Linear Source Modules and Trivial Source Modules^{∗}

Robert Boltje Institut für Mathematik Universität Augsburg 86135 Augsburg Germany email: boltje@math.uni-augsburg.de

November 22, 1996

(revised: February 19, 1997)

Abstract

We consider the Grothendieck ring L_{O}(G) of the category of linear source OG-modules with respect to direct sums, where O is a complete discrete valuation ring of characteristic zero containing enough roots of unity and having a residue field of prime characteristic. We show that L_{O}(G) is semisimple after tensoring with the field of fractions of O, and we determine all its species. Moreover, we show that there is a canonical integral induction formula for L_{O}(G) inducing only lattices of rank 1. As a consequence we obtain similar statements for the ring of trivial source OG-modules.

Introduction

The representation theory of trivial source modules for a finite group G over a complete discrete valuation ring O of characteristic zero which is large enough for G and has a residue field F of characteristic l > 0 is well understood, see for instance [3]. The representation ring T_{O}(G) of trivial source OG-modules is a subring of the Green ring A_{O}(G) and is finitely generated as an abelian group. The study of T_{O}(G) has had an enormous impact on the understanding of the ring A_{O}(G). If K denotes the field of fractions of O, then it is well-known that K ⊗ T_{O}(G) is a semisimple K-algebra, and its species, i.e. the K-algebra maps K ⊗ T_{O}(G) → K, are completely determined. But if one wishes to interpret the representation theory of OG as a link between those of F G and KG, there is a slight drawback to the ring T_{O}(G). While the reduction modulo the radical of O induces a surjecitve ring homomorphism T_{O}(G) → R_{F} (G) onto the ring of Brauer characters of G, the canonical map T_{O}(G) → R_{K}(G) to the character ring of G is not surjective.

∗AMS subject classification: 20C11, 19A22

In this paper we introduce the notion of a linear source OG-module as a slight generalization of a trivial source OG-module. The representation ring L_{O}(G) of these modules is still finitely generated as an abelian group, it covers the character ring by tensoring such modules with K, and the K-algebra K⊗L_{O}(G) is semisimple. Moreover, using canonical induction formulae for R_{K}(G) and R_{F} (G) one can give canonical sections for the surjective maps L_{O}(G) → R_{K}(G) and L_{O}(G) → R_{F} (G).

In Section 1 we introduce the category of linear source OG-modules and prove some elementary facts about them. Section 2 is devoted to the semisimplicity of K ⊗L_{O}(G). We also give a complete description of its species. Moreover, it is shown that two linear source OG-modules are isomorphic, if and only if their restrictions to l-hypo-elementary subgroups contain the same direct summands of rank 1 with the same multiplicities. In Section 3 we recall the notion of a canonical induction formula and show in Section 4 that there are canonical induction formulae for T_{O}(G) and L_{O}(G) using trivial source modules and linear source modules of rank 1 respectively, as already announced in [2].

I would like to thank B. Külshammer for his suggestion to study the trivial source ring and the trivial source modules of rank 1 as a potential example for a canonical induction formula.

Notation For a finite group G we denote by G′ its derived subgroup, by G_{π} the set of elements of π-order in G, and by O_{π}(G) the biggest normal π-subgroup of G, where π can be any set of primes. For a field K, the Grothendieck ring of KGmodules is denoted by R_{K}(G). The notation H < G indicates that H is a proper subgroup of G. For H ≤ G and g ∈ G we set ^{g}H := gHg^{−1}. If k is a commutative ring and V a kG-module, we denote by ker(V) the normal subgroup of G consisting of all elements of G which act as the identity on V. If W is another kG-module, we write W | V to indicate that W is isomorphic to a direct summand of V.

1 Linear source modules

Throughout this paper G denotes a finite group and O denotes a complete discrete valuation ring with residue field F of characteristic l > 0 and field of fractions K of characteristic 0 which is large enough for G, i.e. contains a primitive e-th root of unity where e is the exponent of G. Moreover, all other finite groups occuring in this paper are implicitly assumed to have exponent dividing e so that K is always a splitting field for them.

By OG−lat we denote the category of OG-lattices, i.e. OG-modules M which are free as O-modules and of finite O-rank rk_{O}M. Let A_{O}(G) be the free abelian group on the set of isomorphism classes {M} of objects M ∈ OG−lat modulo the subgroup generated by the elements {M ⊕N}−{M}−{N} for M, N ∈ OG−lat. By the Krull-Schmidt-Azumaya Theorem one can identify A_{O}(G) with the free abelian group on the set of isomorphism classes of indecomposable OG-lattices. The element in A_{O}(G) associated to V ∈ OG−lat will be denoted by [V]. The ring structure of A_{O}(G) is induced by the tensor product V ⊗_{O} W with the diagonal G-action for V, W ∈ OG−lat. In the literature, the ring A_{O}(G) is also called the Green ring of OG. The rings A_{O}(H), H ≤ G, form a Green functor on G, i.e. there are conjugation, restriction, and induction maps

c_{g,H}: A_{O}(H) → A_{O}(^{g}H) ,
res_{U}^{H}: A_{O}(H) → A_{O}(U) ,
ind_{U}^{H}: A_{O}(U) → A_{O}(H) ,

for U ≤ H ≤ G and g ∈ G, satisfying certain axioms, cf. [2] for a definition of a Green functor in the form convenient for our purposes. For H ≤ G, g ∈ G, and V ∈ OH−lat we define ^{g}V ∈ O^{g}H−lat to be the O-module V whose H-action H → End_{O}(V) is replaced by the ^{g}H-action ^{g}H → H → End_{O}(V), where ^{g}H → H is the conjugation map x ↦ g^{−1}xg for x ∈ ^{g}H.

The category OG−lat has as a full subcategory the category OG−per of permutation OG-modules, i.e. OG-modules which have a G-stable finite O-basis. Note that the class of permutation modules is stable under conjugation, restriction, induction, and taking direct sums and tensor products. Thus, the span of the images in A_{O}(H) of objects from OH−per, H ≤ G, form a Green subfunctor P_{O} ⊆ A_{O}, in fact the smallest Green subfunctor of A_{O}.

There is a well-known full subcategory OG−triv of OG−lat which contains OG−per and consists of those OG-lattices all of whose indecomposable direct summands have a trivial source. The objects of OG−triv are called trivial source OG-modules. We summarize some of their basic properties in the following proposition, see for instance [3] for a proof of it.

1.1 Proposition

(a) For V ∈ OG−lat and a Sylow l-subgroup P of G the following are equivalent:

(i) One has V ∈ OG−triv.

(ii) The module res_{P}^{G}(V) is a permutation OP -module.

(iii) The module V is isomorphic to a direct summand of a permutation OGmodule.

(b) For H ≤ G, V, V′ ∈ OG−triv, and W ∈ OH−triv one has:

(i) The OG-lattices V ⊕ V′, V ⊗_{O} V′, and V^{∗} = Hom_{O}(V, O) are again trivial source OG-modules.

(ii) The OH-lattice res_{H}^{G}(V) is a trivial source OH-module.

(iii) The OG-lattice ind_{H}^{G}(W) is a trivial source OG-module.

(iv) Every direct summand of V is a trivial source OG-module.

The images in A_{O}(G) of objects from OG−triv span a subring T_{O}(G) of A_{O}(G) which is the free abelian group on the set of isomorphism classes of indecomposable trivial source OG-modules. Note that in contrast to A_{O}(G) the group T_{O}(G) is always finitely generated. In view of property (ii) in Proposition 1.1 (a), the objects in OG−triv are also called l-permutation OG-modules.

We introduce the category OG−lin of linear source OG-modules as the full subcategory of OG−lat consisting of those OG-lattices all of whose indecomposable direct summands have a source of O-rank 1. Hence, one has inclusions

OG−per ⊆ OG−triv ⊆ OG−lin ⊆ OG−lat

of full subcategories.

An OG-module V is called monomial, if V is a direct sum of OG-lattices isomorphic to OG-lattices of the form ind_{H}^{G}(W) for a subgroup H ≤ G and an OHlattice W of O-rank 1. Similar to Proposition 1.1 one has the following for linear source modules.

1.2 Proposition

(a) For V ∈ OG−lat and a Sylow l-subgroup P of G the following are equivalent:

(i) One has V ∈ OG−lin.

(ii) The module res_{P}^{G}(V) is a monomial OP -module.

(iii) The module V is isomorphic to a direct summand of a monomial OGmodule.

(b) For H ≤ G, V, V′ ∈ OG−lin, and W ∈ OH−lin one has:

(i) The OG-lattices V ⊕ V′, V ⊗_{O} V′, and V^{∗} = Hom_{O}(V, O) are again linear source OG-modules.

(ii) The OH-lattice res_{H}^{G}(V) is a linear source OH-module.

(iii) The OG-lattice ind_{H}^{G}(W) is a linear source OG-module.

(iv) Every direct summand of V is a linear source OG-module.

Proof (a) It suffices to show the equivalences for an indecomposable OG-lattice V. Let H ≤ G be a vertex of V and let S ∈ OH−lat be a source of V. Then V | ind_{H}^{G}(S) and this shows that (i) implies (iii). If V is a direct summand of a monomial OG-module, then res_{P}^{G}(V) is a direct summand of a monomial OP - module by Mackey’s theorem. But by Green’s indecomposability theorem, ind_{Q}^{P}(W) is indecomposable for each Q ≤ P and each W ∈ OQ−lat with rk_{O}(W) = 1. This shows that res_{P}^{G}(V) is monomial, and that (iii) implies (ii). If res_{P}^{G}(V) is monomial, then also ind_{P}^{G}(res_{P}^{G}(V)) is monomial, and contains V (up to isomorphism) as a direct summand, since V is relatively P -projective. Hence, V | ind_{Q}^{G}(W) for some Q ≤ P and some W ∈ OQ−lat with rk_{O}(W) = 1. Since S is a source of V, we obtain S | res_{H}^{G}(V) | res_{H}^{G}(ind_{Q}^{G}(W)). Now, Mackey’s theorem shows that S | ind_{Q′}^{H}(W′) for some Q′ ≤ H and some W′ ∈ OQ−lat with rk_{O}(W′) = 1. But since S (as a source) has vertex H, this implies Q′ = H and S = W′, and we have shown that (ii) implies (i).

(b) Part (iv) holds by the very definition of OG−lin, and the parts (i), (ii), and (iii) follow from (a) (iii), since the class of monomial modules is stable under the constructions in (i), (ii), and (iii).

The span L_{O}(G) of the images in A_{O}(G) of objects from OG−lin is the free abelian group on the set of isomorphism classes of indecomposable linear source OG-modules, and one obtains subring inclusions

P_{O}(G) ⊆ T_{O}(G) ⊆ L_{O}(G) ⊆ A_{O}(G)

which induce inclusions P_{O} ⊆ T_{O} ⊆ L_{O} ⊆ A_{O} of Green subfunctors on G. Note that, since for a given subgroup H of G there are only finitely many OH-lattices of O-rank 1 (up to isomorphism), the group L_{O}(G) is still finitely generated. Note also that each OG-lattice W of O-rank 1 is isomorphic to O_{φ} for some φ ∈ Hom(G, O^{×}), where O_{φ} = O as O-module and g · α = φ(g)α defines the G-action for α ∈ O and g ∈ G. If φ = 1 we write just O instead of O_{1}. Obviously this induces a bijection between the multiplicative group Ĝ := Hom(G, O^{×}) and the set of isomorphism classes of OG-lattices of rank 1 with O_{φψ} ≅ O_{φ} ⊗_{O} O_{ψ} for φ, ψ ∈ Ĝ. It is easy to see that, for φ ∈ Ĝ, each Sylow l-subgroup P is a vertex of O_{φ} and O_{φ|_{P}} is a source of O_{φ}. This implies that O_{φ} is always a linear source module, and that O_{φ} is a trivial source module, if and only if φ|_{P} = 1 which is equivalent to φ lying in Ĝ_{l′}, the l′-part of the abelian group Ĝ.

A finite group H is called l-hypo-elementary, if H/O_{l}(H) is cyclic. The lhypo-elementary groups play an important role for the Green functors P_{O}, T_{O}, L_{O}, and A_{O} as will be explained in the next section.

We will have to consider the set M^{L}(G) of pairs (H, φ) with H ≤ G and φ ∈ Ĥ. Note that M^{L}(G) is a partially ordered set (poset) by defining (U, ψ) ≤ (H, φ), if and only if U ≤ H and φ|_{U} = ψ for (U, ψ), (H, φ) ∈ M^{L}(G), and that G acts on M^{L}(G) by conjugation via poset automorphisms. For (H, φ) ∈ M^{L}(G) we denote by N_{G}(H, φ) the stabilizer of (H, φ) in G.

1.3 Proposition Let H be a finite l-hypo-elementary group with Sylow lsubgroup P.

(a) An indecomposable trivial source OH-module has vertex P, if and only if it is isomorphic to O_{φ} for some φ ∈ Ĥ with φ|_{P} = 1.

(b) If V ∈ OH−lin is indecomposable with vertex P and source O_{ψ}, ψ ∈ P̂, then V ≅ ind_{U}^{H}(O_{φ}), where U = N_{H}(P, ψ) and φ ∈ Û is an extension of ψ. Moreover, if V ≅ ind_{U′}^{H}(O′_{φ}) for some (U′, φ′) ∈ M^{L}(H), then (U, φ) and (U′, φ′) are H-conjugate. Conversely, if ψ ∈ P̂ and U := N_{H}(P, ψ), then ψ has n = [U : P] extensions φ_{1}, …, φ_{n} ∈ Û and V_{i} := ind_{U}^{H}(O_{φ_{i}}) is indecomposable with vertex P and source O_{ψ} for each i = 1, …, n. Furthermore, V_{1}, …, V_{n} are pairwise non-isomorphic.

Proof (a) This follows from the decomposition

ind_{P}^{H}(O) ≅ ⨁_{φ∈Ĥ_{l′}} O_{φ} .

(b) Let ψ ∈ P̂ and U := N_{G}(P, ψ). Then ψ can be extended to U, since U/P centralizes P/ ker(ψ) so that U/ ker(ψ) is abelian. Moreover, there are exactly n = [U : P] such extensions φ_{1}, …, φ_{n} ∈ Û and

ind_{P}^{U}(O_{ψ}) ≅ ⨁_{i=1}^{n} O_{φ_{i}} .  (1.3.a)

The module V_{i} = ind_{U}^{H}(O_{φ_{i}}), i = 1, …, n, is indecomposable, since ind_{U}^{H}(K_{φ_{i}}) is an irreducible KH-module by Clifford theory. By Mackey’s theorem one has

res_{P}^{H}(V_{i}) ≅ ⨁_{h∈H/U}^{n} ^{h}O_{ψ} .  (1.3.b)

If Q ≤ P is a vertex and S ∈ OQ−lat is a source of V_{i}, then V_{i} | ind_{Q}^{H}(S) and therefore O_{ψ} | res_{P}^{H}(ind_{Q}^{H}(S)). Now Mackey’s theorem and Green’s indecomposability theorem imply O_{ψ} = ind_{P ∩^{h}Q}^{P}(S′) for some h ∈ H and some O(P ∩ ^{h}Q)-module S′, hence P = Q and O_{ψ} is a source of V_{i} by (1.3.b). Moreover, V_{1}, …, V_{n} are pairwise non-isomorphic, since K ⊗_{O} V_{i}, i = 1, …, n, are pairwise non-isomorphic by classical Clifford theory.

Now if V ∈ OH−lin is indecomposable with vertex P and source O_{ψ}, then

V | ind_{P}^{H}(O_{ψ}) ≅ ⨁_{i=1}^{n} ind(O_{φ_{i}}) = ⨁_{i=1}^{n} V_{i}

by (1.3.a), so that V ≅ ind_{U}^{H}(O_{φ_{i}}) for some i = 1, …, n. If additionally V ≅ ind_{U′}^{H}(O_{φ′}) for any pair (U′, φ′) ∈ M^{L}(H), then by considering O-ranks we get U = U′, and by restricting V to U we obtain φ′ = ^{h}φ_{i} for some h ∈ H.

By A′_{O}(G) (resp. A′′_{O}(G)) we define the span of the isomorphism classes [V] of indecomposable OG-lattices V whose vertices are the Sylow l-subgroups (resp. are not the Sylow l-subgroups) of G and by A_{O}^{ab}(G) we denote the span of the elements [O_{φ}], φ ∈ Ĝ. Then,

A_{O}(G) = A′_{O}(G) ⊕ A′′_{O}(G)

and A′_{O}(G) splits further into A_{O}^{ab}(G) and the span of the isomorphism classes of indecomposable OG-lattices whose vertices are the Sylow l-subgroups of G, but which are not of O-rank 1. According to those decompositions we obtain projections

p_{G}^{A} : A_{O}(G) ⟶^{q_{G}^{A}} A′_{O}(G) ⟶^{t_{G}^{A}} A_{O}^{ab}(G) .

Similarly we define decompositions

L_{O}(G) = L′_{O}(G) ⊕ L′′_{O}(G) and T_{O}(G) = T′_{O}(G) ⊕ T′′_{O}(G)

by intersecting A′_{O}(G) and A′′_{O}(G) with L_{O}(G) and T_{O}(G). Moreover, we define

L_{O}^{ab} := A_{O}^{ab} ⊆ L′_{O}(G) and T_{O}^{ab} := A_{O}^{ab}(G) ∩ T′_{O}(G) ⊆ T′_{O}(G) .

Then L_{O}^{ab}(G) is generated by [O_{φ}], φ ∈ Ĝ, and T_{O}^{ab}(G) by [O_{φ}], φ ∈ Ĝ_{l′}. By restriction we obtain projection maps

p_{G}^{A}: A_{O}(G) ⟶^{q_{G}^{A}} A′_{O}(G) ⟶^{t_{G}^{A}} A_{O}^{ab}(G)
⋃  ⋃  ⋃
p_{G}^{L}: L_{O}(G) ⟶^{q_{G}^{L}} L′_{O}(G) ⟶^{t_{G}^{L}} L_{O}^{ab}(G)
⋃  ⋃  ⋃
p_{G}^{T}: T_{O}(G) ⟶^{q_{G}^{T}} T′_{O}(G) ⟶^{t_{G}^{T}} T_{O}^{ab}(G) .

Note that in general A′′_{O}(G), L′′_{O}(G), and T′′_{O}(G) are ideals in A_{O}(G), L_{O}(G), and T_{O}(G) respectively, and that A_{O}^{ab}, L_{O}^{ab}, and T_{O}^{ab} are subrings of A_{O}(G).

1.4 Corollary Let H be l-hypo-elementary.

(a) One has T′_{O}(H) = T_{O}^{ab}(H), the subgroups L′_{O}(H) and T′_{O}(H) are subrings of A_{O}(H), and the projection maps q_{H}^{L} : L_{O}(H) → L′_{O}(H) and q_{H}^{T} : T_{O}(H) → T′_{O}(H) are ring homomorphisms.

(b) One has isomorphisms L′_{O}(H) ≅ L′_{O}(H/O_{l}(H)′) ≅ R_{K}(H/O_{l}(H)′) induced by inflation and by tensoring with K over O.

(c) For each U ≤ H with O_{l}(H) ≤ U one has q_{U}^{L} ∘ res_{U}^{H} = res_{U}^{H} ∘ q_{H}^{L}.

Proof (a) The equation T′_{O}(H) = T_{O}^{ab}(H) follows from Proposition 1.3 (a). Then with T_{O}^{ab}(H), also T′_{O}(H) is a ring. From the explicit description of indecomposable linear source OH-modules with vertex O_{l}(H) in Proposition 1.3 (b) one can easily see that the tensor product of two such modules is isomorphic to a direct sum of modules of the form ind_{U}^{H}(O_{φ}) with O_{l}(H) ≤ U ≤ H and φ ∈ Û. Moreover, each indecomposable direct summand V of ind_{U}^{H}(O_{φ}) has vertex O_{l}(H). In fact, if Q is a vertex and S ∈ OQ−lat is a source of V, then V | ind_{Q}^{H}(S) and V | ind_{U}^{H}(O_{φ}), and restriction to P = O_{l}(H) yields ^{h}O_{φ|_{P}} ≅ ind_{P ∩^{k}Q}^{P}(S′) for some h, k ∈ H and

S′ ∈ O(P ∩ ^{k}Q)−lat, which implies P = Q.

Since L′_{O}(H) and T′_{O}(H) are subrings, the projection onto them is a ring homomorphism.

(b) Obviously O_{l}(H)′ acts trivially on every indecomposable V ∈ OH−lin with vertex O_{l}(H). Therefore, the inflation map L′_{O}(H/O_{l}(H)′) → L′_{O}(H) is an isomorphism. Moreover, by the explicit description of the indecomposable linear source H/O_{l}(H)′-modules with vertex O_{l}(H)/O_{l}(H)′ one can easily check that tensoring with K over O sets them into bijective correspondence to the irreducible characters of H/O_{l}(H)′.

(c) Let O_{l}(H) ≤ U ≤ H and let V ∈ OH−lin be indecomposable. If V has vertex O_{l}(H), so have all indecomposable direct summands of res_{O_{l}(H)}^{H}(V) by the explicit description of V. If V has vertex strictly smaller than O_{l}(H), so have all indecomposable direct summands by general considerations.

If G is an l-elementary group, i.e. if G ≅ P × C for an l-group P and a cyclic group C, then the following proposition extends Proposition 1.3 to a complete overview of all indecomposable linear source OG-modules.

1.5 Proposition Let G = P × C be l-elementary with Sylow l-subgroup P. Then for each indecomposable module V ∈ OG−lin there is a pair (H, φ) ∈ M^{L}(G), unique up to G-conjugacy, such that V = ind_{H}^{G}(O_{φ}). Moreover, H contains C and rk_{O}(V) is an l-power. Conversely, if (H, φ) ∈ M^{L}(G) with C ≤ H, then V := ind_{H}^{G}(O_{φ}) is an indecomposable linear source OG-module with vertex Q := H ∩ P and source O_{φ|_{Q}}.

Proof First let V ∈ OG−lin be indecomposable with vertex Q ≤ P and source O_{ψ}, ψ ∈ Q̂. Then, with H := QC, ind_{Q}^{H}(O_{ψ}) is the direct sum of all O_{φ} with φ ∈ Ĥ extending ψ, and we have

V | ind_{Q}^{G}(O_{ψ}) ≅ ⨁_{φ∈Ĥ, φ|_{Q}=ψ} ind_{H}^{G}(O_{φ}) .

Since ind_{H}^{G}(O_{φ}) is indecomposable for all φ ∈ Ĥ by Green’s indecomposability theorem, this implies V = ind_{H}^{G}(O_{φ}) for some φ ∈ Ĥ with φ|_{Q} = ψ. If, at the same time, V = ind_{H′}^{G}(O_{φ′}) for some (H′, φ′) ∈ M^{L}(G), then [G : H′] = rk_{O}(V) = [G : H] is an l-power, so H′ contains C. We have

O_{φ} | res_{H}^{G}(ind_{H}^{G}(O_{φ})) ≅ res_{H}^{G}(ind_{H′}^{G}(O_{φ})) ≅ ⨁_{g∈H\G/H′} ind_{H∩^{g}H′}^{H}(O_{(^{g}φ′)|_{H∩^{g}H′}}) ,

and since C ≤ H ∩ ^{g}H′ for all g ∈ G, Green’s indecomposability theorem yields that O_{φ} is isomorphic to one of the last summands above. This shows that H ∩ ^{g}H′ = H and ^{g}φ′|_{H∩^{g}H′} = φ for some g ∈ G, which implies (H, φ) = ^{g}(H′, φ′).

Finally, if (H, φ) ∈ M^{L}(G) with C ≤ H, then V := ind_{H}^{G}(O_{φ}) is indecomposable by Green’s indecomposability theorem, and the statement about the vertex and source of V follows from the discussion in the first part of the proof and the unicity of (H, φ).

2 Detection theorems and species

2.1 For any Green functor A on G, a subgroup H of G is called coprimordial for A, if

⋂_{U<H} ker (res_{U}^{H}: A(H) → A(U)) ≠ {0} .

We denote the set of coprimordial subgroups for A by C(A). If A(H) is a torsion free abelian group for all H ≤ G, then obviously C(A) = C(K′ ⊗ A) for any field K′ of characteristic zero. Moreover, if A ⊆ A′ is an inclusion of Green functors on G and A′(H) is also torsion free for all H ≤ G, then C(A) = C(K′⊗A) = C(K′⊗A′) = C(A′) for all fields K′ of characteristic zero (see for instance [2, Proposition 6.2]).

It is well-known that the coprimordial subgroups for ℚ ⊗ T_{O} are just the l-hypo-elementary subgroups of G. In fact, if H ≤ G is coprimordial for ℚ ⊗ T_{O}, then H is l-hypo-elementary by Conlon’s induction theorem, cf. [4, Cor. 80.61]. Conversely, if H ≤ G is l-hypo-elementary, then one can produce an element x ∈ ℚ ⊗ T_{O}(H) with res_{U}^{H}(x) = 0 for all U < H, by taking the image x of the primitive idempotent ε_{H} ∈ ℚ⊗Ω(H), whose marks are zero for proper subgroups of H and 1 for H, under the canonical map ℚ ⊗ Ω(H) → ℚ ⊗ T_{O}(H) (where Ω(H) denotes the Burnside ring of H), and by applying [4, Lemma 81.28].

If H = H_{G} denotes the set of l-hypo-elementary subgroups of G, then the above discussion shows that

C(A) = C(ℚ ⊗ A) = C(K ⊗ A) = H

for any Green subfunctor A of A_{O}; in particular for A = T_{O}, L_{O}, and A_{O}.

2.2 Let A be a Green subfunctor of A_{O}. A K-species of A(G) is a ring homomorphism s: A(G) → K. Note that if t: A(H) → K is a K-species of A(H) for some H ≤ G, then t ∘ res_{H}^{G} is a K-species of A(G). For any element g ∈ G we have a K-species s_{g}: A_{O}(G) → K by defining s_{g}([V]) for V ∈ OG−lat as the value of the character of the KG-module K ⊗_{O} V at g.

For each H ∈ H and h ∈ H we may define a species s_{(H,h)} on L_{O}(G) by

s_{(H,h)} : = s_{h} ∘ q_{H}^{L} ∘ res_{H}^{G}: L_{O}(G) → L_{O}(H) → L′_{O}(H) → K ,

since q_{H}^{L} is a ring homomorphism by Corollary 1.4 (a). Restricting this species to T_{O}(G) we obtain a K-species

s_{(H,h)} : = s_{h} ∘ q_{H}^{T} ∘ res_{H}^{G}: T_{O}(G) → T_{O}(H) → T′_{O}(H) → K

on T_{O}(G). By Corollary 1.4 (c), we have s_{(H,h)} = s_{(U,h)} for U = ⟨O_{l}(H), h⟩. Therefore, we can restrict our attention to pairs (H, h) with H ∈ H and h ∈ H such that ⟨O_{l}(H), h⟩ = H. In this case, one can see from Proposition 1.3 that

s_{(H,h)} = s_{h} ∘ p_{H}^{L} ∘ res_{H}^{G} ,  (2.2.a)

since the character value of ind_{U}^{H}(φ) at h is zero for all O_{l}(H) ≤ U < H and all φ ∈ Û.

Part (b) of the following proposition is well-known (see for instance [1, Section 2.13]), but it can be proved along with part (a) without additional effort.

2.3 Proposition Let H, H̃ ∈ H, h ∈ H, and h̃ ∈ H̃ with ⟨O_{l}(H), h⟩ = H and ⟨O_{l}(H̃), h̃⟩ = H̃.

(a) One has s_{(H,h)} = s_{(H̃,h̃)} : L_{O}(G) → K, if and only if there is some g ∈ G such

that ^{g}H̃ = H and ^{g}h̃O_{l}(H)′ = hO_{l}(H)′.

(b) One has s_{(H,h)} = s_{(H̃,h̃)} : T_{O}(G) → K, if and only if there is some g ∈ G such

that ^{g}H̃ = H and ^{g}h̃O_{l}(H) = hO_{l}(H).

Proof It is obvious that the conditions in (a) and (b) are sufficient for s_{(H,h)} = s_{(H̃,h̃)}, since conjugation doesn’t change the species, and since O_{l}(H)′ (resp. O_{l}(H)) acts trivially on any indecomposable linear source (resp. trivial source) OH-module H with vertex O_{l}(H).

First we calculate s_{(U,u)}([ind_{I}^{G}(O_{φ})]) for U ∈ H, u ∈ U with ⟨O_{l}(U), u⟩ = U , I ≤ G, and φ ∈ Î:

s_{(U,u)}([ind_{I}^{G}(O_{φ})]) = (s_{u} ∘ q_{U}^{L}) ∘ (res_{U}^{G}([ind_{I}^{G}(O_{φ})]))

= ∑_{g∈U\G/I}(s_{u} ∘ q_{U}^{L})([ind_{U∩^{g}I}^{U}(O_{(^{g}φ)|_{U∩^{g}I}})])

= ∑_{g∈U\G/I, O_{l}(U)≤U∩^{g}I}(s_{u}([ind_{U∩^{g}I}^{U}(O_{(^{g}φ)|_{U∩^{g}I}} )])

= ∑_{g∈U\G/I, U∩^{g}I=U}(^{g}φ)(u)

= ∑_{g∈U\G/I, U≤^{g}I}(^{g}φ)(u).

Let us assume that s_{(H,h)} = s_{(H̃,h̃)} holds on T_{O}(G). Then evaluation at [ind_{H}^{G}(O)] yields [N_{G}(H): H] = |{g ∈ H̃\G/H | H̃ ≤ ^{g}H}|. Hence, there exists g′ ∈ G with H̃ ≤ ^{g′}H. Similarly, evaluation at [indG ; H̃(O)] yields the existence of some g ∈ G with H ≤ ^{g}H̃. This implies H = ^{g}H̃, and we may assume (after conjugation with g) that H = H̃. Then we have to show that some conjugate of h̃ under N_{G}(H) and h lie in the same coset of H/O_{l}(H)′ for part (a), and in the same coset of H/O_{l}(H) for part (b).

To prove part (a) we apply s_{(H,h)} = s_{(H,h̃)} to [ind_{I}^{G}(O_{φ})] for all O_{l}(H) ≤ I ≤ H and all φ ∈ Î. Then we obtain by the above calculation

s_{(H,h)}([ind_{I}^{G}(O_{φ})]) = ∑_{g∈H\G/I, H≤^{g}I}(^{g}φ)(h)

= \begin{cases}0, & if I < H, \\ ∑_{N_{G}(H)/H}(^{g}φ)(h), & if I = H, \end{cases}

= ∑_{g∈N_{G}(H)/H} ^{g}(ind_{I}^{H}(φ))(h),

and similarly for s_{(H,h̃)}. This implies that

∑_{g∈N_{G}(H)/H}(^{g}χ)(h) = ∑_{g∈N_{G}(H)/H}(^{g}χ)(h̃)

for all irreducible characters χ of H with ker(χ) ≥ O_{l}(H)′, since each such χ is of the form ind_{I}^{H}(φ) with I and φ as above, cf. Proposition 1.3 (b) and Corollary 1.4 (b). This implies that (after dividing out the trivial O_{l}(H)′-action), for all N_{G}(H)/Hstable characters χ of H/O_{l}(H)′, we have χ(hO_{l}(H)′) = χ(h̃O_{l}(H)′). Now the result follows from the next lemma.

To prove part (b) we apply s_{(H,h)} = s_{(H,h̃)} to all [ind_{H}^{G}(O_{φ})] with φ ∈ Ĥ_{l′}. Then we obtain

∑_{g∈N_{G}(H)/H}(^{g}φ)(h) = ∑_{g∈N_{G}(H)/H}(^{g}φ)(h̃)

for all φ ∈ Ĥ_{l′}. This implies that (after dividing out the trivial O_{l}(H)-action), for all N_{G}(H)/H-stable characters χ of H/O_{l}(H), we have χ(hO_{l}(H)) = χ(h̃O_{l}(H)). And again the result follows from the next lemma.

2.4 Lemma Let G and S be finite groups and let S act on G by group automorphisms. If g and g̃ are elements of G such that χ(g) = χ(g̃) for all S-stable characters of G, then there is some s ∈ S such that g and ^{s}g̃ are conjugate in G.

Proof We consider the character table of G as an m × m-matrix M, the rows indexed by the irreducible characters of G, the columns by the conjugacy classes of elements of G. The subgroup R_{K}(G)^{S} of S-fixed points of R_{K}(G) has as a basis the sums of S-conjugate irreducible characters of G. Let χ_{1}, …, χ_{n} denote this basis. If we consider the m×n-matrix M′ which arises from M by adding up the rows indexed by S-conjugate irreducible characters, then the rows are still linearly independent, and M′ has rank n. Obviously χ_{i}(g) = χ_{i}(^{s}g) for all i = 1, …, n, s ∈ S, and g ∈ G. Hence, the columns of M′ belonging to S-conjugate conjugacy classes of G coincide, and we may omit these repetitions in M′ to obtain a matrix M′′ which still has rank n. By [5, Satz V 13.5 (b)], there are as many S-orbits of irreducible characters of G as there are S-orbits of conjugacy classes of G. Therefore M′′ is an n × n-matrix with rank n, and the conjugacy classes of g and g̃ have to lie in the same S-orbit.

2.5 Let H^{L} = H_{G}^{L} be the set of all pairs (H, c) with H ∈ H and c ∈ H/O_{l}(H)′ such that ⟨c⟩ = H/O_{l}(H). Each element (H, c) of H^{L} then determines a K-species s_{(H,c)}^{L}: L_{O}(G) → K by setting s_{(H,c)}^{L} = s_{(H,h)} for any h ∈ H with hH′ = c. By Proposition 2.3 this species is well-defined. Similarly, let H^{T} = H_{G}^{T} be the set of all pairs (H, c) with H ∈ H and c ∈ H/O_{l}(H) such that ⟨c⟩ = H/O_{l}(H). Again each (H, c) ∈ H^{T} determines a well-defined K-species s_{(H,c)}^{T}: T_{O}(G) → K by setting s_{(H,c)}^{T} = s_{(H,h)} for any h ∈ H with hO_{l}(H) = c. The sets H^{L} and H^{T} are G-sets via the obvious conjugation action and, by Proposition 2.3, we have for (H, c), (H̃, c̃) ∈ H^{L} (resp. (H, c), (H̃, c̃) ∈ H^{T}) the equation s_{(H,c)}^{L} = s_{(H̃,c̃)}^{L} (resp. s_{(H,c)}^{T} = s_{(H̃,c̃)}^{T}), if and only if (H, c) and (H̃, c̃) are G-conjugate. Note that

mapping (H, hO_{l}(H)′) to (H, hO_{l}(H)) defines a surjective morphism H^{L} → H^{T} of G-sets and that s_{(H,hO_{l}(H)′)}|_{T_{O}(G)} = s_{(H,hO_{l}(H))}. Therefore, we obtain a commutative diagram of K-algebra homomorphisms.

\begin{matrix}K ⊗ T_{O}(G) & ⊆ & K ⊗ L_{O}(G) \\ s^{T} = (s_{(H,c)})↓ & & ↓s^{L} = (s_{(H,c)}) \\ ( ∏_{(H,c)∈H^{T}} K)^{G} & ↪ & ( ∏_{(H,c)∈H^{L}} K)^{G} , \end{matrix}  (2.5.a)

where we consider the products of copies of K as G-sets via the G-action on the index sets H^{T} and H^{L}, and where the lower embedding is induced by the canonical map H^{L} → H^{T}. Of course, the G-fixed points of the products in Diagram (2.5.a) are again isomorphic to the products of copies of K, indexed by representatives for the G-orbits of H^{T} and H^{L} respectively. Since the species attached to such representatives are pairwise destinct by Proposition 2.3, they are K-linearly independent by the lemma of Dedekind, and therefore, their collections form surjective maps into the products of copies of K. This implies that the maps s^{T} and s^{L} in Diagram (2.5.a) are surjective.

The part concerning s^{T} in the following proposition is well-known.

2.6 Proposition The maps s^{T} and s^{L} in Diagram (2.5.a) are isomorphisms. In particular, the K-algebras K ⊗ T_{O}(G) and K ⊗ L_{O}(G) are semisimple and their species are given by s_{(H,c)}^{T}, (H, c) ∈ H^{T} /G (resp. by s_{(H,c)}^{L}, (H, c) ∈ H^{L}/G).

Proof We only have to show the injectivity of s^{L}. The injectivity of s^{T} then follows from the commutativity of Diagram (2.5.a).

It suffices to show that for all H ∈ H and all x ∈ K ⊗ L′_{O}(H) with

(s_{u} ∘ res_{U}^{H})(x) = 0  (2.6.a)
for all O_{l}(H) ≤ U ≤ H and all u ∈ U with ⟨O_{l}(U), u⟩ = U,

one has x = 0. In fact, if y ∈ K ⊗L_{O}(G) is in the kernel of s^{L}, we have to show that y = 0, which is equivalent to res_{H}^{G}(y) = 0 for all H ∈ H. But, since y ∈ ker(s^{L}), the element x: = q_{H}^{L}(res_{H}^{G}(y)), for H ∈ H, satisfies the condition in (2.6.a) by Corollary 1.4 (c):

0 = (s_{u} ∘ q_{U}^{L} ∘ res_{U}^{H} ∘ res_{H}^{G})(y) = (s_{u} ∘ res_{U}^{H} ∘ q_{H}^{L} ∘ res_{H}^{G})(y) = (s_{u} ∘ res_{U}^{H})(x) .

So let H ∈ H and x = ∑ α_{i}[V_{i}] ∈ K ⊗ L′_{O}(G), where V_{1}, …, V_{n} ∈ OH−lin are representatives of the isomorphism classes of indecomposable linear source OH-modules with vertex O_{l}(H), and α_{1}, … α_{n} ∈ K, such that (2.6.a) holds. Let χ_{1}, …, χ_{n} be the characters of V_{1}, …, V_{n}. Then, by Corollary 1.4 (b), the characters χ_{1}, …, χ_{n} are precisely the irreducible characters of H/O_{l}(H)′, and (2.6.a) just states that the function α_{1}χ_{1} +…+α_{n}χ_{n}: H/O_{l}(H)′ → K vanishes on each uO_{l}(H)′, u ∈ H, since for each u ∈ H we may choose U as ⟨O_{l}(H), u⟩. This implies α_{1} = … = α_{n} = 0, and the proof is complete.

2.7 Corollary The collections of maps

r_{G}^{L} = (p_{H}^{L} ∘ res_{H}^{G})_{H∈H}: L_{O}(G) → ∏_{H∈H} L_{O}^{ab}(H)

and

r_{G}^{T} = (p_{H}^{T} ∘ res_{H}^{G})_{H∈H}: T_{O}(G) → ∏_{H∈H} T_{O}^{ab}(H)

are injective.

Proof It suffices to prove the injectivity of r_{G}^{L}, since r_{G}^{T} is the restriction of r_{G}^{L}. By (2.2.a) we have a commutative diagram

L_{O}(G)  ⟶^{s^{L}}  ∏_{(H,c)∈H^{L}} K
r_{G}^{L} ↘  ↗

∏_{H∈H} L_{O}^{ab}(H)

in which the right diagonal map is defined by mapping an element of L_{O}^{ab}(H), H ≤ G, into the components indexed by (H, hO_{l}(H)′) with the different allowed choices for hO_{l}(H)′ via the maps s_{h}: L_{O}^{ab}(H) → K. Now the injectivity of r_{G}^{L} follows from the injectivity of s^{L}.

2.8 Remark It is easy to see that the species of the different Green functors are related by the following commutative diagram

K ⊗ Ω(G) ↠ K ⊗ P_{O}(G) ↪ K ⊗ T_{O}(G) ↪ K ⊗ L_{O}(G) ↠ K ⊗ R_{K}(G)
≀↓(s_{H})  ≀↓(s_{H}^{P})  ≀↓(s_{(H,c)}^{T})  ≀↓(s_{(H,c)}^{L})  ≀↓(s_{g})
( ∏_{H∈S} K)^{G} ↠ ( ∏_{H∈H} K)^{G} ↪ ( ∏_{(H,c)∈H^{T}} K)^{G} ↪ ( ∏_{(H,c)∈H^{L}} K)^{G} ↠ ( ∏_{g∈G} K)^{G}

whose ingredients we have to explain. By S = S_{G} we denote the G-set of subgroups of G. For H ∈ S, the map s_{H}: Ω(G) → ℤ denotes the mark homomorphism with respect to H which maps the class of a finite G-set X to the number |X^{H}| of Hfixed points. The kernel of the surjective map K ⊗ Ω(G) → K ⊗ P_{O}(G) is precisely the K-span of the primitive idempotents ε_{U}, U ∈ S ∖ H, of K ⊗ Ω(G) which are given by s_{H}(ε_{U}) = 1 if H and U are conjugate, and s_{H}(ε_{U}) = 0 otherwise, cf. [4, Lemma 81.28]. Therefore the mark homomorphisms s_{H} with H ∈ H induce morphisms s_{H}^{P}: K ⊗ P_{O}(G) → K whose collection forms an isomorphism onto the G fixed points of ∏_{H∈H} K. For g ∈ G the map s_{g}: R_{K}(G) → K is defined by evaluating a character of G at g. We still have to explain the lower horizontal maps. Each of the K-spaces is of the form (∏_{x∈R} K)^{G} for a G-set R. We can obviously identify this K-space with the K-space Hom_{G}(R, K) of G-equivariant maps to K, where we consider K as a trivial G-set. Then the lower maps are induced by the morphisms of G-sets G → H^{L}, g ↦ (⟨g⟩, g), H^{L} → H^{T}, (H, hO_{l}(H)′) ↦ (H, hO_{l}(H)), H^{T} → H, (H, hO_{l}(H)) ↦ H, and the inclusion H → S. It is not difficult to see that the squares are commutative.

3 Canonical induction formulae

For the reader’s convenience we will recall the basic facts about canonical induction formulae from [2].

A conjugation functor X on G over a commutative ring k consists of a family of k-modules X(H), H ≤ G, together with k-linear maps c_{g,H}: X(H) → X(^{g}H), the conjugation maps, for g ∈ G and H ≤ G. These maps are required to be transitive with respect to composition and the multiplication in G. Moreover, it is required that c_{g,H} = id_{X(H)} if g ∈ H. Then c_{g,H} is an isomorphism with inverse c_{g^{−1},^{g}H}. We frequently write ^{g}x instead of c_{g,H}(x) for g ∈ G, H ≤ G, and x ∈ X(H). A morphism f: X → Y between k-conjugation functors is a family of k-linear maps f_{H}: X(H) → Y (H), H ≤ G, which commute with the conjugation maps.

A k-conjugation functor A on G together with k-linear maps res_{U}^{H}: A(H) → A(U), the restriction maps, for U ≤ H ≤ G, is called a k-restriction functor on G, if the restriction maps are transitive with respect to subgroup towers, commute with conjugation maps, and are the identity for U = H. A morphism f: A → B of k-restriction functors is a morphism of the underlying k-conjugation functors which additionally commutes with the restriction maps.

If M is a k-restriction functor with additional k-linear maps ind_{U}^{H}: M(U) → M(H), the induction maps, for U ≤ H ≤ G, which are transitive with respect to subgroup towers, commute with conjugations, are the identity for U = H, and which satisfy the Mackey double coset formula for induction followed by restriction, then M is called a k-Mackey functor on G. A morphism between k-Mackey functors is a morphism between the underlying k-restriction functors, whose maps commute also with the induction maps.

3.1 The examples we consider in this paper all satisfy Hypothesis 8.1 of [2], namely that we have

• a Mackey functor M on G such that M(H) is a free abelian group for all H ≤ G,

• a restriction subfunctor A ⊆ M, i.e. a family of subgroups A(H) ⊆ M(H), stable under conjugation and restriction,

• and a family B = (B(H))_{H≤G} of ℤ-bases B(H) of A(H) for all H ≤ G which is stable under conjugation maps, and whose positive span (i.e. the linear combinations with non-negative coefficients) is stable under restriction maps.

3.2 Under the assumptions in 3.1, one can form new Mackey functors A^{+} and A_{+} from A by setting

A^{+}(H) := ( ∏_{U≤H} A(U))^{H} and A_{+}(H) := ( ⨁_{U≤H} A(U))_{H} ,

for H ≤ G, where we consider ∏_{U≤H} A(U) and ⨁_{U≤H} A(U) as being endowed with the H-action coming from the conjugation maps c_{h,U}: A(U) → A(^{h}U) for each h ∈ H and U ≤ H, and where the exponent H denotes taking invariants and the index H denotes taking coinvariants under the H-action. If we write [U, a]_{H} ∈ A_{+}(H) for the class of the element a ∈ A(U) in A_{+}(H), we can define the structure maps on A_{+}(H), H ≤ G, by

c_{+g,H}: A_{+}(H) → A_{+}(^{g}H), [U, a]_{H} ↦ [^{g}U, ^{g}a]_{^{g}H} ,
res_{+}_{I}^{H}: A_{+}(H) → A_{+}(I), [U, a]_{H} ↦ ∑_{h∈I\H/U}[I ∩ ^{h}U, res_{I∩^{h}U}^{^{h}U}(^{h}a)]_{I} ,
ind_{+}_{I}^{H}: A_{+}(I) → A_{+}(H), [J, b]_{I} ↦ [J, b]_{H} ,

for I ≤ H ≤ G, U ≤ H, J ≤ I, a ∈ A(U), b ∈ A(J). Since we don’t need the structure maps of the Mackey functor A^{+}, we omit their definition.

If, for H ≤ G, we define M(H) as the set of monomial pairs (U, a) with U ≤ H and a ∈ B(U), then M(H) is an H-set via the conjugation action, and also a poset by defining (J, b) ≤ (U, a), if and only if J ≤ U and b occurs in res_{J}^{U}(a) with respect to the basis B(J). The H-action on M(H) respects this partial order, and the set Δ(M(H)) of strictly increasing chains in M is again an H-set. If (U, a) ∈ M(H)/H runs through a set of representatives for the H-orbits of M(H), then the elements [U, a]_{H} form a ℤ-basis of A_{+}(H).

In order to avoid confusion we should mention that, as free ℤ-modules, the groups A_{+}(H) and A^{+}(H) are isomorphic with obvious bases indexed by M(H)/H. Only the structure maps as Mackey functors are different, if we want to identify these two groups by identifying these bases. Nevertheless, their Mackey functor structures are related by the more complicated map ρ^{A} defined below. Although we could have replaced the direct product in the definition of A^{+}(H) with a direct sum (as for A_{+}(H)), we prefer the product, because it underlines the dual nature of the two constructions, one being a limit and the other a colimit in the categorical sense.

If for each H ≤ G the group A(H) is a ring, and if conjugation maps and restriction maps for A are ring homomorphisms, then A_{+}(H) is again a ring and also an A(H)-module by the definition b · [U, a]_{H} := [U, res_{U}^{H}(b) · a]_{H} for U ≤ H, a ∈ A(U), and b ∈ A(H), cf. [2, 2.2].

There is a morphism ρ^{A}: A_{+} → A^{+} of Mackey functors given by

ρ_{H}^{A} := (π_{U}^{A} ∘ res_{+}_{U}^{H})_{U≤H}: A_{+}(H) → ( ∏_{U≤H} A(U))^{H} ,

where the projection π_{H}^{A}: A_{+}(H) → A(H), for H ≤ G, is defined on a basis element [U, a]_{H} ∈ A_{+}(H), (U, a) ∈ M(H), by

π_{H}^{A}([U, a]_{H}) := \begin{cases}a, & if U = H, \\ 0, & if U < H. \end{cases}

Then ρ^{A} is injective and induces an isomorphism ρ^{A}: ℚ⊗A_{+} → ℚ⊗A^{+} of ℚ-Mackey functors on G, see [2, Prop. 2.4].

3.3 Let M, A, and B, be given as in 3.1. There is a morphism of Mackey functors on G,

b^{M,A}: A_{+} → M ,

defined by b_{H}^{M,A}([U, a]_{H}) = ind_{U}^{H}(a) for all U ≤ H ≤ G and a ∈ A(U). A canonical induction formula for M from A is a section of b^{M,A} in the category of restriction functors, i.e. a family of maps a_{H}: M(H) → A_{+}(H), H ≤ G, which commutes with conjugation and restriction maps and which satisfies b_{H} ∘ a_{H} = id_{M(H)} for all H ≤ G.

There is a bijection between the set of morphisms a: ℚ ⊗ M → ℚ ⊗ A_{+} of ℚ-restriction functors and the set of morphisms

p: ℚ ⊗ M → ℚ ⊗ A ,  (3.3.a)

of ℚ-conjugation functors. If a and p are in correspondence, then they determine each other by the commutative diagrams

\begin{aligned}ℚ ⊗ M(H) & ⟶^{a_{H}} & ℚ ⊗ A_{+}(H) \\ p_{H} ↘ & ↓π_{H}^{A} \\ & ℚ ⊗ A(H) \end{aligned}  (3.3.b)

and

\begin{aligned}ℚ ⊗ M(H) & ⟶^{a_{H}} & ℚ ⊗ A_{+}(H) \\ r_{H} = (p_{U} ∘ res_{U}^{H})_{U≤H} ↘ & ≀↓ρ_{H}^{A} \\ & ℚ ⊗ A^{+}(H) , \end{aligned}  (3.3.c)

for H ≤ G, see [2, Cor. 5.4 (iii)]. Actually, the commutativity of each one of the two previous diagrams for all H ≤ G is equivalent to the commutativity of the other one for all H ≤ G. Using an explicit inverse of ρ_{H}^{A} one obtains from Diagram (3.3.c) the explicit formula

a_{H}(m) = \frac{1}{|H|} ∑_{H_{0}<···<H_{n}}(−1)^{n}|H_{0}|[H_{0}, (res_{H_{0}}^{H_{n}} ∘ p_{H_{n}} ∘ res_{H_{n}}^{H})(m)]_{H}  (3.3.d)

for all H ≤ G and all m ∈ M(H), where the sum runs over all strictly increasing chains of subgroups of H, cf. Equation (6.1.a) in [2].

If p is chosen as in (3.3.a), there arise two interesting questions, namely:

(i) Is a: ℚ ⊗ M → ℚ ⊗ A_{+} a section of b^{M,A}: ℚ ⊗ A_{+} → ℚ ⊗ M, i.e. is a an induction formula for ℚ ⊗ M from ℚ ⊗ A?

(ii) Is a integral, i.e. is a_{H}(M(H)) ⊆ A_{+}(H) for all H ≤ G?

For a positive answer to Question (i) we have necessary and sufficient conditions, and in case of Question (ii) we at least can give conditions which imply a positive answer, see [2, Prop. 6.4 and Thm. 9.3]. We will need the following particular result on the integrality of a later.

3.4 Theorem ([2, Thm. 9.3 and Cor. 9.4]) Let M, A, and B be given as in 3.1, let p: ℚ ⊗ M → ℚ ⊗ A be given as in (3.3.a) with p_{H}(M(H)) ⊆ A(H) for all H ≤ G, and let a: ℚ ⊗ M → ℚ ⊗ A_{+} be the morphism of restriction functors associated to p via Diagram (3.3.c). Assume further that for a set of primes π the following condition holds:

(∗_{π})

For all U ⊴ H ≤ G such that H/U is a cyclic π-group and all ψ ∈ B(U) which are fixed under H, the coefficients of ψ with respect to the basis B(U) in res_{U}^{H}(p_{H}(ϑ)) and p_{U}(res_{U}^{H}(ϑ)) coincide for all ϑ ∈ M(H).

Then

|H|_{π′} · a_{H}(M(H)) ⊆ A_{+}(H)

for all H ≤ G. If furthermore, res_{U}^{H}(B(H)) ⊆ B(U) for all U ≤ H ≤ G, then one has for all H ≤ G and m ∈ M(H) the explicit formula

a_{H}(m) = \frac{1}{|H|} ∑_{σ=((H_{0},φ)<···<(H_{n},φ_{n}))∈Δ(M(H))/H}(−1)^{n} \frac{|(N_{H}(σ)/H_{0})_{π}|}{|N_{H}(σ)/H_{0}|} ×
× m_{φ_{n}}(res_{H_{n}}^{H}(m))[H_{0}, φ_{0}]_{H} ,

where M(H) is the H-poset of monomial pairs as defined in 3.2, the sum runs over a set of representatives σ for the H-orbits of strictly increasing chains in M(H), N_{H}(σ) denotes the stabilizer of σ in H, and where for U ≤ H, φ ∈ B(U), and m′ ∈ M(U) the term m_{φ}(m′) denotes the coefficient of φ in the element p_{U}(m′) ∈ A(U) with respect to the basis B(U).

4 Canonical induction formulae for linear source modules

4.1 Example We will give some examples for a Mackey functor M, a restriction subfunctor A ⊆ M, a ℤ-basis B(H) of A(H) for all H ≤ G, and a morphism p: M → A of ℚ-conjugation functors such that M, A, and B satisfy the hypotheses in 3.1. Of course, p then induces a morphism of ℚ-conjugation functors as in (3.3.a), such that p(M(H)) ⊆ A(H) for all H ≤ G, and we obtain an associated morphism a: ℚ ⊗ M → ℚ ⊗ A_{+} of ℚ-restriction functors. As in 3.2 we also have a resulting set of monomial pairs M(H) for each H, which is an H-poset.

(a) Let M := L_{O}, A := L_{O}^{ab}, and B(H) := {[O_{φ}] | φ ∈ Ĥ} for H ≤ G. We identify B(H) with Ĥ (i.e. we often write φ instead of [O_{φ}]) and obtain as set of monomial pairs M(H) = M^{L}(H) as defined in Section 1. The maps p_{H}^{L}: L_{O}(H) → L_{O}^{ab}(H), H ≤ G, form a morphism of conjugation functors and we will denote the associated morphism of ℚ-restriction functors by a^{L}: ℚ ⊗ L_{O} → ℚ ⊗ L_{O}^{ab}_{+}. The aim of this section is to show that a^{L} is an integral induction formula for L_{O} from L_{O}^{ab}.

(b) Let M := T_{O}, A := T_{O}^{ab}, and B(H) := {[O_{φ}] | φ ∈ Ĥ_{l′}} for H ≤ G. Then the set of monomial pairs M(H), H ≤ G for this choice of B is given as the H-subposet M^{T} (H) of M^{L}(H) consisting of all those pairs (U, φ) with U/ ker(φ) being an l′- group. In section 1 we defined a morphism p^{T} : T_{O} → T_{O}^{ab} of conjugation functors on G. We denote the associated morphism of restriction functors by a^{T} : ℚ ⊗ T_{O} → ℚ ⊗ T_{O}^{ab}_{+}. Note that T_{O}^{ab}_{+} can be considered as a Mackey subfunctor of L_{O}^{ab}_{+} by identifying the ℤ-basis [U, φ]_{H}, (U, φ) ∈ M^{T} (H)/H, in the obvious way with a subset of the basis [U, φ]_{H}, (U, φ) ∈ M^{L}(H)/H, of L_{O}^{ab}_{+}(H) for H ≤ G. Since p^{T} is the restriction of p^{L}, we obtain the commutative diagram

ℚ ⊗ L_{O} ⟶^{a^{L}} ℚ ⊗ L_{O}^{ab}_{+} ⟶^{b^{L_{O} ,L{ab \atop O}}} ℚ ⊗ L_{O}
⋃  ⋃  ⋃
ℚ ⊗ T_{O} ⟶^{a^{T}} ℚ ⊗ T_{O}^{ab}_{+} ⟶^{b^{T_{O} ,T{ab \atop O}}} ℚ ⊗ T_{O} ,

see [2, Prop. 6.11]. Hence, if a^{L} is an integral canonical induction formula for L_{O} from L_{O}^{ab}, also a^{T} is an integral canonical induction formula for T_{O} from T_{O}^{ab}.

(c) In order to show that a^{L} is integral, we will need another set of data. Let M = L_{O}, and for H ≤ G let B^{el}(H) be the set of isomorphism classes of indecomposable linear source OH-modules V with H/ ker(V) being l-elementary. We define L_{O}^{el}(H) ⊆ L_{O}(H) as the span of B^{el}(H). Then L_{O}^{el} is a restriction subfunctor of L_{O} and the projection map p^{el} : L_{O} → L_{O}^{el} which maps the class of an indecomposable module to itself, if it is in B^{el}, and to zero otherwise, is a morphism of conjugation functors. We denote the set of monomial pairs of this example by M^{el}(H), H ≤ G. The associated morphism of ℚ-restriction functors will be denoted by a^{el} : ℚ ⊗ L_{O} → ℚ ⊗ L_{O}^{el}_{+}.

4.2 Before we can state the next theorem we need to introduce the following notions and recall some facts from [2]. For H ≤ G, V ∈ OH−lin, and φ ∈ Ĥ we define

m_{φ}(V) ∈ ℕ_{0}

as the multiplicity of O_{φ} as a direct summand in V. Recall from 3.2 that L_{O}(H) and L_{O}^{ab}_{+}(H) are both L_{O}^{ab}(H)-modules for all H ≤ G. Note also that for any ring automorphism σ: O → O we have a Galois action on L_{O}(H) for all H ≤ G, induced by composing a representation H → GL_{n}(O) with the induced group homomorphism σ: GL_{n}(O) → GL_{n}(O). The subrings L_{O}^{ab}(H), H ≤ G, are stable under this Galois action, and one obtains an induced Galois action on the groups L_{O}^{ab}_{+}(H), H ≤ G. For H ≤ G, a module V ∈ OH−lin, and (U, φ) ∈ M^{L}(H) we define α_{(U,φ)}^{H}([V]) ∈ ℚ as the coefficient of the basis element [U, φ]_{H} in a_{H}^{L}([V]) ∈ L_{O}^{ab}_{+}(H), i.e.

a_{H}^{L}([V]) = ∑_{(U,φ)∈M^{L}(H)/H} α_{(U,φ)}^{H}([V])[U, φ]_{H}.

For any homomorphism of finite groups f: I → G one has a ring homomorphism res_{f}: L_{O}(G) → L_{O}(I) induced by composing a representation G → GL_{n}(O) with the homomorphism f, since a monomial OG-module is still monomial as OI-module. There is also a ring homomorphism

res_{+}_{f}: L_{O}^{ab}_{+}(G) → L_{O}^{ab}_{+}(I) ,

given on a basis element [H, φ]_{G} of L_{O}^{ab}_{+}(G) by

res_{+}_{f}([H, φ]_{G}) = ∑_{x∈f(I)\G/H}[f^{−1}(^{x}H), (^{x}φ) ∘ f]_{I},

cf. [2, 10.2]. Finally we abbreviate b^{L_{O},L_{O}^{ab}} by b^{L}, b^{T_{O},T_{O}^{ab}} by b^{T}, and b^{L_{O},L_{O}^{el}} by b^{el}.

4.3 Theorem Assume the notation of 4.1 and 4.2.

(i) For each H ≤ G one has b_{H}^{L} ∘ a_{H}^{L} = id_{ℚ⊗L_{O}(H)}.

(ii) The morphism a^{L} is integral, i.e. for each H ≤ G one has a_{H}^{L}(L_{O}(H)) ⊆ L_{O}^{ab}_{+}(H).

(iii) For each H ≤ G the map a_{H}^{L} is L_{O}^{ab}(H)-linear.

(iv) The morphism a^{L} commutes with the Galois action.

(v) For each H ≤ G and φ ∈ Ĥ one has a_{H}^{L}([O_{φ}]) = [H, φ]_{H}.

(vi) The morphism a^{L} is given explicitly by

a_{H}^{L}([V]) = ∑_{σ∈((H_{0},φ_{0})<···<(H_{n},φ_{n}))∈Δ(M^{L}(H))/H}(−1)^{n} \frac{|(N_{H}(σ)/H_{0})_{l′}|}{|N_{H}(σ)/H_{0}|} ×
× m_{φ_{n}}(res_{H_{n}}^{H}(V))[H_{0}, φ_{0}]_{H} ,

for H ≤ G and V ∈ OH−lin, where the sum runs over a set of representatives σ for the H-conjugacy classes of chains in the poset M^{L}(H) and N_{H}(σ) denotes the stabilizer of σ.

(vii) For all H ≤ G, V ∈ OH−lin, and (U, φ) ∈ M^{L}(H) one has

m_{φ}(res_{U}^{H}(V)) = 0 ⟹ α_{(U,φ)}^{H}([V]) = 0.

(viii) For all H ≤ G, (U, φ) ∈ M^{L}(H), and all indecomposable modules V ∈ OH−lin one has α_{(U,φ)}^{H}([V]) = 0, unless a Sylow l-subgroup of U is contained in some vertex of V.

(ix) Let f: I → G be a homomorphism of finite groups. Then the diagram

L_{O}(G) ⟶^{a_{G}^{L}} L_{O}^{ab}_{+}(G)
res_{f} ↓  ↓res_{+}_{f}
L_{O}(I) ⟶^{a_{I}^{L}} L_{O}^{ab}_{+}(I)

is commutative.

Proof The most elaborate part is to prove (ii). We postpone the proof of part (ii) to the end and first establish all the other results.

(iii) This follows immediately from [2, Prop. 6.10], since p_{H}: L_{O}(H) → L_{O}^{ab}(H) is L_{O}^{ab}(H)-linear for all H ≤ G.

(iv) This follows from the commutativity of Diagram (3.3.c), since p_{U}, res_{U}^{H},

res_{+}_{U}^{H}, and π_{U}^{L_{O}^{ab}} commute with the Galois action.

(v) This follows immediately from [2, Prop 6.12], since

res_{U}^{H}(p_{H}([O_{φ}])) = [O_{φ|_{U}}] = p_{U}(res_{U}^{H}([O_{φ}]))

for all φ ∈ Ĥ.

(vi) The formula for a_{H}^{L}([V]) follows from Theorem 3.4, since condition (∗_{π}) is satisfied for the set π of primes distinct from l. In fact, let U ⊴ H ≤ G be such that H/U is a cyclic l′-group, and let ψ ∈ Û be stable under H. Let furthermore V ∈ OH−lin be indecomposable. If rk_{O}(V) = 1, then p_{U}(res_{U}^{H}([V])) = res_{U}^{H}([V]) = res_{U}^{H}(p_{H}([V])); in particular, (∗_{l′}) holds for V in this case. If rk_{O}(V) > 1, then p_{H}([V]) = 0, and we have to show that O_{ψ} | res_{U}^{H}(V) leads to a contradiction. Since [H : U] is an l′-number, V is relatively U-projective, and there is some indecomposable module W ∈ OU−lin with V | ind_{U}^{H}(W). So O_{ψ} | res_{U}^{H}(V) implies

O_{ψ} | res_{U}^{H}(V) | res_{U}^{H}(ind_{U}^{H}(W))

≅ ⨁_{h∈H/U} ind_{U∩^{h}U}^{U}(res_{U∩^{h}U}^{^{h}U}(^{h}W)) ≅ ⨁_{h∈H/U} ^{h}W .

Therefore, the module O_{ψ} is isomorphic to ^{h}W for some h ∈ H, and, since ψ is H-stable, this implies O_{ψ} ≅ W. Hence,

V | ind_{U}^{H}(O_{ψ}) ≅ ⨁_{φ∈Ĥ, φ|_{U} =ψ} O_{φ} ,

which leads to the desired contradiction, since rk_{O}(V) > 1.

(vii) This follows immediately from the explicit formula in part (vi).

(viii) This follows from the explicit formula in part (vi). In fact, if the multiplicity m_{φ_{n}}(res_{H_{n}}^{H}(V)) is non-zero, i.e. O_{φ_{n}} | res_{H_{n}}^{H}(V), then some vertex of O_{φ_{n}} is contained in a vertex of V. In particular, some Sylow l-subgroup of H_{0} is contained in a vertex of V.

(ix) This follows immediately from [2, Prop. 10.3].

(i) We first show that

p_{H}^{L} ∘ b_{H}^{L} ∘ a_{H}^{L} = p_{H}^{L}: ℚ ⊗ L_{O}(H) → ℚ ⊗ L_{O}^{ab}(H)  (4.3.a)

holds for all l-hypo-elementary subgroups H of G. In fact, let V ∈ OH−lin be indecomposable. If rk_{O}(V) = 1, then the above equation holds after evaluation at [V] by part (v). From now on we assume that rk_{O}(V) > 1. Let P denote the Sylow l-subgroup of H. If the vertices of V are strictly contained in P, then α_{(U,φ)}^{H}([V]) = 0 for all (U, φ) ∈ M^{L}(H) with P ≤ U by part (viii). On the other hand, for all (U, φ) ∈ M^{L}(H) with P ≰ U one has p_{H}^{L}([ind_{U}^{H}(O_{φ})]) = 0. Hence, (p_{H}^{L} ∘ b_{H}^{L} ∘ a_{H}^{L})([V]) = 0, and we are left with the case that V has vertex P. Let O_{ψ} be a source of V with ψ ∈ P̂. Then by the same argument as above it suffices to show that

p_{H}^{L}(α_{(U,φ)}^{H}([V]) · ind_{U}^{H}(ind([O_{φ}]))) = 0

for all (U, φ) ∈ M^{L}(H) with P ≤ U. By part (vii) we may assume that O_{φ} | res_{U}^{H}(V), since otherwise α_{(U,φ)}^{H}([V]) = 0. In this case we have

ind_{U}^{H}(O_{φ}) | ind_{U}^{H}(res_{U}^{H}(V)) | ind_{U}^{H}(res_{U}^{H}(ind_{P}^{H}(O_{ψ})))

≅ ind_{U}^{H}( ⨁_{h∈H/U} ind_{P}^{U}(O_{(^{h}ψ)})) ⨁_{h∈H/U} ind_{P}^{H}(O_{(^{h}ψ)}) .

Since rk_{O}(V) > 1, the element ψ ∈ P̂ is not H-stable (cf. Proposition 1.3 (b)). Again by Proposition 1.3 (b), we see that each module ind_{P}^{H}(O_{(^{h}ψ)}), h ∈ H, splits into indecomposable OH-modules of O-rank strictly bigger than 1. Therefore, also ind_{U}^{H}(O_{φ}) contains no direct summand of O-rank 1, and so is annihilated by p_{H}^{L}.

In order to prove the equation b^{L}∘a^{L} = id_{ℚ⊗L_{O}}, it suffices to show that b_{H}^{L} ∘a_{H}^{L} = id_{ℚ⊗L_{O}(H)} for all l-hypo-elementary subgroups H of G, since b^{L} and a^{L} commute with restrictions, and since C(ℚ ⊗ L_{O}) consists precisely of these subgroups. So let H ≤ G be l-hypo-elementary. Since

r_{H}^{L} = (p_{U}^{L} ∘ res_{U}^{H})_{U≤H}: L_{O}(H) → ∏_{U≤H} L_{O}^{ab}(U)

is injective by Corollary 2.7, it suffices to show for each U ≤ H, that p_{U}^{L} ∘ res_{U}^{H} ∘ b_{H}^{L} ∘ a_{H}^{L} = p_{U}^{L} ∘ res_{U}^{H}. Since a^{L} and b^{L} commute with restrictions, this follows from Equation (4.3.a).

(ii) In the proof of part (vi) we showed that condition (∗_{l′}) in Theorem 3.4 is satisfied in the present situation. Hence, we have

|H|_{l} · a_{H}(L_{O}(H)) ⊆ L_{O}^{ab}_{+}(H)

for all H ≤ G. In the sequel we will show that also |H|_{l′} · a_{H}(L_{O}(H)) ⊆ L_{O}^{ab}_{+}(H) for all H ≤ G.

4.4 Lemma Let V ∈ OG−lin be indecomposable such that G/ ker(V) is lelementary. Then there exists (H, φ) ∈ M^{L}(G), unique up to G-conjugacy, such that V ≅ ind_{H}^{G}(O_{φ}). Morever, [G : H] is an l-power and one has ker(V) ≤ H. In particular, rk_{O}(V) is an l-power.

Proof The existence of (H, φ) ∈ M^{L}(G) with V ≅ ind_{H}^{G}(O_{φ}) follows from Proposition 1.5 and also the uniqueness follows from the uniqueness statement of Proposition 1.5 provided that we can show that, for any such (H, φ), we have ker(V) ≤ ker(φ) (≤ H). Let U := ker(V), then V ≅ ind_{H}^{G}(O_{φ}) implies

O | res_{U}^{G}(V) ≅ ⨁_{g∈U\G/H} ind_{U∩^{g}H}^{U}(res_{U∩^{g}H}^{^{g}H}(O_{(^{g}φ)|_{U∩^{g}H}})) .

Hence, O ∈ OU−lin is a direct summand of one of the last summands above. Since the l-Sylow subgroups of U are the vertices of O, this implies that there is some g ∈ G such that [U : U ∩ ^{g}H] is an l′-number. On the other hand, [U : U ∩ ^{g}H] = [U ·^{g}H : ^{g}H] is an l-power, since [G : ^{g}H] = rk_{O}(V) is an l-power by Proposition 1.5. Therefore, U ∩ ^{g}H = U and ^{g}φ|_{U} = 1. Since U is normal in G, this implies U ≤ H and φ|_{U} = 1.

For each U ≤ G we define a map

η_{U}: L_{O}^{el}_{+}(U) → L_{O}^{ab}_{+}(U)

which maps the basis element [I, [V]]_{U}, where I ≤ U and V ∈ OI−lin is an indecomposable module such that I/ ker(V) is l-elementary, to [H, φ]_{U} ∈ L_{O}^{ab}_{+}(U), where (H, φ) ∈ M^{L}(I) is the (uniquely up to I-conjugacy determined) element with V ≅ ind_{H}^{I}(O_{φ}).

4.5 Lemma The maps η_{U}: L_{O}^{el}_{+}(U) → L_{O}^{ab}_{+}(U), H ≤ G, form a morphism of ℚ-restriction functors on G, and the diagram

ℚ ⊗ L_{O} ⟶^{a^{el}} ℚ ⊗ L_{O}^{el}_{+} ⟶^{b^{el}} ℚ ⊗ L_{O}
a^{L} ↘  ↓η  ↗ b^{L}
ℚ ⊗ L_{O}^{ab}_{+}

is commutative. In particular b^{el} ∘ a^{el} = id_{ℚ⊗L_{O}}.

Proof First we show that η is a morphism of restriction functors. Obviously η commutes with conjugation maps. Now let U′ ≤ U ≤ G, let I ≤ U and V ∈ OI−lin be indecomposable such that I/ ker(V) is l-elementary, and let (H, φ) ∈ M^{L}(I) be chosen such that V ≅ ind_{H}^{I}(O_{φ}). Then we have

res_{+}_{U′}^{U}(η_{U}([I, [V]]_{U})) = res_{+}_{U′}^{U}([H, φ]_{U}) = ∑_{u∈U′\U/H}[U′ ∩ ^{u}H, res_{U′∩^{u}H}^{^{u}H}(^{u}φ)]_{U}

and

η_{U′}(res_{+}_{U′}^{U}([I, [V]]_{U})) = ∑_{x∈U′\U/I} η_{U′}([U′ ∩ ^{x}I, res_{U′∩^{x}I}^{^{x}I}(^{x}[V])]_{U′}) .

For x ∈ U we have

res_{U′∩^{x}I}^{^{x}I}(^{x}V) ≅ res_{U′∩^{x}I}^{^{x}I}(ind _{^{x}H}^{^{x}I}(O_{(^{x}φ)}))

≅ ⨁_{y∈U′∩^{x}I\^{x}I/^{x}H} ind_{U′∩^{x}I∩^{yx}H}^{U′∩^{x}I}(res_{U′∩^{x}I∩^{yx}H}^{^{yx}H}(O_{(^{yx}φ)}))

≅ ⨁_{y∈U′∩^{x}I\^{x}I/^{x}H} ind_{U′∩^{yx}H}^{U′∩^{x}I}(res_{U′∩^{yx}H}^{^{yx}H}(O_{(^{yx}φ)})) .

Each of the summands in the last sum is indecomposable by Green’s indecomposability theorem. In fact, let C ≤ H ≤ I be the subgroup with C/ ker(V) = O_{l′}(I/ ker(V)). Then C is normal in I and [I : C] is a power of l. Hence, U′ ∩ ^{yx}C is normal in U′ ∩ ^{yx}I of l-power index, and U′ ∩ ^{yx}C ≤ U′ ∩ ^{yx}H ≤ U′ ∩ ^{yx}I, so that Green’s theorem can be applied. Moreover, U′ ∩ ^{yx}H ∩ker(V) ≤ U′ ∩ ^{yx}H ∩ker(^{yx}φ) and (U′ ∩ ^{yx}H)/(U′ ∩ ^{yx}H ∩ ker(V)) is l-elementary. Therefore, by the definition of η, we obtain for each x ∈ U:

η_{U′}([U′ ∩ ^{x}I, res_{U′∩^{x}I}^{^{x}I}(^{x}[V])]_{U′}) = ∑_{y∈U′∩^{x}I\^{x}I/^{x}H}[U′ ∩ ^{yx}H, (^{yx}φ)|_{U′∩^{yx}H}]_{U′} .

Summing up over x ∈ U′\U/I we therefore obtain exactly the above expansion of res_{+}_{U′}^{U}(η_{U}([I, [V]]_{U})), since the elements yx run through a set of representatives for U′\U/H, if x runs through a set of representatives for U′\U/I, and, for each x, y runs through a set of representatives for U′ ∩ ^{x}I\^{x}I/^{x}H. Hence, η is a morphism of restriction functors.

Next we show that b^{L} ∘ η = b^{el} : ℚ ⊗ L_{O}^{el}_{+} → ℚ ⊗ L_{O}. Let I ≤ U ≤ G, V ∈ OI−lin, and (H, φ) ∈ M^{L}(I) be given as in the previous paragraph. Then we have

b_{U}^{L}(η_{U}([I, [V]]_{U})) = b_{U}^{L}([H, φ]_{U}) = ind_{H}^{U}([O_{φ}])
= ind_{I}^{U}(ind_{H}^{I}([O_{φ}])) = ind_{I}^{U}([V]) = b_{U}^{el}([I, [V]]_{U}) .

Finally, we show that η ∘ a^{el} = a^{L}: ℚ ⊗ L_{O} → ℚ ⊗ L_{O}^{ab}_{+}. We write π instead

of π^{L_{O}^{ab}} : L_{O}^{ab}_{+} → L_{O}^{ab} and ρ_{U} instead of ρ_{U}^{L_{O}^{ab}} = (π_{I} ∘ res_{+}_{I}^{U})_{I≤U} for U ≤ G, cf. 3.2. Since ρ_{U}: ℚ ⊗ L_{O}^{ab}_{+}(U) → ℚ ⊗ L_{O}^{ab}^{+}(U) is injective, it suffices to show that π_{I} ∘ res_{+}_{I}^{U} ∘ η_{U} ∘ a_{U}^{el} = π_{I} ∘ res_{+}_{I}^{U} ∘ a_{U}^{L} for all I ≤ U ≤ G, and since η, a^{el}, and a^{L} are morphisms of restriction functors and since π_{U} ∘ a_{U}^{L} = p_{U}^{L} (cf. Diagram (3.3.b)), it suffices to show that

π_{U} ∘ η_{U} ∘ a_{U}^{el} = p_{U}^{L}

for all U ≤ G. Let V ∈ OU−lin be indecomposable. If V ≅ O_{φ} for some φ ∈ Û, then a_{U}^{el}([V]) = [U, [O_{φ}]]_{U} by [2, Prop. 6.12] (since res_{I}^{U}(p_{U}^{el}([O_{φ}])) = p_{I}^{el}(res_{I}^{U}([O_{φ}])) for all I ≤ U), and we have

(π_{U} ∘ η_{U} ∘ a_{U}^{el})([V]) = [Oφ] = p_{U}^{L}([V]) .

If rk_{O}(V) > 1 but U/ ker(V) is l-elementary, then still by the same argument we have a_{U}^{el}([V]) = [U, [V]]_{U}, and η_{U}([U, [V]]_{U}) = [H, φ]_{U} for some (H, φ) ∈ M^{L}(H) with H < U. Hence, [V] is mapped to zero by π_{U} ∘ η_{U} ∘ a_{U}^{el} as well as by p_{U}^{L}. Finally, if U/ ker(V) is not l-elementary, then, by Equation (3.3.d), a_{U}^{el}([V]) is a linear combination of elements [I, [W]]_{U} with (I, [W]) ∈ M^{el}(U) such that I < U. Hence, η_{U} ∘ a_{U}^{el} maps [V] to a linear combination of elements [H, φ]_{U} with (H, φ) ∈ M^{L}(U) such that H < U, and therefore, [V] lies in the kernel of π_{U} ∘ η_{U} ∘ a_{U}^{el} as well as in the kernel of p_{U}^{L}. This completes the proof of the Lemma.

The proof of part (ii) of Theorem 4.3 will be completed by showing that L_{O}, L_{O}^{el}, B^{el}, and p^{el} satisfy condition (∗_{l}) in Theorem 3.4. In fact, then Theorem 3.4 implies that |H|_{l′} · a_{H}^{el}(L_{O}(H)) ⊆ L_{O}^{el}_{+}(H) for all H ≤ G, and we obtain with Lemma 4.5 that

|H|_{l′} · a_{H}^{L}(L_{O}(H)) = |H|_{l′} · (η_{H} ∘ a_{H}^{el})(L_{O}(H)) ⊆ η_{H}(|H|_{l′} · a_{H}^{el}(L_{O}(H))
⊆ η_{H}(L_{O}^{el}_{+}(H)) ⊆ L_{O}^{ab}_{+}(H) .

On the other hand, we already know that |H|_{l} · a_{H}^{L}(L_{O}(H)) ⊆ L_{O}^{ab}_{+}(H) so that we have a_{H}^{L}(L_{O}(H)) ⊆ L_{O}^{ab}_{+}(H) for all H ≤ G.

So let U ⊴ H ≤ G be given such that H/U is a cyclic l-group, let V ∈ OH−lin be indecomposable, and let W ∈ OU−lin be indecomposable and H-stable such that U/ ker(W) is l-elementary. If H/ ker(V) is l-elementary, then

p_{U}^{el}(res_{U}^{H}([V])) = res_{U}^{H}([V]) = res_{U}^{H}(p_{U}^{el}([V]))

and condition (∗_{l}) is satisfied in this case. If H/ ker(V) is not l-elementary, then p_{H}^{el}([V]) = 0, and we have to show that W does not occur as a direct summand in res_{U}^{H}(V). This follows from the next Lemma.

4.6 Lemma Let U ⊴ H ≤ G be such that H/U is an l-group, and let W ∈ OU−lin be indecomposable such that W is H-stable and U/ ker(W) is l-elementary. Then, for each indecomposable module V ∈ OH−lin with W | res_{U}^{H}(V), also H/ ker(V) is l-elementary.

Proof The subgroup I := ker(W) ≤ U is normal in H, since W is H-stable. First we show that H/I is again l-elementary. In fact, let I ≤ P, C ≤ U be intermediate groups with C/I = O_{l′}(U/I) and P/I = O_{l}(U/I), hence, U/I = P/I × C/I. Then W ≅ ind_{J}^{U}(O_{λ}) for some C ≤ J ≤ H and some λ ∈ Ĵ by Lemma 4.4. Since I = ker(ind_{J}^{U}(O_{λ})), and C/I is cyclic, one also has I = ker(λ|_{C}). By the uniqueness part of Lemma 4.4, for each element h ∈ H, there is some element u ∈ U with h(J, λ) = ^{u}(J, λ). Therefore, we have ^{h}(λ|_{C}) = ^{u}(λ|_{C}) = λ|_{C}, since U/I centralizes C/I. This shows that H/I centralizes C/I and that H/I = C/I × P′/I for a Sylow l-subgroup P′/I of H/I. Hence, H/I is l-elementary.

Now let V ∈ OH−lin be indecomposable with W | res_{U}^{H}(V). It suffices to show that then I ≤ ker(V).

Let Q be a vertex of V and let O_{ψ} ∈ OQ−lin be a source of V with ψ ∈ Q̂. Then, there exists an indecomposable module V′ ∈ O(IQ)−lin with V′ | ind_{Q}^{IQ}(O_{ψ}) and V | ind_{IQ}^{H}(V′). We will show that V′ ≅ O_{ψ′} for an extension ψ′ ∈ ÎQ of ψ with ψ′|_{I} = 1. Then ker(V) ≥ I. We have O | res_{I}^{IQ}(V′), since

[figure: O_{ψ}; Q; V′; IQ; UQ; V; H; I ∩ Q; I; U; W]

O | res_{I}^{U}(W) | res_{I}^{U}(res_{U}^{H}(V)) ≅ res_{I}^{H}(V) | res_{I}^{H}(ind_{IQ}^{H}(V′))

≅ ⨁_{h∈I\H/IQ} ind_{I∩^{h}(IQ)}^{I}(res_{I∩^{h}(IQ)}^{^{h}IQ}(^{h}V′)) ≅ ⨁_{h∈H/IQ} ^{h}(res_{I}^{IQ}(V′)) .

From

O | res_{I}^{IQ}(V′) | res_{I}^{IQ}(ind_{Q}^{IQ}(O_{ψ}))

≅ ⨁_{x∈I\IQ/Q} ind_{I∩^{x}Q}^{I}(res_{I∩^{x}Q}^{^{x}Q}(O_{(^{x}ψ)})) ≅ ind_{I∩Q}^{I}(O_{(ψ|_{I∩Q})})

we see that I ∩ Q contains a Sylow l-subgroup of I. Therefore, Q is a Sylow lsubgroup of IQ. Moreover, we obtain from O | ind_{I∩Q}^{I}(O_{(ψ|_{I∩Q})}) that ψ|_{I∩Q} = 1. Therefore, we can extend ψ ∈ Q̂ to ψ′ ∈ ÎQ with ψ′|_{I} = 1. Since the multiplicity of O in

res_{I}^{IQ}(ind_{Q}^{IQ}(O_{ψ})) = ind_{I∩Q}^{I}(O)

equals 1, there is exactly one indecomposable summand X of ind_{Q}^{IQ}(O_{ψ}) with O | res_{I}^{IQ}(X). But both V′ and O_{ψ′} have this property. Hence, V′ ≅ O_{ψ′}, and the proof is complete.

This completes the proof of Theorem 4.3.

4.7 Remark (i) For the canonical induction formula a^{T} for T_{O} from T_{O}^{ab} arising from Example 4.1 (b) a theorem analogous to Theorem 4.3 holds. This follows from the commutative diagram in 4.1 (b). Since there is an isomorphism of Mackey functors T_{O} ≅ T_{F} for the residue field F of O, we also obtain a canonical induction formula for T_{F} from T_{F}^{ab} by transporting everything via this isomorphism.

(ii) By Lemma 4.5 we know that also a^{el} is a canonical induction formula. We haven’t examined whether a^{el} is integral. We used a^{el} only to show that a^{L} is integral.

(iii) In a subsequent paper we will construct Adams operations on L_{O}(G) using the canonical induction formula a^{L} and the obvious Adams operations on L_{O}^{ab}(G). This construction makes use of the general results in [2, §4].

References

[1] D. Benson, Lambda and psi operations on Green rings, J. Algebra 87 (1984), 360–367.

[2] R. Boltje, A general theory of canonical induction formulae, Preprint 1996.

[3] M. Broué, On Scott-modules and p-permutation modules: An approach through the Brauer morphism, Proc. Amer. Math. Soc. 93 (1985), 401–408.

[4] C. W. Curtis, I. Reiner, Methods of representation theory, vol. 2, J. Wiley & Sons, New York 1987.

[5] B. Huppert, Endliche Gruppen, vol. 1, Springer-Verlag, 1967.