a natural transformation D → Lp, and its restriction to Dp(G) defines a p p natural transformation β : D → T , since (Om)ϕ is a trivial source OH-module, 0 p p p if ϕ has p -order. The maps βG : D(G) → L (G) and βG : D (G) → T (G) are surjective, since there exist induction formulae for Lp(G) and T p(G) using p p ∼ D(G) and D (G), cf. [Bo98a]. If we use the identifications D (G) = DF (G) p ∼ p p and T (G) = TF (G), then the map β : D (G) → T (G) is given by [H, ϕ]G 7→ G p p [IndH (Fϕ)] and the natural inclusion ιG : T (G) → L (G) has the natural left inverse ρG : LOm (G) → LFm (G) given by reduction modulo pm. It is easy to check that the fifth and sixth equation in (1.1.b) and the fifth equation in (1.1.c) hold.

1.6 We denote by R(G) (resp. Rp(G)) the character ring (resp. the Brauer character ring) of G. We identify R(G) (resp. Rp(G)) with the Grothendieck × 0 ring of KGmod (resp. FGmod). For that reason we identify F with the p -part of the group of roots of unity of K via the natural surjection O → O/p = F . Hence, we may assume that the Brauer characters as well as the characters have p their values in OK ⊂ K. Then the decomposition map δG : R(G) → R (G) is 0 given by restriction of a character to the p -elements of G. δG is a surjective ring homomorphism which commutes with induction and restrictions resf with respect to arbitrary group homomorphisms f. In short, δ : R → Rp is a natural transformation between the ring valued functors R and Rp (see [Se78, §14-§18] p for more details). The map δG has a left inverse, the Brauer lift λG : R (G) → R(G), which extends a class function χ on the set of p0-elements of G to the class function on G that maps g ∈ G to χ(gp0 ). The Brauer lift is natural in G. With Brauer’s characterization of virtual characters it is easy to see that λG(χ) ∈ R(G). It is also easy to see that the last two equations in (1.1.b) and the last equation in (1.1.c) hold. Moreover, if m ∈ N is such that Km is a splitting ﬁeld for the linear source modules of G, the functors

OmGtriv → FmGmod → FGmod and

Om(G)lin → KmGmod → KGmod , given by scalar extensions, induce natural surjective ring homomorphisms

p p p γ : T (G) qqqqqqqqqqqqqqqq R (G) and γ : L (G) qqqqqqqqqqqqqqqq R(G) , G qqqqqqq qqqqqqq G qqqqqqq qqqqqqq qqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqq since they commute with induction, and since every element in Rp(G) and R(G) is a Z-linear combination of induced one-dimensional characters. p p We still have to deﬁne the map τG : R (G) → T (G). This is the only map in Diagram (1.1.a) that is not obvious. We use the canonical induction

8 p p formula aG : R (G) → D (G) from [Bo98b, Example 9.8] and deﬁne τG as the composition p aG p βG p τ : R (G) qqqqqqqq D (G) qqqqqqqq T . G qqqqqqq qqqqqqq qqqqqqqq qqqqqqqq qqqqqqqqqq qqqqqqqqqq

Since the map aG commutes with restrictions (cf. [Bo98b, Theorem 10.3]), the map τG commutes with restrictions. Moreover, τG is a ring homomorphism. In fact, since elements in T p(G) are determined by their restrictions to p-hypoelementary subgroups (i.e., subgroups H satisfying that H/Op(H) is G cyclic), it suffices to show that resH ◦ τG is a ring homomorphism for each p- G hypoelementary subgroup H of G. But since τG commutes with resH , it suffices to show that τH is a ring homomorphism whenever H is p-hypoelementary. In p × this case, R (H) is the free abelian group on the classes [Fϕ], ϕ ∈ Hom(H,F ), of one-dimensional modules and one has aH ([Fϕ]) = [H, ϕ]H . This implies p τH ([Fϕ]) = [Fϕ] ∈ T (H) and τH is a ring homomorphism. p Similarly, one checks that γG ◦τG is the identity on R (G). In fact, it suffices G G p p 0 to show that resH ◦ γG ◦ τG = resH : R (G) → R (H) for cyclic p -subgroups H of G. Since τ and γ commute with restrictions, it suffices to show that γH ◦ τH p is the identity on R (H), which is easily verified starting with [Fϕ] as above. Finally, λ = γ◦ι◦τ, since it suffices to show this only for cyclic groups H and p for those it follows immediately by inspection of what happens to [Fϕ] ∈ R (H), ϕ ∈ Hom(H,F ×), on both sides of the equation.

2 The species

2.1 In this section we recall the species of the representation rings X(G) where X stands for B, Rp,ab, Rab, Dp, D, T p, Lp, Rp, or R. For any of these choices of X we will introduce a functor

X : gr → set to the category of ﬁnite sets and a pairing

(−, −)G : X(G) × X (G) → OK for every G ∈ gr such that for each group homomorphism f : G → G˜ we have

res (χ), x = χ, f(x) , (2.1.a) f e G e G˜ ˜ ˜ for all χe ∈ X(G) and x ∈ X (G), if we denote the map X (f): X (G) → X (G) again by f. Moreover in all cases, for ﬁxed x ∈ X (G), the map

X(G) sx : X(G) → OK , χ 7→ (χ, x)G , will be a ring homomorphism with the property

X(G) X(G˜) sx ◦ resf = sf(x) , (2.1.b)

9 whenever f : G → G˜ is a group homomorphism, by (2.1.a). Note that, since X is a functor, the conjugation maps s: G → G, g 7→ sgs−1, for s ∈ G, provide X with the structure of a G-set. For any x1, x2 ∈ X (G) we will have sX(G) = sX(G) ⇐⇒ x = gx for some g ∈ G. (2.1.c) x1 x2 1 2

X(G) X(G) Moreover the collection s of maps sx , where x runs through a set of representatives for the G-orbits X (G)/G, induces an isomorphism Y sX(G) : K ⊗ X(G) → K (2.1.d) x∈X (G)/G

X(G) of K-algebras. Therefore, the maps sx , x ∈ X (G), constitute all the ring homomorphisms X(G) → K. They are called the species of X(G). We may identify the last product with the K-algebra HomG(X (G),K) in the obvious way, where HomG stands for G-equivariant maps and K is considered as G-set with trivial action. Then the isomorphism sX(G) has the form

X(G) s : K ⊗ X(G) → HomG(X (G),K) , (2.1.e)

X(G) X(G) and we have (s (a⊗χ))(x) = asx (χ) = a(χ, x)G for χ ∈ X(G), x ∈ X (G), and a ∈ K. The natural transformations α: X → Y between the representation ring functors in Diagram (1.1.a) will be induced by natural transformations α∗ : Y → X after identifying K ⊗X(G) and K ⊗Y (G) under the isomorphism (2.1.e) with the K-algebras of G-equivariant functions on X (G) and Y(G) such that we have

∗ αG : HomG(X (G),K) → HomG(Y(G),K) , f 7→ f ◦ αG . (2.1.f)

∗ If we denote by αG : Y(G)/G → X (G)/G the induced map on G-orbits, it is easy to verify that one has the following relations:

∗ ∗ αG injective ⇒ αG injective ⇐⇒ αG surjective (2.1.g) and ∗ ∗ αG surjective ⇐⇒ αG surjective ⇐⇒ αG surjective.

2.2 Remark (a) If X : gr → K alg is a contravariant functor with the property that, for each G ∈ gr, X(G) is isomorphic to a ﬁnite product of copies of K, then one obtains a covariant functor X : gr → set by setting

X (G) := K alg(X(G),K) and X (f): K alg(X(G),K) → K alg(X(G˜),K) , s 7→ s ◦ X(f) , for any f : G → G˜ in gr. Conversely, if X : gr → set is a covariant functor, then one obtains a contravariant functor X : gr → K alg by setting X(G) := set(X (G),K)

10 and X(f): set(X (G˜),K) → set(X (G),K) , ε 7→ ε ◦ X (f) , for any f : G → G˜ in gr. Obviously, X(G) is isomorphic to a ﬁnite product of copies of K. It is not diﬃcult to see that the composition of these two constructions (in both orders) yields a functor that is naturally isomorphic to the original one. In fact, if we start with X, then M X(G) = K · es , (2.2.a)

s∈K alg(X(G),K) where es is the unique primitive idempotent of X(G) with s(es) = 1 and we have a natural isomorphism ∼ X(G) = set K alg(X(G),K),K in K alg sending es ∈ X(G) to the characteristic map εs with εs(t) := δs,t and sending ε ∈ set( alg(X(G),K),K) to P ε(s)·e . Also, if we start K s∈K alg(X(G),K) s with X , then we have a natural bijection ∼ X (G) = K alg set(X (G),K),K sending x ∈ X (G) to s with s(ε) := ε(x) for ε ∈ set(X (G),K) and sending s ∈ K alg(set(X (G),K),K) to the unique x ∈ X (G) with s(εx) = 1, where εx : X (G) → K is the characteristic function on {x}.

(b) If Y : gr → K alg is a second contravariant functor with the property that Y (G) is isomorphic to a ﬁnite product of copies of K for every G ∈ gr and Y : gr → set is the associated covariant functor according to part (a), then one obtains a bijection Nat(X,Y ) =∼ Nat(Y, X ) between these sets of natural transformations as follows. We map ϕ ∈ Nat(X,Y ) to ψ ∈ Nat(Y, X ) with

ψG : Y(G) → X (G) , s 7→ s ◦ ϕG , for G ∈ gr, and we map ψ ∈ Nat(Y, X ) to ϕ ∈ Nat(X,Y ) with X ϕG : X(G) → Y (G) , es 7→ et , −1 t∈ψG (s) where we use the notation from part (a) and Equation (2.2.a). (c) If X : gr → set is a covariant functor then we denote by X : gr → set the covariant functor deﬁned by

X (G) := X (G)/G and ˜ ˜ X (f): X (G)/G → X (G)/G, [x]G 7→ [(X (f))(x)]G˜ ,

11 which is well-defined. Here the G-set structure on X (G) comes by functoriality from the inner automorphisms of G. Assume that (X, X ) is as in Subsection 2.1. Tensoring with K yields a contravariant functor K ⊗ X : gr → K alg which takes values in K-algebras that are isomorphic to finite products of copies of K. On the other hand one obtains a covariant functor X : gr → set by the above construction. By Equations (2.1.b), (2.1.c), and (2.1.d), the functors K⊗X and X correspond under the construction in part (a). If also (Y, Y) is as in Subsection 2.1 we obtain a diagram ∼ Nat(X,Y ) Nat(K ⊗ X,K ⊗ Y ) = Nat(Y, X ) ← Nat(Y, X ) . The first injection is defined by tensoring with K, the middle isomorphism is the one from part (b), and the last map is given by sending ψ ∈ Nat(Y, X ) to ψ ∈ Nat(Y, X ) which is defined by

ψG : Y(G)/G → X (G)/G , [y]G 7→ [ψG(y)]G . It is easy to see that ψ is again a natural transformation. By (2.1.f), the natural transformations α ∈ Nat(X,Y ) we considered in Diagram (1.1.a) come from the natural transformations α∗ ∈ Nat(Y, X ) through the above diagram. We do not know if the map Nat(Y, X ) → Nat(Y, X ) is injective. Therefore, the notation α∗ is abusive. What one can see from the above considerations is the following: In order to construct a natural transformation X → Y , one can start with a natural transformation Y → X and try to prove that the associated natural transformation K ⊗ X → K ⊗ Y via the above diagram has an integral form, i.e., that it maps X(G) to Y (G) for each G. In this case we call the original natural transformation Y → X also integral. Note that while not every natural transformation in Nat(X,Y ) comes that way from one in Nat(Y, X ) it does come from one in Nat(Y, X ). Also, while the functor X is determined by X up to functorial isomorphism, the functor X is not. At one point (for X = T p) we will have to give two diﬀerent choices for X in order to obtain all natural transformations in Nat(X,Y ) as induced from some in Nat(Y, X ).

2.3 We deﬁne R: gr → set as the forgetful functor, R(G) := G, and

(χ, g)G := χ(g) (2.3.a) for χ ∈ R(G) and g ∈ G. We obtain the species

R(G) sg : R(G) → OK , χ 7→ χ(g) ,

p 0 for g ∈ G. Similarly, we define R (G) := Gp0 , as the set of elements of p - order in G. This defines a subfunctor Rp ⊂ R. Again we define a pairing p p p (−, −)G : R (G) × R (G) → OK by (2.3.a) for χ ∈ R (G) and g ∈ Gp0 and obtain as species the evaluation maps

Rp(G) p sg : R (G) → OK , χ 7→ χ(g) ,

12 for g ∈ Gp0 . p It is obvious that the decomposition map δG : R(G) → R (G) is induced by ∗ p the natural transformation δG : R (G) → R(G) that is given by the inclusion. ∗ Moreover, the Brauer lift is induced by the natural transformation λG : R(G) → p 0 R (G) that maps g ∈ G to its p -part gp0 . Similarly, we deﬁne functors Rab, Rp,ab : gr → set by Rab(G) := Gab and p,ab ab 0 R (G) := (G )p0 and pairings (χ, gG )G 7→ χ(g) for g ∈ G or for g ∈ G ab 0 ab R (G) ab with gG ∈ (G )p0 to obtain species sgG0 : R (G) → OK , χ 7→ χ(g) and p,ab R (G) p,ab p sgG0 : R (G) → OK , χ 7→ χ(g). As for R(G) and R (G), the inclu- p,ab ab ∗ ab p,ab 0 0 sion ι: R → R is induced by ιG : R (G) → R (G), gG 7→ gp0 G = 0 ab p,ab (gG )p0 , and the reduction map ρ: R → R is induced by the inclusion ∗ p,ab ab ρG : R (G) → R (G). It is well-known that all the assertions in Subsection 2.1 hold for X ∈ {R,Rp,Rab,Rp,ab}, cf. [Se78, §14-§18].

2.4 In the case of the Burnside ring B(G) we denote by B(G) the set of subgroups of G. Obviously, this defines a functor B : gr → set if we define B(f): B(G) → B(G˜), H 7→ f(H), for every morphism f : G → G˜ in gr. For H H H 6 G and S ∈ Gset we set ([S],H)G := |S | ∈ Z, where S denotes the set of H-fixed points of S. This induces a map B(G) × B(G) → Z such that B(G) sH : B(G) → Z is just the mark homomorphism with respect to G. Clearly, all the claims in Subsection 2.1 are satisfied, cf. [CR87, §80].

2.5 Next we determine the species of D(G) and Dp(G). We set

0 D(G) := {(H, hH ) | H 6 G, h ∈ H} and p 0 0 ab D (G) := {(H, hH ) | H 6 G, h ∈ H with hH ∈ (H )p0 } , where H0 denotes the derived subgroup of H. Obviously, we have a surjective map

∗ p 0 0 0 ιG : D(G) → D (G) , (H, hH ) 7→ (H, (hH )p0 ) = (H, hp0 H ) and an injective map

∗ p 0 0 ρG : D (G) → D(G) , (H, hH ) 7→ (H, hH ) . For a group homomorphism f : G → G˜ in gr we deﬁne

D(f): D(G) → D(G˜) , (H, hH0) 7→ (f(H), f(h)f(H)0) , and similarly Dp(f): Dp(G) → Dp(G˜). With this deﬁnition, D and Dp are ∗ ∗ functors gr → set and ιG and ρG from above deﬁne natural transformations. For χ ∈ D(G) and (H, hH0) ∈ D(G) we set

0 Rab(H) G χ, (H, hH ) G := (shH0 ◦ πH ◦ resH )(χ)

13 ab p with πH : D(H) → R (H) as deﬁned in (1.4.a). Similarly, for χ ∈ D (G) and (H, hH0) ∈ Dp(G) we set

0 Rp,ab G χ, (H, hH ) G := (shH0 ◦ πH ◦ resH )(χ) . From that we obtain species

ab G R (H) D(G) resH πH ab shH0 s : D(G) qqqqqqqq D(H) qqqqqqqq R (H) qqqqqqqq O 0 qqqqqqq qqqqqqq qqqqqqq K qqqqqqqq qqqqqqqq qqqqqqqq (H,hH ) qqqqqqqqqq qqqqqqqqqq qqqqqqqqqq and

Rp,ab(H) p G D (G) p resH p πH p,ab shH0 s : D (G) qqqqqqqq D (H) qqqqqqqq R (H) qqqqqqqq O . 0 qqqqqqq qqqqqqq qqqqqqq K qqqqqqqq qqqqqqqq qqqqqqqq (H,hH ) qqqqqqqqqq qqqqqqqqqq qqqqqqqqqq

It follows from considering the injectivity of the map ρG in [Bo98b, Propo- D(G) D(G) sition 2.4] that two maps s 0 and s 0 coincide if and only if (H1,h1H1) (H2,h2H2) 0 0 (H1, h1H1) and (H2, h2H2) are conjugate under G. Moreover the surjectivity of D(G) the map ρG in [Bo98b, Proposition 2.4] implies that the number of maps s(H,hH0) is the same as the Z-rank of D(G). Since ring homomorphisms into domains are always linearly independent, by Dedekind’s lemma, we obtain that (2.1.d) is an isomorphism for X = D. The same arguments hold for Dp. Note that ab p p,ab D(G) is R+ (G) and D (G) is equal to R+ (G) in the terminology of [Bo98b, Proposition 2.4]. It is now easy to see that the natural inclusion ι: Dp → D is induced by the natural surjection ι∗ from above and that the natural surjection ρ: D → Dp is induced by the natural embedding ρ∗ from above. The natural embedding η : Rab → D and the natural surjection π : D → Rab ∗ ab 0 0 are induced by the natural surjection ηG : D(G) → R (G), (H, hH ) 7→ hG ∗ ab 0 0 and the natural embedding πG : R (G) → D(G), gG 7→ (G, gG ). Similarly, the natural embedding η : Rp,ab → Dp and the natural surjection π : Dp → Rp,ab ∗ p p,ab 0 0 are induced by the natural surjection ηG : D (G) → R (G), (H, hH ) 7→ hG ∗ p,ab p 0 0 and the natural embedding πG : R (G) → D (G), gG 7→ (G, gG ). The natural transformations between Dp and B are induced by the natural ∗ p 0 ∗ p transformations ιG : D (G) → B(G), (H, hH ) 7→ H and πG : B(G) → D (G), H 7→ (H, 1 · H0).

2.6 Proposition The Equation (2.1.a) holds for (X, X ) = (D, D) and (X, X ) = (Dp, Dp). Proof Let f : G → G˜ be in gr and let (H, hH0) ∈ D(G). We have to show that

Rab(H) G Rab(f(H)) G˜ shH0 ◦ πH ◦ resH ◦ resf = s(f(H),f(h)f(H)0) ◦ πf(H) ◦ resf(H)

˜ G as maps from D(G) to OK . Since D is a functor, one has resH ◦ resf = G˜ resf : H→f(H) ◦ resf(H). Also, in the right hand side of the above equation we can use Rab(f(H)) Rab(H) s(f(H),f(h)f(H)0) = s(H,hH0) ◦ resf : H→f(H) ,

14 by Equation (2.1.b) for Rab and Rab, so that it suﬃces to show that

πH ◦ resf : H→f(H) = resf : H→f(H) ◦ πf(H) .

But this is easily veriﬁed: Let [U, ϕ]f(H) be a basis element of D(f(H)) and assume ﬁrst that U < f(H). Then πf(H)([U, ϕ]f(H)) = 0 and −1 −1 resf : H→f(H)([U, ϕ]f(H)) = [f (U), ϕ ◦ f]H with f (U) < H so that also the left hand side evaluates to 0. If U = f(H), then both sides evaluate to ϕ◦f. This proves the proposition for (X, X ) = (D, D). The proof for (X, X ) = (Dp, Dp) is similar, or even a special case of the one above.

2.7 Finally we recall from [Bo98a, §2] the species of Lp(G) and T p(G). We set p 0 L (G) := {(H, hOp(H) ) | H 6 G, hhOp(H)i = H/Op(H)} and p T (G) := {(H, hOp(H)) | H 6 G, hhOp(H)i = H/Op(H)} .

Note that the second condition implies that H/Op(H) is cyclic, i.e., that H is p-hypoelementary. We obtain functors Lp, T p : gr → set if, for f : G → G˜ in gr, 0 p p (H, hOp(H) ) ∈ L (G), and (H, hOp(H)) ∈ T (G), we set 0 0 f(H, hOp(H) ) = (f(H), f(h)Op(f(H)) ) and f(H, hOp(H)) = (f(H), f(h)Op(f(H))) .

Let m be a multiple of n such that Km is a splitting field for the linear source p modules of G. We identify L (G) = LOm (G) and L (G) as explained in Obm Subsection 1.5. We denote by L0(G) ⊆ Lp(G) the subgroup generated by the isomorphism classes [V ] of indecomposable V ∈ lin having the Sylow sub- ObmG groups of G as vertices, and we denote by L00(G) ⊆ Lp(G) the ideal generated as group by elements [V ], where V runs through the remaining indecomposable modules in lin. Similarly, we define the subgroup T 0(G) and the ideal OmG 00 p b L p 0 T p 0 T (G) of T (G). We denote by qG : L (G) → L (G) and qG : T (G) → T (G) the projection corresponding to the decompositions Lp(G) = L0(G) ⊕ L00(G) p 0 00 T L and T (G) = T (G) ⊕ T (G). Then qG is the restriction of qG. If G is p- hypoelementary then L0(G) and T 0(G) are subrings of Lp(G) and T p(G), and L T the maps qG and qG are ring homomorphisms. Now we set 0 R(H) L G [V ], (H, hOp(H) ) G := (sh ◦ γH ◦ qH ◦ resH )([V ]) and Rp(H) T G [W ], (H, hOp(H)) := (s ◦ γH ◦ q ◦ res )([W ]) G hp0 H H 0 p for V ∈ lin and W ∈ triv,(H, hOp(H) ) ∈ L (G), and (H, hOp(H)) ∈ OmG OmG p b b p p T (G). This is well-defined and induces maps L (G) × L (G) → OK and p p T (G) × T (G) → OK by linear extension. As species we obtain

R(H) p G L L (G) p resH p qH 0 γH sh s : L (G) qqqqqqqq L (H) qqqqqqqq L (H) qqqqqqqq R(H) qqqqqqqq O 0 qqqqqqq qqqqqqq qqqqqqq qqqqqqq K qqqqqqqq qqqqqqqq qqqqqqqq qqqqqqqq (H,hOp(H) ) qqqqqqqqqq qqqqqqqqqq qqqqqqqqqq qqqqqqqqqq

15 and

Rp(H) p G T s T (G) p resH p qH 0 γH p hp0 s : T (G) qqqqqqqq T (H) qqqqqqqq T (H) qqqqqqqq R (H) qqqqqqqq O qqqqqqq qqqqqqq qqqqqqq qqqqqqq K qqqqqqqq qqqqqqqq qqqqqqqq qqqqqqqq (H,hOp(H)) qqqqqqqqqq qqqqqqqqqq qqqqqqqqqq qqqqqqqqqq such that all the assertions in Subsection 2.1 hold. It is easy to see that the natural embedding ι: T p → Lp is induced by the natural surjection

∗ p p 0 ιG : L (G) → T (G) , (H, hOp(H) ) 7→ (H, hOp(H)) ,

∗ p p but there is no dual map ρG : T (G) → L (G) for ρG. The only candidate that 0 comes to one’s mind which maps (H, hOp(H)) to (H, (hp0 Op(H) )) is not well- deﬁned. For that reason we have to introduce a second functor T˜ p : gr → set by ˜ p T (G) := {(Q, g) | Q 6 G a p-subgroup, g ∈ NG(Q)p0 } . For a group homomorphism f : G → G˜ we set T˜ p(f)(Q, g) := (f(Q), f(g)). It is well-known and not diﬃcult to see that the map

∗ ˜ p p κG : T (G) → T (G) , (Q, g) 7→ (hQ, gi, gQ) , which is natural in G induces an isomorphism between the G-orbits of T˜ p(G) and T p(G). Again we have a pairing

p ˜ p T G (−, −)G : T (G) × T (G) → OK , (χ, x) 7→ (sg ◦ γH ◦ qH ◦ resH )(χ) , with H := hQ, gi, and obtain an associated species

Rp(H) p G T T (G) p resH p qH 0 γH p sg s : T (G) qqqqqqqq T (H) qqqqqqqq T (H) qqqqqqqq R (H) qqqqqqqq O qqqqqqq qqqqqqq qqqqqqq qqqqqqq K qqqqqqqq qqqqqqqq qqqqqqqq qqqqqqqq (Q,g) qqqqqqqqqq qqqqqqqqqq qqqqqqqqqq qqqqqqqqqq such that their collection induces an isomorphism K ⊗ T p(H) → p p p T (G) T (G) HomG(T (G),K). We have s = s ∗ so that the natural surjection (Q,g) κG(Q,g) κ∗ induces the identity on T (G). Now we are able to give a map

∗ ˜ p p 0 ρG : T (G) → L (G) , (Q, g) 7→ (hQ, gi, gQ ) ,

p p that induces ρG : L (G) → T (G). p p L (G) D(G) The composition of β : D(G) → L (G) with s 0 equals s 0 , (H,hOp(H) ) (H,hH ) since we have a commutative diagram ab G R (H) resH πH ab shH0 D(G) qqqqqqqq D(H) qqqqqqqq qqqqqqqq qqqqqqq qqqqqqq R (H) qqqqqqq OK qqqqqqqq qqqqqqqq qqqqqqqq qqqqqqqqqq qqqqqqqqqq qqqqqqqqqq

qqqqqqqqqqqqqqq qqqqqqqqqq qqqqqqqqqq R(H) βG βG sh

qqqqqqq qqqqqqq qqqqqqq qqqqqqq qqqqqqqqqqqqqqqqqqqq G qqqqqqqqqqqqqqqqqqqq L p q resH p q qH 0 γH L (G) qqqqqqqq L (H) qqqqqqqq L (H) qqqqqqqq R(H) qqqqqqq qqqqqqq qqqqqqq qqqqqqqq qqqqqqqq qqqqqqqq qqqqqqqqqq qqqqqqqqqq qqqqqqqqqq

16 for H 6 G p-hypoelementary and h ∈ H with hhOp(H)i = H/Op(H). In fact, for [U, ϕ]H ∈ D(H) we have

R(H) L H (sh ◦ γH ◦ qH )([IndU (Obm)ϕ]) = 0 , if U < H, since hOp(H) generates H/Op(H), and if U = H, then both maps applied to [U, ϕ]H result in ϕ(h). Similarly, one can see that the composition of p p p p T (G) D (G) β : D (G) → L (G) with s equals s 0 for (H, hOp(H)) ∈ (H,hOp(H)) (H,hp0 H ) T p(G). Therefore, the natural surjections β : Dp → T p and β : D → Lp are induced by the natural embedding

∗ p p 0 βG : T (G) → D (G) , (H, hOp(H)) 7→ (H, hp0 H ) , and the natural transformation

∗ p 0 0 βG : L (G) → D(G) , (H, hOp(H) ) 7→ (H, hH ) . Note that the last map is not injective in general, but that it must be injective on G-orbits by (2.1.g). One can also use the map

∗ ˜ p p 0 βG : T (G) → D (G) , (Q, g) 7→ (H, gH ) , with H := hQ, gi, in order to induce the map β : Dp → T p. Next it is easy to see that the natural surjections γ : T p → Rp and γ : Lp → R are induced by the natural embeddings

∗ p p γG : R (G) → T (G) , g 7→ (hgi, g) , and ∗ p γG : R(G) → L (G) , g 7→ (hgi, g) . Again, one can also use T˜ p and deﬁne

∗ p ˜ p γG : R (G) → T (G) , g 7→ (1, g) , in order to induce the map γ : T p → Rp. p p Finally, the map τG : T (G) → R (G) is induced by ∗ ˜ p p τG : T (G) → R (G) , (Q, g) 7→ g , which deﬁnes a natural surjection τ ∗ : T˜ p → Rp. In this case there is no possible p p map from T (G) to R (G) that induces τG.

2.8 Proposition The Equation (2.1.a) holds for (X, X ) ∈ {(T p, T p), (T p, T˜ p), (Lp, Lp)}. Proof We ﬁrst prove the assertion for (Lp, Lp). Let f : G → G˜ be in gr and let 0 p (H, hOp(H) ) ∈ L (G). We have to show that

R(H) L G R(f(H)) L G˜ sh ◦ γH ◦ qH ◦ resH ◦ resf = sf(h) ◦ γf(H) ◦ qf(H) ◦ resf(H) .

17 G G˜ We transform the left hand side using resH ◦ resf = resf : H→f(H) ◦ resf(H) and R(f(H)) R(H) the right hand side using sf(h) = sh ◦resf : H→f(H) from Equation (2.1.b) for (R, R) so that it suﬃces to show that

L L γH ◦ qH ◦ resf = resf ◦ γJ ◦ qJ , for a surjective map f : H → J in gr. Since γ is a natural transformation we L L have resf ◦ γJ = γH ◦ resf and it suﬃces to show that qH ◦ resf = resf ◦ qJ . Now let V ∈ OmJ lin be indecomposable, where m is large enough such that Km is a splitting ﬁeld for the linear source modules of G. Since H is p-hypoelementary, L also J is, and f(Op(H)) = Op(J). If V has vertex Op(J), then qJ ([V ]) = [V ] L and qH (resf ([V ])) = resf ([V ]), since resf ([V ]) has vertex Op(H), and we are done. If V has vertex smaller than Op(J), then resf ([V ]) has vertex smaller than Op(H) and both sides of the equation we have to prove evaluate to 0 on [V ]. The proof for (X, X ) = (T p, T p) and (X, X ) = (T p, T˜ p) is similar.

2.9 We summarize the preceding constructions in the diagram

ι∗ qqqqq qqqqq qqqqqqqqqqqqqqqqqqqq p,ab qqqqqqqqqqqqqqqq ab R qqqqqqqqqqqqqqqqqqqq R qqqqq qqqqq ∗ qqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqq ρ qqqqqqqqqqqqqqqqqqqq q q qqqqqqqqqqqqqqqqqqqq qqqqqqqq qqqqqqqq qqqqqqqqqqqqqqqqqqqq qqqqqqqq qqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqq qqqqqqqqqq qqqqqqqqqq qqqqqqqqqq qqqqqqqqqq η∗ π∗ η∗ π∗

∗ ∗ qqqqqqqq qqqqqqqq qqqqqqqq qqqqqqqq ι qqqqqqqqqqqqqqqqqqq ι qqqqqqqqqqqqqqqqqqq qqqqq qqqqq qqqqq qqqqq qqqqqqqqqqqqqqqqqqqq p qqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqq B qqqqqqqqqqqqqqqqqqqq D qqqqqqqqqqqqqqqqqqqq D qqqqqqqqq qqqqqqqqq qqqqqqqqq qqqqqqqqq qqqqqqqqq ∗ qqqqqqqqq qqqqqqqqq ∗ qqqqqqqqq qqqqqqqqqq qqqqqqqqqq qqqqqqqqqq qqqqqqqqqq qqqqqq π qqqqqq qqqqqqqqqqqqqqqqqqq qqqqqq ρ qqqqqq q q qqqqqqqqqqqqqqq qqqqqqqqqqqqqqq qqqqqqqqqq qqqqqqqqqq qqqqqqqqqq qqqqqqqqqq β∗ β∗ β∗ (2.9.a)

qqqqqqqqqqqqqqq qqqqqqqqqq qqqqqqqqqq qqqqqqqqqqqqqqqqqqq ∗ q p q κ p p qqqqq qqqqq ˜ T qqqqqqqqqqqqqqqq qqqqqqq qqqqqqq L qqqqq qqqqq T qqqqqqqq ∗ qqqqqqqq qqqqqqqqqqqqqqqqqqqq qqqqqqqq qqqqqqqq qqqq qqqq qqqqqqqqqq ρ qqqqqqqqqq qqqqqqqqqqqqqqqqqqq q q qqqqqqqqqqqqqqq qqqqqqqqqqqqqqq @ qqqqqqqqqq qqqqqqqqqq qqqqqqqqqq qqqqqqqqqq @ γ∗ τ ∗ γ∗ γ∗ @

@ qqqqqqqq qqqqqqqq ∗ qqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqq qqqqqqqqqq qqqqqqqqqq qqqqqqqqqqqqqqqqqqqq qqqqqqqqqq qqqqqqqqqq qqqqqqqqqqqqqqqqqqq qqqqq λ q q qqqqq qqqqq q p qqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqq R qqqqqqqqqqqqqqqqqqqq R qqqqq qqqqq qqqqqqqqqq ∗ qqqqqqqqqq qqqqqqqq qqqqqqqq qqqqqqqqqq δ qqqqqqqqqq of functors gr → set and natural transformations between them. If α∗ : Y → X is an arrow in the above diagram then the associated natural transformation in Diagram (1.1.a) is induced by the rule in (2.1.f), or in terms of species by the rule that the composition

Y (G) αG sy X(G) qqqqqqqq Y (G) qqqqqqqq O qqqqqqq qqqqqqq K qqqqqqqq qqqqqqqq qqqqqqqqqq qqqqqqqqqq X(G) equals the species s ∗ for all y ∈ Y(G). One has the corresponding equations αG(y) as in Diagram (1.1.a) and additionally that β∗ ◦ κ∗ = β∗ and κ∗ ◦ γ∗ = γ∗.

18 Moreover, the arrow ι∗ : Lp → T p is missing to avoid an overloading of the diagram and one has ι∗ ◦ ρ∗ = κ∗.

3 Idempotent formulae

3.1 Each of the representation rings X(G), X ∈ {Rp,ab,Rab,B,Dp,D,T p,Lp,Rp,R} in Diagram (1.1.a) becomes a commutative split semisimple K-algebra after tensoring with K, the components indexed by the G-orbits X (G)/G of the associated G-set X (G) introduced in X(G) Section 2. Thus, for every x ∈ X (G) there exists a primitive idempotent ex of K ⊗ X(G) with the property

( 0 X(G) X(G) 1, if x =G x , sx0 ex = 0 0, if x 6=G x ,

0 X(G) X(G) for x, x ∈ X (G), or, if we denote by εx the image of ex under the isomorphism K ⊗ X(G) → HomG(X (G),K) from (2.1.e), then we have

( 0 X(G) 0 1, if x =G x , εx (x ) = 0 0, if x 6=G x .

Obviously, we have, for x, x0 ∈ X (G),

X(G) X(G) X(G) X(G) 0 ex = ex0 ⇐⇒ εx = εx0 ⇐⇒ x =G x .

X(G) 3.2 In this section we will give explicit formulae for the idempotents ex for all X(G) and all x ∈ X (G) starting from those of K ⊗ Rab(G) by applying general abstracts results. Some of these formulae are well-known, some are new. The following equation, where α: X → Y is one of the arrows in Diagram (1.1.a) and α∗ : Y → X is the associated arrow in Diagram (2.9.a), will be very useful. One has X(G) X Y (G) αG(ex ) = ey , (3.2.a) y∈Y(G)/G α∗ ([y] )=[x] G G G for every x ∈ X (G). In fact, for y0 ∈ Y(G), we have

( ∗ 0 X(G) 0 X(G) ∗ 0 1, if αG(y ) =G x, (αG(εx ))(y ) = εx (αG(y )) = ∗ 0 0, if αG(y ) 6=G x, X Y (G) 0 = εy (y ) . y∈Y(G)/G α∗ (y)∈[x] G G

∗ Note that if αG is injective Equation (3.2.a) simpliﬁes to

X(G) Y (G) αG e ∗ = ey , (3.2.b) αG(y)

19 Y (G) for each y ∈ Y(G). In this case we can therefore derive a formula for ey X(G) from a formula of e ∗ . A glance at Diagram (1.1.a) shows that one can go αG(y) from D(G) to any other representation ring Y (G) by surjective maps so that formulae for the idempotents of K ⊗ D(G) will enable us to derive formulae for the idempotents of all the other representation rings using Equation (3.2.b), since the hypothesis to apply this equation is satisﬁed by (2.1.g). X(G) Also note that Equation (3.2.a) implies that the idempotents ex which ∗ lie in the kernel of αG are precisely those with x∈ / im(αG). This knowledge has important applications, since it provides relations in Y (G) that were used for example in [Bo97a] and [We87].

3.3 Through our approach in Section 2 we also obtain general formulae for the behaviour of idempotents under restriction maps. If f : G → G˜ is a morphism in gr then Equation (2.1.b) implies ( X(G) X(G˜) X(˜x) X(G˜) 1, if f(x) =G˜ x˜, sx resf (ex˜ ) = sf(x) (ex˜ ) = 0, if f(x) 6=G˜ x˜. for x ∈ X (G) andx ˜ ∈ X (G˜), where we wrote f(x) instead of (X (f))(x). This implies

X(G˜) X X 1 res (e ) = eX(G) = eX(G) f x˜ x [G : N (x)] x x∈X (G)/G x∈X (G) G f(x)∈[˜x] f(x)∈[˜x] G˜ G˜

X X 1 X(G) = ex [G : NG(x)] g˜∈G/N˜ (˜x) x∈X (G) G˜ g˜ f(x)= x˜ so that for eachx ˜ ∈ X (G˜) we have

g˜ X(G˜) X X [f(G): Nf(G)( x˜)] X(G) resf ex˜ = ex , [G : NG(x)] g˜∈f(G)\G/N˜ (˜x) x∈X (G) G˜ g˜ f(x)= x˜ by considering the decomposition of the G˜-orbit ofx ˜ into f(G)-orbits. In par- X(G˜) ticular, if [˜x]G˜ ∩ im(f) = ∅, then resf (ex˜ ) = 0.

X(G) ab p p,ab 3.4 First we recall and derive formulae for ex for X ∈ {R,R ,R ,R } and x ∈ X (G). It is well-known and follows from the orthogonality relation that 1 X eR(G) = χ(g−1)χ ∈ K ⊗ R(G) , g |G| χ∈Irr(G) where Irr(G) denotes the set of irreducible characters of G, and that conse- quently Rab(G) 1 X −1 ab e 0 = ϕ(g )ϕ ∈ K ⊗ R (G) , gG |Gab| ϕ∈Gb

20 for g ∈ G, with Gb = Hom(G, K×). Using Equation (3.2.b) for the natural inclusions ρ∗ : Rp → R and ρ∗ : Rp,ab → Rab we immediately obtain

p 1 X eR (G) = δ(eR(G)) = χ(g−1)χ| g g |G| Gp0 χ∈Irr(G) 1 X X = χ(g−1) d · ϑ |G| ϑχ χ∈Irr(G) ϑ∈IBr(G) 1 X X 1 X = d χ(g−1) ϑ = η (g−1)ϑ , |G| ϑχ |G| ϑ ϑ∈IBr(G) χ∈Irr(G) ϑ∈IBr for g ∈ Gp0 . Here, IBr(G) denotes the set of irreducible Brauer characters of G, dϑχ the decomposition number of χ and ϑ, and ηϑ the character of the projective cover of the simple FG-module with Brauer character ϑ. Using the Cartan invariants cβϑ we can also write

p 1 X X eR (G) = c β(g−1) ϑ . g |G| βϑ ϑ∈IBr(G) β∈IBr(G)

0 ab Similarly, for g ∈ G with gG ∈ (G )p0 we obtain

Rp,ab(G) 1 X −1 1 X −1 e 0 = ρ ϕ(g )ϕ = ϕ(g )ϕ 0 gG G |Gab| |Gab| p ϕ∈Gb ϕ∈Gb 1 X −1 = ab ϕ(g )ϕ . |G |p0 ϕ∈Gbp0 × × 0 We identify Gb also with Hom(G, OK ) and Hom(G, O ) and the p -part Gbp0 × with Hom(G, F ) via the inclusions OK ⊂ O ⊂ K and the canonical surjection O → F .

D(G) 0 3.5 Next we derive a formula for e(H,hH0) ∈ K ⊗ D(G) for each (H, hH ) ∈ Rab(G) D(G) using the formula for egG0 . We use [Bo98b, Proposition 2.4] which gives an explicit inverse of the isomorphism

G Y ab G πH ◦ resH : K ⊗ D(G) → K ⊗ R (H) , H6G H6G where G acts on the above product by natural conjugation. Combining this ab Q isomorphism with the isomorphism K ⊗ R (H) → hH0∈Hab K in the H- component gives us the collection of species of D(G). More precisely, assume that for each U 6 G we have an element aU ∈ K ⊗ ab g R (U) such that this family is ﬁxed under G in the sense that (aU ) = agUg−1 g −1 for every U 6 G and g ∈ G, where (−) is deﬁned as resf for f : gUg → U, v 7→ g−1vg. Then [Bo98b, Proposition 2.4] asserts that 1 G X n Hn πU ◦ res (−1) |H0| H0, res (aH ) = aU (3.5.a) U |G| H0 n G H0<···

21 P for each U 6 G. In the above equation, [U, ϕ∈Uˆ αϕϕ]G is defined as P ϕ∈Uˆ αϕ[U, ϕ]G ∈ K ⊗ D(G) for any coefficients αϕ ∈ K. Now we fix 0 (H, hH ) ∈ D(G) and define a family aU , U 6 G, of G-fixed elements ab aU ∈ K ⊗ R (U) as follows. If U is not conjugate to H we set aU := 0. For U = H we set r X Rab(H) aH := ehH0 , 0 r∈NG(H)/NG(H,hH )

ab g R (H) and for arbitrary g ∈ G we set agHg−1 := aH . Then we obtain shH0 (aH ) = 1 ab and sR (H)(a ) = 0 unless hH˜ 0 and hH0 are conjugate under N (H). If we hH˜ 0 H G G denote the argument of πU ◦ resH in Equation (3.5.a) by σG (aU )U6G , then 1 X X X σ (a ) = G U U6G |G| 0 −1 g∈G/NG(H) r∈NG(H)/NG(H,hH ) H0<···

ab H R (H) By the last statement in Subsection 3.3, we have res (e 0 ) = 0 unless H0 hH 0 H0 ∩ hH 6= ∅. Therefore, it suﬃces to sum over chains starting in subgroups ab 0 0 0 R (H) H0 with hH ∈ H0H /H . Together with the explicit formula for ehH0 from Subsection 3.4 we obtain

D(G) 1 X n |H0| X −1 e(H,hH0) = 0 (−1) ab ϕ(h )[H0, ϕ|H0 ]G . |NG(H, hH )| |H | H0<···

1 X n |H0| X −1 = 0 (−1) ab ϕ(h )[H0, ϕ|H0 ]G , |NG(H, hH )| |H |p0 H0<···

22 −1 −1 since ρG([H, ϕ]G) = [H, ϕp0 ]G and ϕ(h ) = ϕp0 (h ) for any ϕ ∈ Hb and 0 ab hH ∈ (H )p0 . From there we can proceed using Equation (3.2.b) applied to π : Dp → B and obtain, for any H 6 G,

B(G) Dp(G) eH = πG(e(H,1H0))

1 X n |H0| X = (−1) ab ϕ(1) [G/H0] |NG(H)| |H |p0 H0<···

ab since |H |p0 = |Hbp0 |. This is exactly the formula obtained by Gluck in [Gl81].

3.6 Next we derive formulae for the idempotents of K ⊗Lp(G) and K ⊗T p(G). p 0 p Equation (3.2.b) applied to β : D → L yields, for (H, hOp(H) ) ∈ L (G),

Lp(G) D(G) e 0 = βG(e 0 ) (H,hOp(H) ) (H,hH ) 1 X |H0| X = (−1)n ϕ(h−1)IndG (O ) , 0 ab H0 m ϕ|H0 |NG(H, hH )| |H | H0<···

T p(G) Dp(G) e = βG(e 0 ) (H,hOp(H)) (H,hp0 H ) 1 X |H0| X = (−1)n ϕ(h−1)IndG F . 0 ab H0 ϕ|H0 |NG(H, hH )| |H |p0 H0<···

0 0 ab 0 Note that, since hp0 H is a p -element in H , the condition H0 ∩ hp0 H 6= ∅ is now equivalent to the condition that |H|p0 divides |H0| and we sum exactly over those chains H0 < ··· < Hn = H with the property that H0 contains a Hall p0-subgroup of H. It should be noted that in contrast to the idempotent formulae for the other representation rings we cannot derive a formula in terms of the canonical basis given by the classes of indecomposable modules.

3.7 Finally, we will derive a second formula for the idempotents of K ⊗ R(G) and K ⊗ Rp(G) from the idempotents of K ⊗ Lp(G) and K ⊗ T p(G).

23 Applying Equation (3.2.b) to γ : Lp → R we obtain for g ∈ G the formula

R(G) ∗ Lp(G) eg = γG(e(hgi,g))

1 X −1 G = ϕ(g )indH (ϕ) , |CG(g)| ϕ∈Hb where H := hgi, since the condition H0 ∩ {g}= 6 ∅ implies that H0 = Hn = H. If ϕ denotes a generator of Hb and ϕ(g−1) = ζ, then we obtain

|H|−1 1 X eR(G) = ζi · indG (ϕi) . g |C (g)| H G i=0

Similarly we obtain for g ∈ G0 the formula

Rp(G) ∗ T p(G) eg = γG(e(hgi,g))

|H|1 1 X 1 X = ϕ(g−1)indG (ϕ) = ζi · indG (ϕi) , |C (g)| H |C (g)| H G G i=0 ϕ∈Hb where again H := hgi, ϕ is a generator of Hb, and ζ = ϕ(g−1).

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