Canonical induction formulae and the defect of a character

Robert Boltje Institut für Mathematik Universität Augsburg 86 135 Augsburg Germany email: boltje@math.uni-augsburg.de

October 30, 1995

Abstract

We explain the idea and the machinery of canonical induction formulae with going as little into details as possible and show that a certain version keeps track of the l-defect d(χ) of an irreducible character χ of a finite group G by mapping χ precisely to the d(χ)-th layer in the filtration A ⊆ l^{−1}A ⊆ l^{−2}A …, for a prime l and a certain free abelian group A.

1 Introduction and preliminaries

Since its publication in 1947 Brauer’s induction theorem (see [6]) which says that each character of a finite group is expressible as an integral linear combination of induced one-dimensional characters of elementary subgroups has become a cornerstone both in group theory and number theory. The idea that there are distinguished ways of expressing a character according to Brauer’s theorem (at least if one allows all subgroups instead of only elementary subgroups to induce from) originated independently with V. Snaith, cf. [14], and the author, cf. [1]. To explain what this means we have to introduce some notation.

1.1 For a finite group G let R(G) denote the character ring of G and R^{≥0}(G) the additive monoid of characters of G, i.e. R(G) is the free abelian group on the set Irr(G) of absolutely irreducible characters of G and R^{≥0}(G) ⊆ R(G) is the set of all non-negative integral linear combinations of elements in Irr(G). Furthermore, let R_{+}^{ab}(G) be the free abelian group on G-conjugacy classes [H, φ]_{G} of the poset M(G) of pairs (H, φ) where H is a subgroup of G and φ is a one-dimensional character of H. The poset structure on M(G) is defined by (K, ψ) ≤ (H, φ), if and only if K ≤ H and ψ = res_{K}^{H}(φ), and it is respected by G-conjugation. The group R_{+}^{ab}(G) has already been introduced by Deligne in [10]. Using methods from topology, Snaith defined in [14] a function

t_{G}: R^{≥0}(G) → R_{+}^{ab}(G)

which is natural with respect to restrictions. In fact, for H ≤ G there is a restriction map

res_{+}_{H}^{G}: R_{+}^{ab}(G) → R_{+}^{ab}(H), [K, ψ]_{G} ↦ ∑_{g∈H\G/K}[H ∩ ^{g}K, res_{H∩^{g}K}^{^{g}K}(^{g}ψ)]_{H},

where the sum runs over representatives g ∈ G of the double cosets H\G/K, gK := gKg^{−1} and ^{g}ψ(gxg^{−1}) := ψ(x) for x ∈ K. This restriction map has been defined by Deligne (cf. [10]) and can also be naturally interpreted by viewing R_{+}^{ab}(G) as the Grothendieck group of a suitable category of monomial Gmodules, cf. [3, Chapter 5]. Snaith’s map t_{G} is a canonical induction formula, where ‘canonical’ means that t_{G} commutes with restrictions, and ‘induction formula’ means that b_{G} ∘ t_{G} = id_{R(G)}, where b_{G} is defined as

b_{G}: R_{+}^{ab}(G) → R(G), [H, φ]_{G} ↦ ind_{H}^{G}(φ),

and also commutes with restrictions. Due to the topological nature of its construction, the map t_{G} is not additive but has a sort of derivation property with respect to addition, cf. [5, Theorem 2.16 (c)], so that it is not clear how to extend it from R^{≥0}(G) to R(G).

Independently of Snaith the author constructed an additive map

a_{G}: R(G) → R_{+}^{ab}(G)

which also is a canonical induction formula, cf. [2]. This was done by purely algebraic methods. In general, the maps a_{G} and t_{G} are different, even on Irr(G). But there is also an algebraic description of t_{G} which allows to compare both maps, cf. [5]. Moreover, there is a geometric construction of the map a_{G} due to P. Symonds (cf. [5, (2.26)] and [16]). Here we will only use the canonical induction formula a_{G}.

1.2 Without going into details we should try to make clear that the group R_{+}^{ab}(G) is a very natural construction. Note that the groups R(H), H ≤ G, are tied together by homomorphisms

c_{g,H}: R(H) → R(^{g}H),
res_{K}^{H}: R(H) → R(K),
ind_{K}^{H}: R(K) → R(H),

for K ≤ H ≤ G, g ∈ G, called conjugation, restriction and induction. These homomorphisms satisfy certain compatibility conditions which are the axioms of a Mackey functor on G. For a precise definition see for example [17]. Let furthermore, R^{ab}(G) ⊆ R(G) denote the span of the set Ĝ of all one-dimensional characters. Then the family of groups R^{ab}(H), H ≤ G, is stable under conjugation and restriction maps. We will call such a structure a restriction functor on G. Obviously, there is a forgetful functor from the category Mack(G) of Mackey functors on G to the category Res(G) of restriction functors on G. This functor has a left adjoint functor

−_{+} : Res(G) → Mack(G),

which associates to a restriction functor A(H), H ≤ G, the family of groups

A_{+}(H) := coinv_{H}( ⨁_{K≤H} A(K)), H ≤ G,  (1.2.1)

where H acts on ⨁_{K≤H} A(K) by the conjugation maps and for an H-module M the abelian group coinv_{H}(M) is the biggest quotient of M which admits trivial H-action. Note that this definition of R_{+}^{ab}(G) coincides with the adhoc definition given earlier, cf. [3, I.2.2, I.2.4, I.4.1]. For example, if A(H) = Z for all H ≤ G and all restriction and conjugation maps are the identity, then A_{+}(G) is the Burnside ring of G.

If we tensor for the moment everything with Q, we obtain a very clear description of the set of all families of maps Q ⊗ R(H) → Q ⊗ R_{+}^{ab}(H), H ≤ G, which commute with restriction and conjugation maps. In fact, Corollary 4.2 (iii) in [3] states that there is a natural isomorphism

Res(G)(A, B_{+}) → Con(G)(A, B),  (1.2.2)

where A and B are restriction functors on G over Q and Con(G) is the category of conjugation functors, requiring only conjugation maps and no restriction or induction maps. In the case A(H) = Q ⊗ R(H), H ≤ G, and B(H) = Q ⊗ R^{ab}(H), H ≤ G, we are left with finding suitable conjugation respecting families of maps

p_{H}: Q ⊗ R(H) → Q ⊗ R^{ab}(H).  (1.2.3)

The isomorphism in (1.2.2) is very explicit. For a given morphism of conjugation functors p: Q ⊗ R → Q ⊗ R^{ab} as in (1.2.3), the associated morphism of restriction functors

a^{p}: Q ⊗ R → Q ⊗ R_{+}^{ab}

is given on the level of a subgroup H of G by

a_{H}^{p}(χ) := \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}(χ)))]_{H}  (1.2.4)

for χ ∈ Q ⊗ R(H), where the sum runs over all chains of subgroups of H. Note that res_{H_{0}}^{H_{n}}(p_{H_{n}}(res_{H_{n}}^{H}(χ))) is an element of Q ⊗ R^{ab}(H_{0}) and that for such an element θ = ∑_{φ∈Ĥ_{0}} a_{φ}φ ∈ Q ⊗ R^{ab}(H_{0}), where a_{φ} are rational numbers, we define [H_{0}, θ]_{H} := ∑_{φ∈Ĥ_{0}} a_{φ}[H_{0}, φ]_{H}.

1.3 For given p as in (1.2.3) there are two natural questions to ask about the associated morphism a^{p}: Q ⊗ R → Q ⊗ R_{+}^{ab}:

(i) Is a^{p} an induction formula, i.e., is b_{H} ∘ a_{H}^{p} = id_{Q⊗R(H)} for all H ≤ G?

(ii) Is a^{p} integral, i.e., is a_{H}^{p}(R(H)) ⊆ R_{+}^{ab}(H) for all H ≤ G?

The best answers we know so far are the following:

1.4 Theorem (cf. [3, Corollary II.2.4]) Let C be the set of cyclic subgroups of a finite group G and let p ∈ Con(Q ⊗ R, Q ⊗ R^{ab}). If p_{H} = id_{Q⊗R(H)} for H ∈ C, then b ∘ a^{p} = id_{Q⊗R}. Furthermore, for H ∈ C and φ ∈ Ĥ we have a_{H}^{p}(φ) = [H, φ]_{H}.

Actually Corollary II.2.4 in [3] is more general. It deals with an arbitrary Mackey functor M on G over Q replacing Q ⊗ R and with a sub-restriction functor A ⊆ M on G over Q replacing Q ⊗ R^{ab} such that there is a set C of subgroups of G with A(H) = M(H) and p_{H} = id_{M(H)} for H ∈ C and such that for all H ≤ G the elements of M(H) are uniquely determined by their restriction to subgroups K ≤ H, K ∈ C. Then the associated morphism a^{p} is a section of b: A_{+} → M and a_{H}(m) is the image of m ∈ M(H) in A_{+}(H) for H ∈ C, where we use the definition of A_{+}(H) in (1.2.1).

1.5 Theorem (cf. [3, Theorem II.4.5]) Let p: R → R^{ab} be a morphism of conjugation functors on G and π a set of primes such that the following condition holds:

(∗_{π}) For each pair of subgroups V ≤ U ≤ H such that V is normal in U and U/V is a cyclic π-group, and for each one-dimensional character ψ ∈ V̂ which is stable under U-conjugation, the coefficients of ψ in the elements (res_{V}^{U} ∘ p_{U})(θ) and (p_{V} ∘ res_{V}^{U})(θ) of R^{ab}(V), coincide for θ ∈ Irr(U).

Then, for H ≤ G and χ ∈ R(H), the explicit formula (1.2.4) for a_{H}^{p}(χ) can be transformed into

a_{H}^{p}(χ) = ∑_{{σ=((H_{0},φ_{0})<···<(H_{n},φ_{n})) \atop modH}}(−1)^{n} \frac{|(N_{H}(σ)/H_{0})_{π}|}{|N_{H}(σ)/H_{0}|}m_{φ_{n}}(res_{H_{n}}^{H}(χ))[H_{0}, φ_{0}]_{H},

where the sum runs over all chains σ taken up to H-conjugation in the poset M(H), N_{H}(σ) denotes the stabilizer of σ in H, (N_{H}(σ)/H_{0})_{π} the set of πelement of N_{H}(σ)/H_{0}, and for any subgroup K ≤ H and λ ∈ R(K) the expansion

p_{H}(λ) = ∑_{ψ∈K̂} m_{ψ}(λ)ψ ∈ R^{ab}(K)

defines the multiplicities m_{ψ}(λ) ∈ Z, ψ ∈ K̂.

Like Theorem 1.4 also Theorem 1.5 has a more general version (cf. [3, Theorem II.4.5]), where R and R^{ab} are replaced by a Mackey functor M on G and a restriction subfunctor A of M on G such that A has a conjugation-stable basis which replaces Ĥ, for H ≤ G.

Since |G_{π}|/|G|_{π} is an integer for any finite group G (cf. [11, V.19.14]), we obtain

1.6 Corollary (cf. [3, Corollary II.4.7]) Let p: R → R^{ab} be a morphism of conjugation functors on G and let π be a set of primes such that condition (∗_{π}) in Theorem 1.5 is satisfied, then |H|_{π′}a_{H}^{p}(χ) ∈ R_{+}^{ab}(H) for all H ≤ G and χ ∈ R(H). In particular, if π is the set of all primes, a^{p} is integral.

2 The canonical Brauer induction formula

2.1 Keeping the notation of the previous section we define a morphism p: R → R^{ab} of conjugation functors on G for H ≤ G as being induced by taking H′-fixed points of an H-module, thus obtaining an H^{ab}-module, where H′ denotes the commutator subgroup and H^{ab} the commutator factor group of H. Note that R^{ab}(H) and R(H^{ab}) can be naturally identified. In other words, p_{H} is defined as

p_{H}: R(H) → R^{ab}(H), Irr(H) ∋ χ ↦ \begin{cases}χ, & if χ ∈ Ĥ, \\ 0, & otherwise. \end{cases}  (2.1.1)

It is easy to see that the hypotheses of Theorem 1.4 and Theorem 1.5 (cf. the proof of Proposition 4.4) with π the set of all primes are satisfied so that the resulting morphism a^{p}: Q ⊗ R → Q ⊗ R_{+}^{ab} of restriction functors gives rise to an integral canonical induction formula

a_{G}: R(G) → R_{+}^{ab}(G)

which we will call the canonical Brauer induction formula. This map is precisely the one defined in [2].

2.2 The advantage of having a canonical Brauer induction formula rather than any expression of a character according to Brauer’s induction theorem is best explained at the example of the Adams operations

Ψ_{G}^{k}: R(G) → R(G), χ ↦ (g ↦ χ(g^{k}))

for k ∈ Z, χ ∈ R(G), g ∈ G. Note that it is obvious that Ψ_{G}^{k}(χ) is again a class function. But one has to work a little bit to see that Ψ_{G}^{k}(χ) is again an element in R(G). Note also that this is immediate for one-dimensional characters, since Ψ_{G}^{k}(φ) = φ^{k} for φ ∈ Ĝ. Finally note that the family Ψ_{H}^{k}, H ≤ G, is a morphism of restriction functors, i.e. commutes with conjugations and restrictions.

Now let

a_{G}(χ) = ∑_{[H,φ]_{G}} α_{[H,φ]_{G}}(χ)[H, φ]_{G}  (2.2.1)

be the expansion of the canonical Brauer induction formula for χ ∈ R(G), thus defining α_{[H,φ]_{G}} as the integral coefficient of the basis element [H, φ]_{G} of R_{+}^{ab}(G) in a_{G}(χ). Since b_{G} ∘ a_{G} = id_{R(G)}, we obtain the expression

χ = ∑_{[H,φ]_{G}} α_{[H,φ]_{G}}(χ)ind_{H}^{G}(φ)  (2.2.2)

for χ ∈ R(G) in a way ensured by Brauer’s Theorem (except for the restriction that H is elementary). By applying Ψ_{G}^{k} to Equation (2.2.2) we obtain

Ψ_{G}^{k}(χ) = ∑_{[H,φ]_{G}} α_{[H,φ]_{G}}(χ)Ψ_{G}^{k}(ind_{H}^{G}(φ)).  (2.2.3)

Now the pity is that Adams operations do not commute with induction, unless k is coprime to the exponent of G. But just for fun we could define an element

Ψ̃_{G}^{k}(χ) := ∑_{[H,φ]_{G}} α_{[H,φ]_{G}}(χ)ind_{H}^{G}(φ^{k})  (2.2.4)

in R(G), thus having defined a homomorphism Ψ̃_{G}^{k}: R(G) → R(G). It turns out that, since the maps a_{H}, H ≤ G, commute with restrictions, also the maps Ψ̃_{H}^{k}, H ≤ G, commute with restrictions. Furthermore, by Theorem 1.4, one has Ψ̃_{H}^{k} = Ψ_{H}^{k} for all cyclic subgroups H of G. Hence, if we only assume that Ψ_{G}^{k} takes values in the group of class functions on G, Ψ̃_{G}^{k} − Ψ_{G}^{k} is a map from R(G) to the group of class function on G, commuting with restrictions to subgroups and being trivial on cyclic subgroups. Therefore, if χ ∈ R(G) and g ∈ G are given, we have

(Ψ̃_{G}^{k} − Ψ_{G}^{k})(χ)(g) = (Ψ̃_{<g>}^{k} − Ψ_{<g>}^{k})(res_{<g>}^{G}(χ)(g)) = 0,

showing that Ψ̃_{G}^{k} = Ψ_{G}^{k}. In particular, Ψ_{G}^{k}(χ) is not only a class function on G but an element of R(G). This means that in the sums in Equations (2.2.3) and (2.2.4) the corresponding summands for themselves may be different, but in the whole sum all the differences cancel due to the naturality of a_{G}.

2.3 The above example shows that the canonical Brauer induction formula can be used to extend maps defined on R^{ab}(G) to maps on R(G). This concept can be used in many situations, yielding a character theoretic definition of the determinant and the Chern class of a character. Moreover, extensions of characters and a correspondence related to the Glauberman correspondence can be defined that way, cf. [3, Chapter IV]. Also in number theory this method was used to define local root numbers (cf. [1, Chapter II]) and conductors of Galois representations (cf. [1, Chapter II] and [4]).

Canonical induction formulae also exist for other representation rings (see [3, Chapter III]), as for example for the Brauer character ring, the trivial source ring, and the linear source ring. Some of the above constructions make sense for one-dimensional representations of those rings, so that we may for example define Chern classes of trivial source modules and other exotic maps whose existence was not at all clear before.

2.4 There are many surprising properties of the map a_{G}. For example, Equation (2.2.2) implies that

∑_{[H,φ]_{G}} α_{[H,φ]_{G}}(χ)[G : H] = χ(1),

but much simpler the degree of χ ∈ R(G) can be extracted from the coefficients α_{[H,φ]_{G}}(χ) as

∑_{[H,φ]_{G}} α_{[H,φ]_{G}}(χ) = χ(1).

This follows immediately from Equation (2.2.4) and the equality of Ψ̃_{G}^{k} and Ψ_{G}^{k} for k = 0 by applying the inner product of R(G) with the trivial character of G on both sides of Equation (2.2.4). For a list of more properties of a_{G} see [2, Theorem 2.1] and [3, Proposition III.1.2].

For further applications of canonical induction formulae see [15].

3 Variations of the canonical Brauer induction formula

3.1 In view of Theorem 1.4 we realize that we still obtain a canonical induction formula by defining p_{H} = 0 for non-cyclic subgroups H ≤ G and p_{H} = id_{Q⊗R(H)} for H ≤ G cyclic. According to (1.2.4) the corresponding canonical induction formula a^{p} has the explicit description

a_{H}^{p}(χ) = \frac{1}{|H|} ∑_{{H_{0}<···<H_{n} \atop H_{n}cyclic}} (−1)^{n}|H_{0}|[H_{0}, res_{H_{0}}^{H}(χ)]_{H}  (3.1.1)

for H ≤ G and χ ∈ R(H), where the sum runs over all chains of cyclic subgroups of H and [H_{0}, res_{H_{0}}^{H}(χ)]_{H} has to be interpreted as explained below (1.2.4). But this formula is not integral. In fact, the smallest non-negative integer e such that e · a_{G}^{p}(χ) ∈ R_{+}^{ab}(G) for all χ ∈ R(G) turns out to be the Artin exponent of G (see [2, Theorem 3.17] for a proof of this statement and [9, § 76] for a definition of the Artin exponent).

The resulting expression after applying b_{G},

χ = \frac{1}{|G|} ∑_{{H_{0}<···<H_{n} \atop H_{n}cyclic}} (−1)^{n}|H_{0}|ind_{H_{0}}^{G}(res_{H_{0}}^{G}(χ)),  (3.1.2)

for χ ∈ R(G) can be transformed by using the Möbius function μ(K, H) of the poset of cyclic subgroups of G (see [12] for generalities about the Möbius function of a poset) and taking into account that μ(K, H) = μ(|H/K|), where the latter μ denotes the usual number theoretic Möbius function, into

χ = \frac{1}{|G|} ∑_{{K≤H \atop Hcyclic}} μ(|H/K|)|K|ind_{K}^{G}(res_{K}^{G}(χ))  (3.1.3)

which is precisely the explicit version found by Brauer (cf. [7]) of Artin’s induction theorem stating that each character can be expressed as a rational linear combination of characters induced from cyclic subgroups.

3.2 In view of Brauer’s induction theorem one might think that the morphism p: Q⊗R → Q⊗R^{ab} of conjugation functors, given by p_{H} = 0 for H ≤ G not elementary, and p_{H} as in (2.1.1) for H ≤ G elementary, results in an integral canonical induction formula a^{p}. In fact a^{p} is an induction formula, since the hypothesis of Theorem 1.4 is satisfied, but in general a^{p} is not integral. Also the condition (∗_{π}) in Theorem 1.5 is not satisfied for the set π of all primes, if V ≤ U is chosen such that V is elementary and U is not, and if ψ is the trivial character of V.

More generally, let H be a set of subgroups of G closed under taking subgroups and conjugates and containing the set of cyclic subgroups of G. Furthermore, let p_{H} = 0 for H ∉ H and let p_{H} be defined as in (2.1.1) for H ∈ H. Then the resulting morphism a^{p} is a canonical induction formula by Theorem 1.4, and we know (see [2, Corollary 3.13] for a proof) precisely in which cases a^{p} is integral, namely if and only if for each pair of subgroups K ≤ H of G such that K is normal in H, H/K is cyclic, and K lies in H, also H lies in H. The smallest such set H is the set of solvable subgroups of G. This fact about the integrality of a^{p} is related to the fact that the idempotent of the Burnside algebra over Q associated with H is integral if and only if the above condition on H is satisfied.

4 Projectification

4.1 Pushing the idea of subsection 3.1 further, we choose a prime l (the letter p being reserved for maps as in (1.2.3)) and consider the morphism of conjugation functors p: R → R^{ab} given by p_{H} = 0, if |H| is divisible by l, and by p_{H} as in (2.1.1), if H is an l′-subgroup of G. By the isomorphism in (1.2.2) we obtain a corresponding morphism

a^{l′}: Q ⊗ R → Q ⊗ R_{+}^{ab}

of restriction functors on G, which is explicitly given by

a_{G}^{l′}(χ) = \frac{1}{|H|} ∑_{{H_{0}<···<H_{n} \atop |H_{n}|_{l}=1}} (−1)^{n}|H_{0}|[H_{0}, res_{H_{0}}^{H_{n}}(p_{H_{n}}(res_{H_{n}}^{H}(χ)))]_{H}  (4.1.1)

for H ≤ G and χ ∈ R(H), cf. (1.2.4), where the sum runs over all chains of l′-subgroups of H. It is clear that in general b_{G} ∘a_{G}^{l′} cannot be the identity, since for an l-group G and χ ∈ R(G) we have only one such chain, and we obtain that b_{G} ∘ a_{G}^{l′}(χ) = (χ(1)/|G|) · ind_{1}^{G}(1) is always a rational multiple of the regular character of G. We are going to describe the map b_{G}∘a_{G}^{l′}: Q⊗R(G) → Q⊗R(G) in a different way.

4.2 Let e_{G}^{l′} be the characteristic function on G of the subset of l′-elements of G. Obviously e_{G}^{l′} is an idempotent class function on G. But it is even an element of Q⊗R(G), since using the expression (3.1.3) for χ = 1, multiplying by e_{G}^{l′}, and using the obvious property res_{H}^{G}(e_{G}^{l′}) = e_{H}^{l′} for H ≤ G, we are reduced to the case where G is cyclic, and in this case we have e_{G}^{l′} = (1/|G|_{l}) · ind_{G′_{l}}^{G}(1), where G_{l′} is the l′-part of G.

4.3 Proposition For a finite group G the map b_{G} ∘ a_{G}^{l′}: Q ⊗ R(G) → Q ⊗ R(G) is multiplication by e_{G}^{l′}. In particular, if χ ∈ R(G) vanishes on all l-singular elements, then b_{G}(a_{G}^{l′}(χ)) = χ.

Proof Let χ ∈ R(G). We have to show that

(b_{G} ∘ a_{G}^{l′})(χ) = e_{G}^{l′} · χ.  (4.3.1)

By Equation (4.1.1) both (b_{G} ∘ a_{G}^{l′})(χ) and e_{G}^{l′} · χ vanish on l-singular elements. Hence, it suffices to show that

res_{H}^{G}((b_{G} ∘ a_{G}^{l′})(χ)) = res_{H}^{G}(e_{G}^{l′} · χ)  (4.3.2)

for all cyclic l′-subgroups H of G. However, since b_{G} and a_{G}^{l′} commute with restriction and since restriction is multiplicative and maps e_{G}^{l′} to e_{H}^{l′}, Equation (4.3.2) is equivalent to the equation

(b_{H} ∘ a_{H}^{l′})(res_{H}^{G}(χ)) = e_{H}^{l′} · res_{H}^{G}(χ)

so that we may assume in Equation (4.3.1) that G is a cyclic l′-group. In this case, the right hand side in Equation (4.3.1) equals χ and by Equation (4.1.1) the left hand side is equal to

\frac{1}{|G|} ∑_{H_{0}<···<H_{n}}(−1)^{n}|H_{0}|ind_{H_{0}}^{G}(res_{H_{0}}^{G}(χ)),

where the sum runs over all chains of subgroups of G. But this sum is equal to χ by Equation (3.1.2). Alternatively, we could use

a_{G}^{l′}(φ) = [G, φ]_{G}

for φ ∈ Ĝ from Theorem 1.4 to see that (b_{G} ∘ a_{G}^{l′})(χ) = χ.

4.4 Proposition For a finite group G and χ ∈ R(G) we have |G|_{l}·a_{G}^{l′}(χ) ∈ R_{+}^{ab}(G).

Proof We will verify the hypothesis of Theorem 1.5 for the set π of all primes different from l. So let U ≤ V ≤ G be such that V is normal in U and U/V is a cyclic l′-group. Furthermore, let ψ ∈ V̂ be U-stable and let θ ∈ Irr(U). We have to show that the inner product

(ψ, res_{V}^{U} (p_{U}(θ)) − p_{V} (res_{V}^{U} (θ)))

vanishes. If V is not an l′-group, then U isn’t either. Hence, p_{U} and p_{V} are trivial and the above inner product vanishes.

So let V be an l′-group. Then also U is an l′-group, and since U stabilizes ψ, the cyclic group U/V acts trivially on the cyclic group V/kerψ. Hence U/kerψ is abelian, and ψ can be extended to U. If θ is one-dimensional, then

res_{V}^{U} (p_{V} (θ)) = res_{V}^{U} (θ) = p_{V} (res_{V}^{U} (θ)),

and we are done. If θ is not one-dimensional, then res_{V}^{U} (p_{U}(θ)) = 0, and we have to show that ψ does not occur as a constituent in p_{V} (res_{V}^{U} (θ)). However, assuming that ψ does occur implies that ψ also occurs in res_{V}^{U} (θ). By Frobenius reciprocity we deduce that θ occurs in ind_{V}^{U} (ψ), which is the sum of all extensions of ψ to U. All of them being one-dimensional gives the desired contradiction.

Applying b_{G} to the assertion in the above Proposition we obtain the following presumable well-known Corollary.

4.5 Corollary Let χ ∈ R(G), then e_{G}^{l′} · χ ∈ (1/|G|_{l}) · R(G). In particular e_{G}^{l′} ∈ (1/|G|_{l}) · R(G).

4.6 Remark Let L be a finite field extension of the field Q_{l} of l-adic numbers containig all |G|-th roots of unity, and let O be the valuation ring of L. We consider the cyclotomic field of the |G|-th roots of unity over Q both contained in L and in C. The functor L ⊗_{O} −: OG − mod → LG − mod induces an isomorphism between the Grothendieck group of finitely generated projective OG-modules and the subgroup P (G) of R(G) consisting of all virtual characters vanishing on l-singular elements (cf. [13, Théorème 37]). The groups P (H), H ≤ G, form a Mackey subfunctor of R on G. For an l′-subgroup H of G we set P^{ab}(H) := R^{ab}(H), and for H ≤ G with l | |H| we set P^{ab}(H) = 0. Then P^{ab} is a restriction subfunctor of R^{ab} and of P. The morphism p: R → R^{ab} from 4.1 can be viewed as a morphism

p: R → P^{ab}

of conjugation functors on G. It is easy to see that P_{+}^{ab}(H) is naturally contained in R_{+}^{ab}(H) for H ≤ G, and that a^{l′} can be considered as a morphism

a^{l′}: R → P_{+}^{ab}  (4.6.1)

of restriction functors on G.

4.7 Theorem We keep the notation of Remark 4.6. The restriction of a^{l′} in (4.6.1) to P^{ab} is an integral canonical induction formula

a^{l′}: P → P_{+}^{ab}.

In particular, a_{H}^{l′}(χ) ∈ P_{+}^{ab}(H) ⊆ R_{+}^{ab}(H) for χ ∈ P (H), H ≤ G.

Proof It follows from Proposition 4.3 that (b_{H} ∘a_{H}^{l′})(χ) = χ for χ ∈ P (H), H ≤ G. We only have to show the integrality part of the statement. Since we have already verified that condition (∗_{π}) in Theorem 1.5 holds for the morphism p from 4.1 and the set of all primes different from l, we obtain by Theorem 1.5 an explicit formula for a_{H}^{l′}(χ), H ≤ G, χ ∈ R(H), and it suffices to prove that

\frac{|(N_{H}(σ)/H_{0})_{l′}|}{|N_{H}(σ)/H_{0}|}m_{φ_{n}}(res_{H_{n}}^{H}(χ))

is an integer for any H ≤ G, any chain σ = ((H_{0}, φ_{0}) < ⋯ < (H_{n}, φ_{n})) in M(H) and any χ ∈ P (H). Since m_{φ_{n}}(res_{H_{n}}^{H}(χ)) = 0 unless H_{n} is an l′-subgroup of H, we may assume that H_{n} is an l′-group and also that χ is a character of a projective indecomposable OH-module M (cf. the notation introduced in Remark 4.6). Note that, since H_{0} is an l′-group, we have

|(N_{H}(σ)/H_{0})_{l′}| = |N_{H}(σ)_{l′}|/|H_{0}|.

Hence, we have to show that

\frac{|N_{H}(σ)_{l′}| · m_{φ_{n}}(res_{H_{n}}^{H}(χ))}{|N_{H}(σ)|_{l′} · |N_{H}(σ)|_{l}}

is an integer.

It follows from [11, V.19.14] that |N_{H}(σ)_{l′}|/|N_{H}(σ)|_{l′} is an integer, and we will show that |N_{H}(σ)|_{l} divides m_{φ_{n}}(res_{H_{n}}^{H}(χ)).

Note that φ_{n} is the character of a projective OH_{n}-module O_{φ_{n}} of rank one, and m_{φ_{n}}(res_{H_{n}}^{H}(χ)) is just the number of summands isomorphic to O_{φ_{n}} in a direct sum decomposition of res_{H_{n}}^{H}(M) into indecomposable OH_{n}-lattices. Let Q be a Sylow l-subgroup of N_{H}(σ), H′ := QH_{n} (note that Q normalizes σ and hence H_{n}), and let N be an indecomposable summand of res_{H′}^{H}(M). It suffices to show that |Q| divides the multiplicity m_{φ_{n}}(res_{H_{n}}^{H′}(θ)), where θ is the character of N, and we may assume that this multiplicity is not zero. Since N is a projective OH′-module, N is a direct summand of ind_{1}^{H′}(O) ≅ ind_{H_{n}}^{H′}(ind_{1}^{H_{n}}(O)). Since H′/H_{n} ≅ Q is an l-group, N is isomorphic to ind_{H_{n}}^{H′}(L) for some indecomposable OH_{n}-lattice L by Green’s indecomposability theorem. Mackey’s decomposition formula then yields

res_{H_{n}}^{H′}(N) ≅ ⨁_{s∈H′/H_{n}} ind_{H_{n}∩^{s}H_{n}}^{H_{n}}(res_{H_{n}∩^{s}H_{n}}^{^{s}H_{n}}(^{s}L)) ≅ ⨁_{s∈H′/H_{n}} ^{s}L.

Since O_{φ_{n}} is a summand of res_{H_{n}}^{H′}(N), and since φ_{n} is stable under H′ ≤ N_{H}(σ), we have O_{φ_{n}} ≅ L, and

res_{H_{n}}^{H′}(N) ≅ ⨁_{i=1}^{|H′/H_{n}|} O_{φ_{n}},

hence m_{φ_{n}}(res_{H_{n}}^{H′}(N)) = |H′/H_{n}| = |Q| as required.

5 Projectification and the defect of an irreducible character

The last theorem showed that a_{G}^{l′}(χ) is integral for χ ∈ P (G) and we already know that a_{G}^{l′}(χ) is not integral in general. Since we have seen that the worst denominator in the canonical induction formula of 3.1 is the Artin exponent, we may expect that the denominators occurring in a_{G}^{l′}(χ) for χ ∈ Irr(G) have some significance for χ. We already know by Proposition 4.4 that we only have to consider l-power denominators.

First we answer the question, what l-power factor is necessary to make (b_{G} ∘ a_{G}^{l′})(χ) = e_{G}^{l′} · χ integral for χ ∈ Irr(G). At least the same l-power is necessary to make a_{G}^{l′}(χ) integral, since b_{G}(R_{+}^{ab}(G)) ⊆ R(G). I’m grateful to Burkhard Külshammer who pointed out to me that the answer is given in a paper of Brauer and Feit.

5.1 Theorem (cf. [8]) For χ ∈ Irr(G) we have

l^{d(χ)} · e_{G}^{l′} · χ ∈ R(G) and l^{d(χ)−1} · e_{G}^{l′} · χ ∉ R(G),

where d(χ) is the l-defect of χ, i.e. l^{d(χ)} ·χ(1)_{l} = |G|_{l}. In particular, for n ∈ N_{0} we have l^{n} · e_{G}^{l′} ∈ R(G) if and only if l^{n} ≥ |G|_{l} (cf. Corollary 4.5).

Proof This theorem is not stated explicitly in [8]. There, the integers

a_{ij} := l^{d} · (χ_{i}, e_{G}^{l′} · χ_{j}), 1 ≤ i, j ≤ m,

are studied, where χ_{1}, …, χ_{m} are the irreducible characters in an l-block of G of defect d. It is easy to see that for χ ∈ Irr(G) the irreducible constituents of e_{G}^{l′} · χ lie in the same l-block as χ. Hence, for j ∈ {1, …, m} and k ≥ 0, we have:

l^{d−k} · e_{G}^{l′} · χ_{j} ∈ R(G) ⟺ l^{k} | a_{ij} for i = 1, …, m.  (5.1.1)

It follows from (8) and the congruence preceeding (7) in [BF] that the l-part of gcd(a_{1j}, …, a_{mj}) is given by the l-height h(χ_{j}) of χ_{j}, which is defined by d(χ_{j}) + h(χ_{j}) = d. Using (5.1.1) this proves the theorem.

The next theorem shows that the map a_{G}^{l′} is strict in the sense that it doesn’t need higher denominators than the least possible ones.

5.2 Theorem For χ ∈ Irr(G) we have

l^{d(χ)} · a_{G}^{l′}(χ) ∈ R_{+}^{ab}(G) and l^{d(χ)−1} · a_{G}^{l′}(χ) ∉ R_{+}^{ab}(G).

Proof First note that b_{G} is injective on a_{G}^{l′}(R(G)). In fact, let χ ∈ R(G) with b_{G}(a_{G}^{l′}(χ)) = 0, then χ vanishes on l′-elements by Proposition 4.3. By the explicit formula (4.1.1) for a_{G}^{l′}(χ) this implies a_{G}^{l′}(χ) = 0. Since e_{G}^{l′} is an idempotent, we have b_{G} ∘ a_{G}^{l′} ∘ b_{G} ∘ a_{G}^{l′} = b_{G} ∘ a_{G}^{l′} by Proposition 4.3, and since b_{G} is injective on a_{G}^{l′}(R(G)), it follows that

a_{G}^{l′} ∘ b_{G} ∘ a_{G}^{l′} = a_{G}^{l′}.

Now let χ ∈ Irr(G). For n ∈ N_{0} we have

a_{G}^{l′}(l^{n} · e_{G}^{l′} · χ) = l^{n} · a_{G}^{l′}(χ)

by the above equation applied to l^{n}·χ. Hence, if l^{n}·e_{G}^{l′}·χ is integral, so is l^{n}·a_{G}^{l′}(χ) by Theorem 4.7. Conversely, if l^{n}·a_{G}^{l′}(χ) is integral, so is b_{G}(l^{n}·a_{G}^{l′}(χ)) = l^{n}·e_{G}^{l′}·χ. Therefore, Theorem 5.1 yields the result.

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