Types of blocks with dihedral or quaternion defect groups

Marc Cabanes DMI-ENS,45 rue d’Ulm,75005 Paris France

Claudine Picaronny DMI-ENS,45 rue d’Ulm,75005 Paris France ENS, 61 av du Président Wilson, 94230 Cachan France

The main purpose of this paper is to prove the following

Theorem 1. If two 2-blocks B, B′ of finite groups G and G′ have the same Brauer category with the same defect group D containing a cyclic subgroup of index 2, then they have the same type.

The terminology used above is taken from [Br2]. The main step toward this theorem consists in defining a perfect isometry between B and B′. A perfect isometry may be viewed as a correspondence with signs between irreducible characters of B and B′ which essentially preserves contribution matrices (see [B2]). The existence of such an isometry has numerous consequences on the block algebras OGB, OG′B′ over a complete valuation ring O with residual field k = O/J(O) of characteristic 2. It may also result from an equivalence of their derived categories over O (see [Br2] 3 and 6)^{1}. Brauer-Olsson theorems ([B3], [O]) on generalized decomposition numbers of characters in blocks with dihedral and generalized quaternion defect provide enough information to define perfect isometries ; this will be our main source and we won’t need to use the methods of [E]. Part of the present paper shows how to restate in a compact way most of Brauer-Olsson results (see II and III below), mainly by use of the Broué-Puig ∗ construction [Br-P1].

The case where D is a generalized quaternion group and G′ = C_{G}(Z(D)) deserves special attention : when B and B′ are the principal blocks, the corresponding block algebras are equal (this is a consequence of Brauer-Suzuki theorem on groups with generalized quaternion Sylow). We show that in the general case the signs in the isotypie may be removed :

Theorem 2. If B is a 2-block of the finite group G with a generalized quaternion defect group D, if H = C_{G}(Z(D)) and b is the block of H inducing up to B, then there is an isotypie I : CF(G, B; O) → CF(H, b; O) which sends Irr(B) onto Irr(b).

The paper is organized as follows. Part I is devoted to background results and notations on local block theory, mainly via the approach of [A-Br], in order to set the main definitions of [Br2]. In part II we recall the results on fusion of subpairs for this kind of defect group. In particular we show how the theory of essential groups leads to a quick determination of Brauer-Olsson cases of fusion.

In part III we show how Brauer-Olsson results on generalized decomposition numbers may be stated in terms of the ∗ construction. This provides a precise parametrization of nonrational characters in Irr(B) by non-rational characters in Irr(D). This is then used in IV to properly define perfect isometries and check Theorem 1.

Typeset by AMS-TEX

1Using Erdmann’s classification of the corresponding modular block algebras ([E]), M. Linckelmann shows that two blocks satisfying the hypothesis of theorem 1 are derived equivalent over k when D is dihedral [Li].

Part V is devoted to the proof of Theorem 2. It consists mainly in imitating what has already been done for principal blocks (Brauer-Suzuki theorem [B1] VII, see also [D] 14) : one studies restrictions to H = C_{G}(Z(D)) of integral combinations of characters in B which are zero on G_{2′}. In our case a truncated restriction provides an isometry on those central functions (Step 1), this is due to the strong condition of control satisfied by H in G. Moreover, we have coherence (Step 2) : this isometry extends to all integral combinations of characters into a perfect isometry. The equality of all signs involved is checked by ad hoc computations mainly using sums of involutions in the group algebra.

Both authors thank Michel Broué for having provided explanations and encouragement. The results of part III were presented in the framework of a DFG program at Essen in May of 86.

I. Notations and background.

Let ℓ be a prime, G and G′ two finite groups, O a complete valuation ring of characteristic zero containing primitive |G| and |G′|-th roots of unity and having residual field k of characteristic ℓ. Let K be the fraction field of O, irreducible characters of G are considered as characters of the group algebra KG, so they are elements of CF(G; O) the set of central functions on G with values in O. We denote similarly CF(G; K) = K⊗CF(G; O); we recall its standard inner product defined by (f_{1}, f_{2})_{G} = |G|^{−1} ∑_{g∈G} f_{1}(g)f_{2}(g^{−1}).

It is well known that the elements of Irr(G)∪ Irr(G′) take values in a finite extension K′ ⊃ ℚ. We denote by Γ the Galois group of K′ over its subfield generated by roots of unity of order prime to ℓ. If σ ∈ Γ, χ ∈ Irr(G) and g ∈ G, the formula σ(χ)(g) = σ(χ(g)) defines an action of Γ. The characters of G fixed by Γ are called “ℓ-rational”.

1. Conjugation. If g ∈ G, we denote by int(g) the interior automorphism defined by int(g)(g′) = gg′g^{−1}. If H is a subgroup of G and if g, g′ are elements of G, we write g =_{H} g′ when g = int(h)(g′) for some h ∈ H.

2. Blocks and characters. Any ℓ-block B of G may be considered as a primitive idempotent in Z(OG), the center of the group algebra over O ; this induces an orthogonal projection f ↦ B.f in CF(G; O) by B.f(g) = f(Bg) (where f has been extended to OG by linearity), the image is denoted by CF(G, B; O). The projection sends CF_{ℓ′}(G; O) = {f ∈CF(G; O); f(G \ G_{ℓ′}) = 0} into itself, so we denote by CF_{ℓ′}(G, B; O) the set of elements of CF(G, B; O) which are zero outside ℓ-regular elements. We use analogous notations CF(G, B; K), CF_{ℓ′}(G; K), CF_{ℓ′}(G, B; K) for corresponding vector spaces over K. Irreducible characters in B are supposed to be characters of KG-modules, their set is denoted by Irr(B) := Irr(G)∩CF(G, B; O), with the same notation for Brauer characters IBr(B) = IBr(G)∩CF_{ℓ′}(G, B; K). One has CF_{ℓ′}(G, B; O) = O[IBr(B)] and CF(G, B; K) = K[Irr(B)].

3. Decomposition map. If x ∈ G_{ℓ}, one has the decomposition map d_{G}^{x} : CF(G; O) → CF_{ℓ′}(C_{G}(x); O) defined by d_{G}^{x}f(g) = f(xg) ; it is onto. If b is an ℓ-block of C_{G}(x), one writes d_{G}^{(x,b)}f = b.d_{G}^{x}f. We shall often abbreviate by omitting the subscript G when there is no ambiguity.

4. Subpairs. We freely use the setting of subpairs, “Brauer elements”, fusion of subpairs, conjugation families, as taken from [A-Br]. If a maximal subpair (D, b_{D}) is given in G, subpairs included in it are just indexed by the corresponding subgroups of D : one writes them (X, b_{X}). If S is a system of representatives of Brauer elements in (D, b_{D}) mod. Gconjugation, the family of maps (d^{(x,b_{x})})_{(x,b_{x})∈S} provides an isometry

(d^{(x,b_{x})})_{(x,b_{x})∈S} : CF(G, b_{1}; O) → ⨁_{(x,b_{x})∈S}^{⊥} CF_{ℓ′}(C_{G}(x), b_{x}; O).

5. Isotypies. As in [Br2], the fact that two ℓ-blocks B and B′ in G and G′ have the same Brauer category means the following : they have a common defect group D, there are maximal subpairs (D, b_{D}) ⊃ (1, B) and (D, b′_{D}) ⊃ (1, B′) in G and G′ such that for all pair X, Y of subgroups of D one has {σ ∈ Hom(X, Y); ∃g ∈ G such that g(X, b_{X})g^{−1} ⊂ (Y, b_{Y}) and σ = int(g)_{|X}} = {σ ∈ Hom(X, Y); ∃g′ ∈ G′ such that g′(X, b_{X})g′^{−1} ⊂ (Y, b_{Y}) and σ = int(g′)_{|X}}. In particular the fusion of subpairs and Brauer elements included in (D, b_{D}) (see [A-Br]) are the same in G and G′.

If B, B′ are ℓ-blocks of G, G′ with same Brauer category, let’s consider a linear bijection I : CF(G, B; K) → CF(G′, B′; K) such that

(i) ∀χ ∈ Irr(B), I(χ) or −I(χ) is in Irr(B′) (ii) ∀u ∈ D \ {1} and ∀χ,ξ ∈ Irr(B), one has (d^{(u,b′_{u})}(I(χ)), d^{(u,b′_{u})}(I(ξ)))_{C_{G′} (u)} = (d^{(u,b_{u})}χ, d^{(u,b_{u})}ξ)_{C_{G}(u)}.

Such a map is an isometry and defines a family of isometries I_{ℓ′}^{<u>} from CF_{ℓ′}(C_{G}(u), b_{u}; K) onto CF_{ℓ′}(C_{G′}(u), b′_{u}; K) by d^{(u,b′_{u})} ∘ I = I_{ℓ′}^{<u>} ∘ d^{(u,b_{u})}, so I is fusion compatible in the sense of [Br2] 4.3.

If, for all u in D \ {1}, there exists an isometry I^{<u>} from CF(C_{G}(u), b_{u}; K) onto CF(C_{G′}(u), b′_{u}; K) such that (i)_{u} ∀χ ∈ Irr(b_{u}), I(χ) or −I(χ) is in Irr(b′_{u}) (ii)_{u} I_{l′}^{<u>} ∘ d_{C_{G}(u)}^{(1,b_{u})} = d_{C_{G′} (u)}^{(1,b′_{u})} ∘ I^{<u>}, then I is a perfect isometry (in the sense of [Br2] 1.4). Moreover, if each I^{<u>} is a perfect isometry, then I is an isotypie from B to B′.

Proofs of the above are easy to derive from [Br2] 4.5 and 4.6.

6. Basic sets and contribution matrices. If (u, b_{u}) is a B-Brauer element, a basic set for b_{u} is any ℤ-basis Φ_{u} of ℤ[IBr(b_{u})] ⊂ CF_{ℓ′}(C_{G}(u), b_{u}; O). The Cartan matrix C(Φ_{u}) for this basic set is such that C(Φ_{u})^{−1} = ((ϕ, ϕ′)_{C_{G}(u)})_{ϕ,ϕ′∈Φ_{u}}.

According to [B2] 5, if χ, ξ ∈ Irr(B), one defines the “contribution of (u, b_{u}) to (χ, ξ)_{G}” as (d^{(u,b_{u})}χ, d^{(u,b_{u})}ξ)_{C_{G}(u)}. Let Δ_{u} = (n_{χϕ})_{χ∈ Irr(B), ϕ∈Φ_{u}} be the generalized decomposition matrix with respect to Φ_{u} : d^{(u,b_{u})}χ = ∑_{ϕ∈Φ_{u}} n_{χϕ}ϕ. Then Δ_{u}^{t}Δ_{u} = C(Φ_{u}) and the contributions are given by the matrix Δ_{u}C(Φ_{u})^{−1}Δ_{u}^{t}.

7. Broué-Puig ∗ construction. If (D, b_{D}) is a maximal subpair for G, a “(G, b_{D})-stable” generalized character of D is any generalized character η of D such that η(u) = η(v) each time there is g ∈ G such that (u, b_{u}) = g(v, b_{v})g^{−1} with both u, v in D. Then, if χ ∈ Irr(b_{1}), one defines the central function χ ∗ η ∈ CF(G, b_{1}; O) by d^{(u,b_{u})}(χ ∗ η) = η(u)d^{(u,b_{u})}χ. The main result in [Br-P1] is that χ ∗ η is a generalized character.

If σ ∈ Γ, then σ(χ ∗ η) = σ(χ) ∗ σ(η).

II. Fusion.

Let us recall the notion of “essential” subpair : Definition. A subpair (U, b) in G is said to be essential if, and only if, b is of defect Z(U) in C_{G}(U) and N = N_{G}(U, b)/UC_{G}(U) contains a proper subgroup M such that ℓ divides |M| and ∀g ∈ N \ M, |M ∩ M^{g}| is prime to ℓ.

The main application to the fusion of subpairs is the following description of conjugation families (see [Br1] 2.9) : if B is an ℓ-block of G, a set of B-subpairs contained in a maximal one (D, b_{D}) is a conjugation family if, and only if, it contains (D, b_{D}) and a conjugate of each essential B-subpair (for a complete study, see [L]).

From now on we assume ℓ = 2, B is a 2-block of G, (D, b_{D}) is a maximal B-subpair and D contains a cyclic subgroup C of index 2. The inspection of essential subpairs is made easy by the following elementary fact applied to subgroups of D : if U has a cyclic subgroup of index ≤ 2, then Aut(U) is a 2-group except when U is kleinian (then Aut is S_{3}) or quaternion of order 8 (then Aut is S_{4}).

Let us denote by x a generator of C, |x| its order and let y be an element of minimal order in D \ C. One denotes z = x^{|x|/2}, it is central. Then, either D is cyclic, generalized quaternion (then y is of order 4), or a semidirect product C ⋊<y> with yxy^{−1} = x (abelian), yxy^{−1} = x^{−1} (dihedral), yxy^{−1} = zx with |x| ≥ 4 (semidihedral) or yxy^{−1} = zx^{−1} with |x| ≥ 8 (quasidihedral). Note that the Klein group is considered both as abelian and dihedral. One checks easily the following : D has kleinian subgroups U with C_{D}(U) = U if, and only if, D is dihedral or quasidihedral, then they are D-conjugate to <z, y> or <z, xy>; D has quaternion subgroups of order 8 if, and only if, D is generalized quaternion or quasidihedral, then they are D-conjugate to <x^{|x|/4}, xy> or <x^{|x|/4}, y>.

One then finds a saturated system of the essential subpairs in (D, b_{D}) mod. G-conjugation as follows :

— if D is dihedral of order ≥ 8 : one takes the pairs (U, b_{U}) such that U =<z, xy> or <z, y> (both kleinian) and N_{G}(U, b_{U})/C_{G}(U) ≃ S_{3},

— if D is quaternion of order ≥ 16 : one takes the pairs (U, b_{U}) such that U =<x^{|x|/4}, xy> or <x^{|x|/4}, y> (both quaternion of order 8) and N_{G}(U, b_{U})/UC_{G}(U) ≃ S_{3},

— if D is quasidihedral of order ≥ 16 : one takes the pairs (U, b_{U}) such that U =<z, y> (kleinian) or <x^{|x|/4}, xy> (quaternion of order 8) and N_{G}(U, b_{U})/UC_{G}(U) ≃ S_{3},

— otherwise there is none.

The cases are labeled (aa) when the two subpairs listed above are essential or when |N_{G}(D)/C_{G}(D)| = 3 (which implies that D is kleinian or quaternion of order 8).

We label (ab) (resp. (ba)) the cases when only the first (resp. the second) is essential. Note that they are different only when D is quasidihedral : otherwise, one may replace y by xy.

The remaining cases for fusion are labeled (bb) (this includes D abelian non kleinian and D semidihedral).

Note that if (U, b_{U}) is essential then N_{G}(U, b_{U}) is transitive on the elements of U of given order^{2}. One then obtains easily in each case a sytem of representatives S of the Brauer elements (u, b_{u}) mod. G-conjugation. For instance, if D is dihedral, a sytem of representatives is {(u, b_{u})}_{u∈S′} where S′ is a system of representatives of C mod. inversion, plus {(y, b_{y})} in case (ab), nothing in case (aa).

2This would prove at once that the saturated system of essential subpairs given above is minimal, hence a system of representatives mod. G-conjugation.

The following is straightforward :

Proposition 0. (i) B is nilpotent if and only if the fusion falls into case (bb), (ii) if u ∈ D \ {1} and (u, b_{u}) is not conjugate to (z, b_{z}), then b_{u} is nilpotent, (iii) if D is dihedral of order ≥ 8, generalized quaternion or quasidihedral, b_{z} has a reduction mod. z denoted b̄_{z} with dihedral defect D/ <z> in C_{G}(z)/ <z>. When D is quasidihedral and case (aa) (resp. (ab)) occurs for B, then case (ab) (resp. (bb)) occurs for b_{z}. When D is dihedral, case (bb) occurs for b_{z}. Otherwise (D generalized quaternion or D quasidihedral with case (ba) or (bb)) C_{G}(z) is a B-control subgroup (see [A-Br] 4.20), so the fusion case for b_{z} is the one labeled the same for B.

III. Brauer-Olsson’s theorems and the ∗ construction.

We now return to our particular 2-blocks. We have seen above that except in the cases (aa), (ab) and (ba), B is nilpotent. Then the problem of perfect isometries and types is solved (see [Br2] 5B, in fact the blocks have the same source algebra thus are Morita equivalent by [P]). So we concentrate on the cases already studied by Brauer-Olsson where D is dihedral, generalized quaternion or quasidihedral. Then D/[D, D] is kleinian. One denotes the four linear characters of D as follows : {λ_{1}, λ_{2}, λ_{3}, λ_{4}} with λ_{1} the trivial character, Ind_{C}^{D}1 = λ_{1} + λ_{2}, λ_{3}(xy) = 1 in the dihedral case, −1 in the others. For technical reasons we denote η_{0} = −2λ_{1} + λ_{2} + λ_{3} + λ_{4} when D is kleinian or quaternion of order 8, η_{0} = −λ_{2} + λ_{3} + λ_{4} otherwise ; then it is easy to check that η_{0} is (G, b_{D})-stable in all cases for fusion.

If μ ∈ Irr(D) \ {λ_{1}, λ_{2}, λ_{3}, λ_{4}} , it is of the form μ_{λ} = Ind_{C}^{D}λ where λ ∈ Irr(C) and |λ| ≥ 4. Moreover μ_{λ} = μ_{λ′} if, and only if, λ = λ′ or λ′^{y}.

We gather next all the information we need on Irr(D). We denote by ϵ the character of C of order 2.

Characters of a dihedral group of order≥8

λ_{1} λ_{2} λ_{3} λ_{4} Ind_{C}^{D}λ_{4≤|λ|<|x|} Ind_{C}^{D}λ_{|λ|=|x|}
1  1 1 1 1  2  2
z  1 1 1 1  2  −2
y  1 −1 −1 1  0  0
xy  1 −1 1 −1  0  0
Res_{C}  1 1 ϵ ϵ λ + λ^{y} λ + λ^{y}

Characters of a generalized quaternion or quasidihedral group

λ_{1} λ_{2} λ_{3} λ_{4} Ind_{C}^{D}λ_{4≤|λ|<|x|/2} Ind_{C}^{D}λ_{|λ|=|x|/2} Ind_{C}^{D}λ_{|λ|=|x|}
1  1 1 1 1  2  2  2
z  1 1 1 1  2  2  −2
x^{|x|/4}_{if |D|≥16}  1 1 1 1  2  −2  0
y  1 −1 1 −1  0  0  0
xy  1 −1 −1 1  0  0  0
Res_{C}  1 1 ϵ ϵ  λ + λ^{y}  λ + λ^{y} λ + λ^{y}

To describe irreducible characters in B and their generalized decomposition numbers, we shall keep the same notations as in [B3] for the dihedral case and [O] 4.6 for the others. By [B3](6C,6H) and [O] 4.6, Irr(B) contains four characters χ_{1}, χ_{2}, χ_{3}, χ_{4} of height zero ; they satisfy d^{x}χ_{i} = δ_{i}ϕ_{x} with δ_{i} a sign and IBr(b_{x}) = {ϕ_{x}} ; if IBr(b_{x^{2}}) has just one element ϕ_{x^{2}} (that is |D| ≥ 8), then d^{(x^{2},b_{x^{2}} )}χ_{i} = γ_{i}ϕ_{x^{2}} where γ_{i} is a sign. The numbering satisfies : γ_{1}δ_{1} = γ_{2}δ_{2} = −γ_{3}δ_{3} = −γ_{4}δ_{4}.

III.1. Dihedral defect.

Assume D is dihedral and case (aa) or (ab).

Lemma 1. Let χ be any irreducible character of height zero in B and η, η′ any (G, b_{D})-stable generalized characters of D. Then we have

(χ ∗ η, χ ∗ η′)_{G} = \{ \begin{matrix}\frac{1}{2}(η, η′)_{C} + \frac{1}{2}η(1)η′(1), & in case (aa) \\ \frac{1}{2}(η, η′)_{C} + \frac{1}{2}(η, η′)_{<y>}, & in case (ab) \end{matrix}

where (η, η′)_{Y} denotes the inner product of restrictions to Y ⊂ D.

Proof. By the isometry in I.4 and the definition of the ∗ construction (I.7), (χ ∗ η, χ ∗ η′)_{G} is ∑_{(u,b_{u})∈S} η(u)η′(u^{−1})(d^{(u,b_{u})}χ, d^{(u,b_{u})}χ)_{C_{G}(u)}. When u ≠ 1 b_{u} is nilpotent (II.0.(i)) and the generalized decomposition number of χ is a sign, so ((d^{(u,b_{u})}χ, d^{(u,b_{u})}χ)_{C_{G}(u)} = |D(u)|^{−1} where D(u) is a defect group of b_{u}. In case (aa) one may take u ∈ C, so D(z) = D and D(u) = C for others. In case (ab) one has in addition D(y) =<z, y>. This determines all (d^{(u,b_{u})}χ, d^{(u,b_{u})}χ)_{C_{G}(u)}’s for u ≠ 1, then (d^{1}χ, d^{1}χ)_{G} = 1 − ∑_{u≠1}(d^{(u,b_{u})}χ, d^{(u,b_{u})}χ)_{C_{G}(u)} making |D|^{−1} + \frac{1}{2} in case (aa), resp. |D|^{−1} + \frac{1}{4} in case (ab). This implies the formulas of Lemma 1.

Let’s consider the following (G, b_{D})-stable generalized characters of D : if μ = Ind_{C}^{D}λ with 4 ≤ |λ| ≤ |x|, let

η_{μ} = \{ \begin{matrix}μ − λ_{2}, & if 4 ≤ |λ| < |x|, \\ μ + λ_{2} − 2.λ_{1}, & if |λ| = |x| in case (aa) , \\ μ − λ_{3} + λ_{4} − λ_{1}, & if |λ| = |x| in case (ab) . \end{matrix}

Proposition 1. There is a parametrization μ ↦ χ_{μ} of the elements of Irr(B) of height ≠ 0 by Irr(D) \ {λ_{1}, λ_{2}, λ_{3}, λ_{4}} such that

\begin{aligned}δ_{1}χ_{1} ∗ η_{0} & = −δ_{2}χ_{2} − δ_{3}χ_{3} − δ_{4}χ_{4}, \\ δ_{1}χ_{1} ∗ η_{μ} & = χ_{μ} − δ_{2}χ_{2}. \end{aligned}

In case (ab), one may assume moreover δ_{1}δ_{3} = δ_{2}δ_{4} = −1 and χ_{1} ∗ λ_{3} = χ_{3}.

Proof. We compute (χ_{1} ∗ η, χ_{1} ∗ η′)_{G} using Lemma 1 for the described (D, b_{D})-generalized characters η, η′.

χ_{1} ∗ λ_{1} χ_{1} ∗ η_{0} χ_{1} ∗ η_{μ} χ_{1} ∗ λ_{3}

only in case (ab)

χ_{1} ∗ λ_{1}  1  0  0  0
χ_{1} ∗ η_{0}  0  3  1  1
χ_{1} ∗ η_{μ′}  0  1  1 + δ_{μ,μ′}  0

χ_{1} ∗ λ_{3}

only in case (ab)

0  1  0  1

So δ_{1}χ_{1} ∗ η_{0} is a linear combination with signs of three distinct irreducible characters all different from χ_{1}. If they are not all of height zero, only one is and, since δ_{1}χ_{1}∗η_{0} is 2-rational, the other two must be the elements of F_{1}, the only class of cardinality 2 under the action of the Galois group Γ on Irr(B) (this forces |D| ≥ 16) ; in particular, the rational part of their generalized decomposition numbers on (x, b_{x}) is zero and, as η_{0}(x) = −3, this contradicts the fact that decomposition numbers of characters of height zero on (x, b_{x}) are signs (see [B3] 4C, 4E). So these three are χ_{2}, χ_{3}, χ_{4} and the study of decomposition numbers at (x, b_{x}) shows that δ_{1}χ_{1} ∗ η_{0} = − ∑_{i=2}^{4} δ_{i}χ_{i}. Assume now |D| ≥ 8. The δ_{1}χ_{1} ∗ η_{μ}’s, being of square norm 2, are each a linear combination with signs of two distinct characters. Moreover the inner products with − ∑_{i=2}^{4} δ_{i}χ_{i} is 1, so δ_{1}χ_{1} ∗ η_{μ} = ϵ_{μ}χ_{μ} − δ_{i_{0}}χ_{i_{0}} for i_{0} ∈ {2, 3, 4} and ϵ_{μ} ∈ {±1}. The mutual inner products are 1 and (δ_{1}χ_{1}∗η_{μ}+δ_{i_{0}}χ_{i_{0}})(1) = δ_{1}χ_{1}(1)+δ_{i_{0}}χ_{i_{0}}(1) has a constant sign, so one may write δ_{1}χ_{1} ∗ η_{μ} = ϵχ_{μ} − δ_{i_{0}}χ_{i_{0}} with distinct χ_{μ}’s in Irr(B) \ {χ_{1}, χ_{2}, χ_{3}, χ_{4}}. This provides a bijection μ ↦ χ_{μ} between Irr(D) \ {λ_{1}, λ_{2}, λ_{3}, λ_{4}} and the characters of height ≠ 0 by [B3] Theorem 1. When μ = Ind_{C}^{D}λ with |λ| = 4, then χ_{μ} must be the only 2rational character which is not of height zero and the results on the generalized decomposition numbers ([B3] 6C) at (x^{2}, b_{x^{2}}) readily imply ϵ = 1 and i_{0} = 2.

In case (ab), δ_{1}δ_{3} = δ_{2}δ_{4} = −1 and d^{1}χ_{1}, d^{1}χ_{2} are independent with d^{1}χ_{3} = d^{1}χ_{1} and d^{1}χ_{4} = d^{1}χ_{2} by [B3] 6H. On the other hand χ_{1} ∗ λ_{3} is orthogonal to χ_{1}, of norm 1, height zero and same d^{1} as χ_{3}, so it is χ_{3}.

III.2. Quaternion defect.

Assume D is generalized quaternion and case (ab) or (aa). We keep the notations of [O]. In particular, we take a numbering of the characters of height zero χ_{1}, χ_{2}, χ_{3}, χ_{4} satisfying [O] 4.6. We write F_{n−2} = {χ_{5}} , F_{n−1} = {χ_{6}} (characters of height n − 2 where |x| = 2^{n−1}) when they exist. Also we shall use the signs ϵ_{1}, κ, ρ defined in [O] 4.6.

Lemma 2. Let χ be any irreducible character of height zero in B and η, η′ any (G, b_{D})-stable generalized characters of D. Then we have

(χ ∗ η, χ ∗ η′)_{G} = \{ \begin{matrix}\frac{1}{2}(η, η′)_{C} + \frac{1}{2}(η, η′)_{<z>}, & in case (aa) \\ \frac{1}{2}(η, η′)_{C} + \frac{1}{2}(η, η′)_{<y>}, & in case (ab) \end{matrix}

where (η, η′)_{Y} denotes the inner product of restrictions to Y ⊂ D.

Proof. The idea of the proof is the same as for Lemma 1. To determine (χ ∗ η, χ ∗ η′)_{G}, one must moreover compute the contribution (d^{(z,b_{z})}χ, d^{(z,b_{z})}χ)_{C_{G}(z)}. It can be determined from [O] 4.6 by the formula recalled in I.6 : (d^{(z,b_{z})}χ, d^{(z,b_{z})}χ)_{C_{G}(z)} equals |D|^{−1} + \frac{1}{4} in case (aa), resp. |D|^{−1} + \frac{1}{8} in case (ab).

Let’s consider the following (G, b_{D})-stable generalized characters of D : If μ = Ind_{C}^{D}λ with 4 ≤ |λ| ≤ |x|, let

η_{μ} = \{ \begin{matrix}μ − λ_{2}, & if 4 ≤ |λ| < |x|/2, \\ μ + λ_{2} − 2λ_{1}, & if |λ| = |x|/2 in case (aa) , \\ μ + λ_{3} − λ_{4} − λ_{1}, & if |λ| = |x|/2 in case (ab) , \\ μ, & if |λ| = |x|. \end{matrix}

We set F = {μ; μ = Ind_{C}^{D}λ with 4 ≤ |λ| < |x|} and F′ = {μ; μ = Ind_{C}^{D}λ with |λ| = |x|}

Proposition 2. There is a parametrization μ ↦ χ_{μ} of characters of height 1 by Irr(D) \ {λ_{1}, λ_{2}, λ_{3}, λ_{4}} such that (notations of [O] 4.6) :

δ_{1}χ_{1} ∗ η_{0} = −δ_{2}χ_{2} − δ_{3}χ_{3} − δ_{4}χ_{4},
δ_{1}χ_{1} ∗ η_{μ} = χ_{μ} − δ_{2}χ_{2}, if μ ∈ F
δ_{1}χ_{1} ∗ η_{μ} = \{ \begin{matrix}χ_{μ} + ϵ_{1}κχ_{5} + ϵ_{1}ρχ_{6} & in case (aa) \\ χ_{μ} − ϵ_{1}κχ_{5} & in case (ab) \end{matrix} if μ ∈ F′

In case (ab), one may assume δ_{1}δ_{4} = δ_{2}δ_{3} = −1 and χ_{1} ∗ λ_{4} = χ_{4}.

Proof. As in Proposition 1, we compute (δ_{1}χ_{1} ∗ η, δ_{1}χ_{1} ∗ η′)_{G} by use of Lemma 2. The results are in {0, 1, 2, 3} and are precisely the inner products of the expressions given in the statement.

The proof of the equalities of the proposition then goes as in Proposition 1, making use of [O] 4.6 to recognize classes of characters under the action of the Galois group Γ and to check the decomposition numbers at x, x^{2} and z ; it should be noted here that in the table of [O] 4.6 for the decomposition numbers at z, when l(b_{z}) = 3, the rows corresponding to the characters of height zero should be multiplied by -1 (otherwise the first column would not be orthogonal to a_{0}^{(n−2)}).

III.3. Quasidihedral defect.

Assume D is quasidihedral and case (aa), (ab) or (ba). Keeping as in III.2 the numbering of characters of height zero and the notations of [O] 4.6 , we write F_{n−2} = {χ_{5}} (character of height n − 2 where |x| = 2^{n−1}).

As in the dihedral and quaternion cases, one proves the following :

Lemma 3. Let χ be any irreducible character of height zero in B and η, η′ any (G, b_{D})-stable generalized characters of D. Then we have

(χ ∗ η, χ ∗ η′)_{G} = \{ \begin{matrix}\frac{1}{2}(η, η′)_{C} + \frac{3}{8}η(1)η′(1) + \frac{1}{8}η(z)η′(z), & in case (aa) \\ \frac{1}{2}(η, η′)_{C} + \frac{1}{4}η(1)η′(1) + \frac{1}{4}η(xy)η′(xy), & in case (ab) \\ \frac{1}{2}(η, η′)_{C} + \frac{1}{2}(η, η′)_{<z,y>}, & in case (ba) \end{matrix}

where (η, η′)_{Y} denotes the inner product of restrictions to Y ⊂ D.

Consider the following (G, b_{D})-stable generalized characters of D : If μ = Ind_{C}^{D}λ with 4 ≤ |λ| ≤ |x|, let

η_{μ} = \{ \begin{matrix}μ − λ_{2}, & if 4 ≤ |λ| < |x|/2 Êor if |λ| = |x|/2 Êin case (ab), \\ μ + λ_{3} − λ_{4} − λ_{1}, & if |λ| = |x|/2 in cases (aa) and (ba), \\ μ + λ_{4} − λ_{3} − λ_{1}, & if |λ| = |x| in case (ab), \\ μ + λ_{2} − λ_{3} − λ_{1}, & if |λ| = |x| in case (aa), \\ μ, & if |λ| = |x| in case (ba) . \end{matrix}

Lert F = {μ; μ is in one of the above first three cases} and F′ be the remaining cases, that is F′ = {μ; μ = Ind_{C}^{D}λ with |λ| = |x| in case (aa) or (ba)}

Proposition 3. There is a parametrization of characters of height 1 by Irr(D) \ {λ_{1}, λ_{2}, λ_{3}, λ_{4}} such that (notations of [O] 4.6):

δ_{1}χ_{1} ∗ η_{0} = − δ_{2}χ_{2} − δ_{3}χ_{3} − δ_{4}χ_{4},
δ_{1}χ_{1} ∗ η_{μ} =χ_{μ} − δ_{2}χ_{2} if μ ∈ F,
δ_{1}χ_{1} ∗ η_{μ} =\{ \begin{matrix}χ_{μ} + δ_{4}χ_{4} − ϵ_{1}κχ_{5} & in case (aa), \\ χ_{μ} − ϵ_{1}κχ_{5} & in case (ba), \end{matrix} if μ ∈ F′.

In case (ab) (resp. (ba)), one may assume δ_{1}δ_{4} = δ_{2}δ_{3} = −1 and χ_{1} ∗ λ_{3} = χ_{4} (resp. χ_{1} ∗ λ_{4} = χ_{4}).

Proofs. The inner products (δ_{1}χ_{1} ∗ η_{i}, δ_{1}χ_{1} ∗ η_{j})_{G} for i, j ∈ {μ} ∪ {0} can be computed using Lemma 3. The outcomes are in {−1, 0, 1, 2, 3} and coincide with the inner products of the expressions given in the statement.

The proof of the equalities then goes as in Proposition 1, making use of [O] 4.6 to recognize classes of characters under the action of the Galois group and to check the decomposition numbers at x, x^{2} and z.

III.4. The case when z is central (|D| ≥ 8).

The following proposition is used in IV.2 and V. We assume one of the cases described in 1, 2, 3 occurs and we keep the same notation. We consider b_{z} and determine the characters with z in their kernel.

Proposition 4. If z ∈ Z(G) then {χ | χ ∈ Irr(b_{z}), χ(z) = χ(1)} = {χ_{i}, χ_{μ} | i = 1, 2, 3, 4, μ = Ind_{C}^{D}λ, 4 ≤ |λ| ≤ |x|/2} with notations of III.1,2,3.

Proof. The block b̄_{z} of C_{G}(z)/ <z>, being of dihedral defect (II.0.(iii)), has four characters of height zero. If one makes them into characters of C_{G}(z) with z in their kernel, they are in b_{z} with same degrees, so they remain of height zero. Since there are four of them, they are all the characters of height zero in b_{z}.

Now let θ the endomorphism of CF(G; O) defined by θ(f)(h) = f(zh). If χ ∈ Irr(b_{z}) then θ(χ) ∈ {±χ}. So θ(f) = f if and only if (f, χ)_{G} = 0 for each χ such that θ(χ) = −χ. This implies that the components of χ_{1} ∗ η_{μ} satisfy θ(χ) = χ when μ = Ind_{C}^{D}λ with 4 ≤ |λ| ≤ |x|/2 and θ(χ) = −χ when |λ| = |x|. This finishes the proof by Propositions 1,2,3 above.

IV. Proof of Theorem 1.

We now prove Theorem 1. Let B, B′ be as in the hypotheses of the theorem, with characters χ, χ′ respectively. We keep the notations of the preceding section except that we put a prime ′ on each character or sign for B′. We choose for B, B′, b̄_{z}, b̄′_{z} a parametrization of irreducible characters satisfying Propositions 1, 2, 3.

IV.1. Fusion compatible isometries. Let

I : CF(G, B; K) → CF(G′, B′; K)

defined by I(δ_{i}χ_{i}) = δ′_{i}χ′_{i} for i = 1, 2, 3, 4, I(ϵ_{1}κχ_{5}) = ϵ′_{1}κ′χ′_{5}, I(ϵ_{1}ρχ_{6}) = ϵ′_{1}ρ′χ′_{6} when they exist and I(χ_{μ}) = χ′_{μ} for each μ ∈ Irr(D) \ {λ_{1}, λ_{2}, λ_{3}, λ_{4}}.

Let us show that I is a fusion compatible isometry. We must check the equalities of the inner products (ii) given in I.5, or equivalently that the matrix of mutual inner products of the d^{(u,b_{u})}’s of δ_{1}χ_{1}, δ_{2}χ_{2}, δ_{3}χ_{3}, δ_{4}χ_{4}, (χ_{μ})_{μ∈Irr(D)\{λ_{1},λ_{2},λ_{3},λ_{4}}}, ϵ_{1}κχ_{5}, ϵ_{1}ρχ_{6} is the same in G and G′. Let Δ_{u} be the decomposition matrix of δ_{1}χ_{1}, δ_{2}χ_{2}, δ_{3}χ_{3}, δ_{4}χ_{4}, (χ_{μ})_{μ∈Irr(D)\{λ_{1},λ_{2},λ_{3},λ_{4}}}, ϵ_{1}κχ_{5}, ϵ_{1}ρχ_{6} at (u, b_{u}) with respect to a basic set. Then, the Cartan matrix of this basic set is still equal to Δ_{u}^{t}Δ_{u} since changing the rows of Δ_{u} by signs does not affect Δ_{u}^{t}Δ_{u}. So, the matrix we seek is Δ_{u}(Δ_{u}^{t}Δ_{u})^{−1}Δ_{u}^{t}. It is clear from [B3] 6C, 6H and [O] 4.6, 4.8 that the generalized decomposition matrices of δ_{1}χ_{1}, δ_{2}χ_{2}, δ_{3}χ_{3}, δ_{4}χ_{4}, ϵ_{1}κχ_{5}, ϵ_{1}ρχ_{6} and δ′_{1}χ′_{1}, δ′_{2}χ′_{2}, δ′_{3}χ′_{3}, δ′_{4}χ′_{4}, ϵ′_{1}κ′χ′_{5}, ϵ′_{1}ρ′χ′_{6} with respect to a suitable basic set are the same up to a sign (this sign is ϵ_{m}ϵ′_{m} when u ∈ C is of order 2^{m}, otherwise it is 1). It is also the case at the row μ by the formulas of Propositions 1, 2, 3. This gives Δ_{u}(Δ_{u}^{t}Δ_{u})^{−1}Δ_{u}^{t} = Δ′_{u}(Δ′_{u}^{t}Δ′_{u})^{−1}Δ′_{u}^{t} as required.

IV.2. Isotypies. Let’s show how to extend those isometries into isotypies. As said in I.5 we have to extend each I_{2′}^{<u>}: CF_{2′}(C_{G}(u), b_{u}; K) → CF_{2′}(C_{G′}(u), b′_{u}; K). If b_{u} is nilpotent, then CF_{2′}(C_{G}(u), b_{u}; O) = Oϕ_{u} where {ϕ_{u}} = IBr(b_{u}) (see [Br-P]). One has d^{(u,b_{u})}χ_{i} = ±ϕ_{u} for i = 1, 2, 3, 4, so I_{2′}^{<u>}(ϕ_{u}) = ϵ_{u}ϕ′_{u} with ϵ_{u} ∈ {±1}. Choose any χ (resp. χ′) of height zero in Irr(b_{u}) (resp. Irr(b′_{u})), then ϕ_{u} = d^{1}χ. Thus I^{<u>} defined by I^{<u>}(χ ∗ η) = ϵ_{u}χ′ ∗ η extends I_{2′}^{<u>} and it is a fusion compatible perfect isometry (see [Br2] 5B) satisfying (i)_{u} and (ii)_{u} of I.5.

If b_{u} is not nilpotent with u ≠ 1, then u = z and D is generalized quaternion with fusion (aa) or (ab), or quasidihedral with fusion (aa) or (ba), b_{z} has defect D and |D| ≥ 8 (Proposition 0). Let (χ̃_{i})_{i=1,...,6}, (χ̃_{μ})_{μ} be the elements of Irr(b_{z}) in a numbering satisfying Proposition 2 or 3, with associated signs (δ̃_{i})_{i=1,2,3,4}, ϵ̃_{1}, κ̃, ρ̃. When reducing mod. <z> one gets a block b̄_{z} of C_{G}(z)/ <z> with dihedral defect D/ <z>. The characters χ̃_{1}, χ̃_{2}, χ̃_{3}, χ̃_{4} have z in their kernel and one denotes by χ̄_{1}, χ̄_{2}, χ̄_{3}, χ̄_{4} the corresponding characters in b̄_{z} (Proposition 4). Then Proposition 1 is satisfied since the numbering of the characters of height zero is determined by the products of generalized decomposition numbers at x and x^{2} and those are preserved. Their associated signs are δ̄_{i} = δ̃_{i}.

Now let’s take the following basic sets for b̄_{z} in C_{G}(z)/<z>:

ϕ_{1} = d^{1}(δ̄_{1}χ̄_{1} + δ̄_{2}χ̄_{2}), ϕ_{2} = −d^{1}(δ̄_{2}χ̄_{2}), when |IBr(b_{z})| = 2,

ϕ_{1} = d^{1}(δ̄_{1}χ̄_{1} + δ̄_{2}χ̄_{2}), ϕ_{2} = −d^{1}(δ̄_{4}χ̄_{4}), ϕ_{3} = −d^{1}(δ̄_{1}χ̄_{1}), when |IBr(b_{z})| = 3 and |D| = 8, ϕ_{1} = d^{1}(δ̄_{1}χ̄_{1} + δ̄_{2}χ̄_{2}), ϕ_{2} = −d^{1}(δ̄_{3}χ̄_{3}), ϕ_{3} = −d^{1}(δ̄_{1}χ̄_{1}), when |IBr(b_{z})| = 3 and |D| ≥ 16. The relations of III.1 for C_{G}(z)/ <z> show that we have defined basic sets and allow to

compute the decomposition and Cartan matrices. One finds the Cartan matrix ( \begin{matrix}\frac{|D|}{8} + 1 & 2 \\ 2 & 4 \end{matrix} )

or ( \begin{matrix}\frac{|D|}{8} + 1 & −1 & 1 \\ −1 & 2 & 0 \\ 1 & 0 & 2 \end{matrix} ) for |IBr(b_{z})| = 2 or 3 respectively. So they are twice the above when

(ϕ_{i})_{i} is considered as basic set of b_{z} in CF(C_{G}(z), b_{z}; O). Thus we obtain the Cartan matrices used in [O] p. 227 and 229. Olsson proved that if the Cartan matrix is as above, then the generalized decomposition matrix at z of χ_{1}, χ_{2}, χ_{3}, χ_{4}, (χ_{μ})_{μ}, χ_{5}, χ_{6} is one of the following :

( \begin{matrix}δ_{1}ϵ_{1} & 0 & 0 & −δ_{4}ϵ_{1} & ϵ_{1} & … & ϵ_{1} & −ϵ_{1} & … & −ϵ_{1} & κ \\ δ_{1}ϵ_{1} & −δ_{2}ϵ_{1} & δ_{3}ϵ_{1} & −δ_{4}ϵ_{1} & 0 & … & 0 & 0 & … & 0 & 2κ \end{matrix} )^{t} ,

( \begin{matrix}0 & δ_{2}ϵ_{1} & 0 & −δ_{4}ϵ_{1} & ϵ_{1} & … & ϵ_{1} & −ϵ_{1} & … & −ϵ_{1} & κ & 0 \\ 0 & 0 & −δ_{3}ϵ_{1} & δ_{4}ϵ_{1} & 0 & … & 0 & 0 & … & 0 & −κ & ρ \\ −δ_{1}ϵ_{1} & δ_{2}ϵ_{1} & 0 & 0 & 0 & … & 0 & 0 & … & 0 & κ & ρ \end{matrix} )^{t} ,

( \begin{matrix}0 & δ_{2}ϵ_{1} & 0 & −δ_{4}ϵ_{1} & ϵ_{1} & … & ϵ_{1} & −ϵ_{1} & … & −ϵ_{1} & κ & 0 \\ δ_{1}ϵ_{1} & −δ_{2}ϵ_{1} & 0 & 0 & 0 & … & 0 & 0 & … & 0 & −κ & −ρ \\ 0 & 0 & δ_{3}ϵ_{1} & −δ_{4}ϵ_{1} & 0 & … & 0 & 0 & … & 0 & κ & −ρ \end{matrix} )^{t} ,

with ϵ_{1} written \frac{|D|}{8} − 1 times and −ϵ_{1} written \frac{|D|}{8} times.

When |IBr(b_{z})| = |IBr(b′_{z})| = 2, this implies that d^{z}(δ_{1}χ_{1}) = ϵ_{1}(ϕ_{1}+ϕ_{2}), d^{z}(δ_{2}χ_{2}) = −ϵ_{1}ϕ_{2}. So, the isometry I described in 1 above thus satisfies I_{2′}^{<z>}(ϵ_{1}ϕ_{i}) = ϵ′_{1}ϕ′_{i} for i = 1, 2. If the generalized decomposition matrix of Irr(B) and Irr(B′) correspond to the same of the above last two cases, then similarly I_{2′}^{<z>}(ϵ_{1}ϕ_{i}) = ϵ′_{1}ϕ′_{i} for i = 1, 2, 3. Otherwise I_{2′}^{<z>}(ϵ_{1}ϕ_{1}) = −ϵ′_{1}ϕ′_{1}, I_{2′}^{<z>}(ϵ_{1}ϕ_{2}) = ϵ′_{1}ϕ′_{3}, I_{2′}^{<z>}(ϵ_{1}ϕ_{3}) = ϵ′_{1}ϕ′_{2}.

When I_{2′}^{<z>}(ϵ_{1}ϕ_{1}) = ϵ′_{1}ϕ′_{1}, define I^{<z>} : CF(C_{G}(z), b_{z}; K) → CF(C_{G′}(z), b′_{z}; K) by I^{<z>}(δ̃_{i}χ̃_{i}) = ϵ_{1}ϵ′_{1}δ̃′_{i}χ̃′_{i} for i = 1, 2, 3, 4, I^{<z>}(χ̃_{μ}) = ϵ_{1}ϵ′_{1}χ̃′_{μ}, I^{<z>}(ϵ̃_{1}κ̃χ̃_{5}) = ϵ_{1}ϵ′_{1}ϵ̃′_{1}κ̃′χ̃′_{5}, I^{<z>}(ϵ̃_{1}ρ̃χ̃_{6}) = ϵ_{1}ϵ′_{1}ϵ̃′_{1}ρ̃′χ̃′_{6}. Then IV.1 above tells us that I^{<z>} extends I_{2′}^{<z>} into a fusion compatible isometry. Then, by I.5, I is perfect and so is I^{<z>}. Thus I is an isotypie and this completes the proof of Theorem 1.

When I_{2′}^{<z>}(ϵ_{1}ϕ_{1}) = −ϵ′_{1}ϕ′_{1}, it suffices to compose the map I^{<z>} defined above with a perfect isometry σ :CF(C_{G}(z), b_{z}; O) → CF(C_{G}(z), b_{z}; O) such that σ^{2} = Id, σ(δ̃_{1}χ̃_{1}) = δ̃_{4}χ̃_{4} and σ(δ̃_{2}χ̃_{2}) = δ̃_{3}χ̃_{3} if |D| = 8, resp. σ(δ̃_{1}χ̃_{1}) = δ̃_{3}χ̃_{3} and σ(δ̃_{2}χ̃_{2}) = δ̃_{4}χ̃_{4} if |D| ≥ 16. This clearly implies σ(ϕ_{2}) = ϕ_{3}, σ(ϕ_{3}) = ϕ_{2} and also σ(ϕ_{1}) = −ϕ_{1} since ϕ_{1} = d^{1}(δ̃_{1}χ̃_{1} + δ̃_{2}χ̃_{2}) = −d^{1}(δ̃_{3}χ̃_{3} + δ̃_{4}χ̃_{4}). The existence of such an isometry is checked as in IV.1.

Remark : In the generalized quaternion case (cf. Proposition 0 (iii), this includes the above case when |IBr(b_{z})| = 3) another proof is as follows. We prove independently in V below that there is an isotypie between CF(G, B; O) and CF(C_{G}(z), b_{z}; O). So it remains to find an isotypie between CF(C_{G}(z), b_{z}; O) and CF(C_{G′}(z), b′_{z}; O). But in this case of a central z, it is easily checked that the isometry I of IV.1 is an isotypie : the Brauer elements (u, b_{u}) with u ≠ 1, z still give no trouble while on the other hand I_{2′}^{<z>} = I_{2′}^{1} which is extended by I.

V. The quaternion case.

From now on assume that B is a 2-block of G with generalized quaternion defect group D and set (D, b_{D}) ⊃ (Z(D), b_{z}) ⊃ ({1}, B). We write H = C_{G}(z).

Let us define a linear map :

R : CF(G, B; O) → CF(H, b_{z}; O)
f ↦ b_{z}. Res_{H}^{G}(f).

Let CF_{0}(G) (resp. CF_{0}(H)) denote the subspace of CF(G, B; O) (resp. CF(H, b_{z}; O)) equal to ker d^{1}. Then CF(G, B; O) = CF_{2′}(G, B; O)⊕^{⊥} CF_{0}(G) and CF(H, b_{z}; O) = CF_{2′}(H, b_{z}; O)⊕^{⊥} CF_{0}(H) (see I.4). Moreover, one has R(CF_{2′}(G, B; O)) ⊂ CF_{2′}(H, b_{z}; O) and R(CF_{0}(G)) ⊂ CF_{0}(H) (see I.2).

We will prove the following additional properties.

Step 1. If f ∈ CF(G, B; O) and f_{0} ∈ CF_{0}(G), then (f, f_{0})_{G} = (R(f), R(f_{0}))_{H}. Thus R induces an isometry from CF_{0}(G) ∩ ℤ[Irr(B)] onto CF_{0}(H) ∩ ℤ[Irr(b_{z})] whose inverse map coincides with B.Ind_{H}^{G}.

Step 2. (coherence) There exists an isotypie R̃ : CF(G, B; O) → CF(H, b_{z}; O) which coincides with R on CF_{0}(G).

Step 3. R̃(Irr(B)) = Irr(b_{z}) or −Irr(b_{z}).

It is obvious that Theorem 2 follows from 2 and 3 above.

The following shows that the hypothesis on D and fusion of subpairs is a bit stronger than control by the subgroup H. We denote D^{#} = {u ∈ D / z ∈<u>} = D \ {1} and H^{#} = {h ∈ H / h_{2} ∈ D^{#}}.

Lemma 4. If u ∈ D^{#}, f ∈ CF(G, B; O) and h ∈ H^{#}, then (i) the block b_{u} is the same for G and H, (ii) d^{(u,b_{u})}f = d^{(u,b_{u})}Rf over C_{G}(u) = C_{H}(u), (iii) f(h) = Rf(h).

Proof. We first show that if b is a block of C_{G}(u) = C_{H}(u) then the inclusion (1, B) ⊂ (<u>, b) in G is equivalent to the inclusion (1, b_{z}) ⊂ (<u>, b) in H. This proves (i) at once. Since < z >⊂< u >, we only need to show this for u = z. So let b be a block of H such that (1, B) ⊂ (<z>, b), then there exists g ∈ G such that g(<z>, b)g^{−1} ⊂ (D, b_{D}). But z is the only involution in D, so g ∈ H and b = b_{z}. Clearly, (i) implies (ii) which implies (iii).

We now check Step 1.

By I.4, one has (f, f_{0})_{G} = ∑_{(u,b_{u})∈S}(d^{(u,b_{u})}f, d^{(u,b_{u})}f_{0})_{C_{G}(u)} for S a system of representatives of Brauer elements (u, b_{u}) in (D, b_{D}) mod. G-conjugacy. One has d^{(1,B)}f_{0} = 0 and d^{(1,b_{z})}Rf_{0} = 0. Then (f, f_{0})_{G} = ∑_{(u,b_{u})∈S}(d^{(u,b_{u})}f, d^{(u,b_{u})}f_{0})_{C_{H}(u)} = (Rf, Rf_{0})_{G} since S is a system of representatives for subpairs of H in (D, b_{D}) by Lemma 4 (i) and Proposition 0 (iii).

Then R induces an isometry from CF_{0}(G) into CF_{0}(H).

The map I = B.Ind_{H}^{G} : CF(H, b_{z}; O) →CF(G, B; O) is adjoint to R. Then I ∘ R fixes each element of CF_{0}(G) since (I ∘ R(f_{0}), f)_{G} = (R(f_{0}), R(f))_{H} = (f_{0}, f)_{G}. On the other hand rk_{O}CF_{0}(G) = ∑_{(u,b_{u})∈S,u≠1} |IBr(b_{u})| = rk_{O}CF_{0}(H). This implies that R( CF_{0}(G)) = CF_{0}(H), I(CF_{0}(H)) = CF_{0}(G), and R, I give inverse isometries on those spaces. They also give rise to inverse isometries on generalized characters in CF_{0}’s since R and I clearly preserve characters. This establishes Step 1.

We now turn to Step 2. We use the notations of III.2 for the elements of Irr(b_{z}) and associated signs : (χ_{i})_{i=1,2,3,4,5,6}, (χ_{μ})_{μ∈ Irr(D)\{1,λ_{2},λ_{3},λ_{4}}} are the characters of b_{z} (in IV.2, they were denoted by χ̃ to avoid confusion with Irr(B)).

In case (aa), the following are generalized characters in CF_{0}(H) : δ_{1}χ_{1}+δ_{2}χ_{2}+δ_{3}χ_{3}+δ_{4}χ_{4}, χ_{μ} − δ_{1}χ_{1} − δ_{2}χ_{2} for μ ∈ Irr(D) \ {λ_{1}, λ_{2}, λ_{3}, λ_{4}}, ϵ_{1}κχ_{5} − δ_{1}χ_{1} − δ_{4}χ_{4}, ϵ_{1}ρχ_{6} − δ_{1}χ_{1} − δ_{3}χ_{3}. Concerning the last two this comes from the decompositon at z (see [O] 4.6) and Proposition 4. The others come from Proposition 2. In case (ab), one checks similarly that the following are in CF_{0}(H) : δ_{1}χ_{1} + δ_{4}χ_{4}, δ_{2}χ_{2} + δ_{3}χ_{3}, χ_{μ} − δ_{1}χ_{1} − δ_{2}χ_{2} for μ ∈ Irr(D) \ {λ_{1}, λ_{2}, λ_{3}, λ_{4}}, ϵ_{1}κχ_{5} + δ_{1}χ_{1} − δ_{2}χ_{2}. Let us consider the images by I :

Lemma 5. In case (aa) there are ψ_{1}, ψ_{2}, ψ_{3}, ψ_{4}, ψ_{5}, ψ_{6}, (ψ_{μ})_{μ∈Irr(D)\{λ_{1},λ_{2},λ_{3},λ_{4}}} in Irr(B)∪ −Irr(B) and corresponding to distinct characters, such that :

\begin{aligned}I(δ_{1}χ_{1} + δ_{2}χ_{2} + δ_{3}χ_{3} + δ_{4}χ_{4}) & = ψ_{1} + ψ_{2} + ψ_{3} + ψ_{4}, \\ I(χ_{μ} − δ_{1}χ_{1} − δ_{2}χ_{2}) & = ψ_{μ} − ψ_{1} − ψ_{2}, \\ I(ϵ_{1}κχ_{5} − δ_{1}χ_{1} − δ_{4}χ_{4}) & = ψ_{5} − ψ_{1} − ψ_{4}, \\ I(ϵ_{1}ρχ_{6} − δ_{1}χ_{1} − δ_{3}χ_{3}) & = ψ_{6} − ψ_{1} − ψ_{3}. \end{aligned}

Lemma 6. In case (ab) there are ψ_{1}, ψ_{2}, ψ_{3}, ψ_{4}, ψ_{5}, (ψ_{μ})_{μ∈Irr(D)\{λ_{1},λ_{2},λ_{3},λ_{4}}} in Irr(B) ∪ −Irr(B) and corresponding to distinct characters, such that :

\begin{aligned}I(δ_{1}χ_{1} + δ_{4}χ_{4}) & = ψ_{1} + ψ_{4}, \\ I(δ_{2}χ_{2} + δ_{3}χ_{3}) & = ψ_{2} + ψ_{3}, \\ I(χ_{μ} − δ_{1}χ_{1} − δ_{2}χ_{2}) & = ψ_{μ} − ψ_{1} − ψ_{2}, \\ I(ϵ_{1}κχ_{5} + δ_{1}χ_{1} − δ_{2}χ_{2}) & = ψ_{5} + ψ_{1} − ψ_{2}. \end{aligned}

Proofs. The proofs are very similar to what was done in III : the inner products of the results are known since I is an isometry. Let’s take case (aa). Then I(δ_{1}χ_{1} + δ_{2}χ_{2} + δ_{3}χ_{3} + δ_{4}χ_{4}) is a generalized character of square norm 4. Its value at 1 is 0, so it cannot be ± twice a character, hence I(δ_{1}χ_{1} + δ_{2}χ_{2} + δ_{3}χ_{3} + δ_{4}χ_{4}) is of the form announced in the lemma. The other images have square norm 3 and inner product -2 with ψ_{1} + ψ_{2} + ψ_{3} + ψ_{4}. So there exist ψ_{μ}, ψ_{5}, ψ_{6} in ±Irr(B) such that I(χ_{μ} − δ_{1}χ_{1} − δ_{2}χ_{2}) − ψ_{μ}, I(ϵ_{1}κχ_{5} − δ_{1}χ_{1} − δ_{4}χ_{4}) − ψ_{5}, I(ϵ_{1}ρχ_{6} − δ_{1}χ_{1} − δ_{3}χ_{3}) − ψ_{6} are each a sum of two elements in {−ψ_{1}, −ψ_{2}, −ψ_{3}, −ψ_{4}} with mutual inner products 1 or 2. Numbering ψ_{1}, ψ_{2}, ψ_{3}, ψ_{4} such that the one common to I(ϵ_{1}κχ_{5} − δ_{1}χ_{1} − δ_{4}χ_{4}) − ψ_{5} and I(ϵ_{1}ρχ_{6} − δ_{1}χ_{1} − δ_{3}χ_{3}) − ψ_{6} is −ψ_{1}, then −ψ_{4} in I(ϵ_{1}κχ_{5} − δ_{1}χ_{1} − δ_{4}χ_{4}) − ψ_{5} and −ψ_{3} in I(ϵ_{1}ρχ_{6} − δ_{1}χ_{1} − δ_{3}χ_{3}) − ψ_{6}, we obtain the desired result.

The case (ab) goes along the same line.

Definition 7. Let R̃ : CF(G, B; K) → CF(H, b_{z}; K) defined by ψ_{i} ↦ δ_{i}χ_{i} for i = 1, 2, 3, 4, ψ_{μ} ↦ χ_{μ} for μ ∈ Irr(D) \ {λ_{1}, λ_{2}, λ_{3}, λ_{4}}. ψ_{5} ↦ ϵ_{1}κχ_{5}, ψ_{6} ↦ ϵ_{1}ρχ_{6}.

Then we have

Lemma 8. If f ∈ CF(G, B; K) and h ∈ H^{#} then R̃(f)(h) = R(f)(h).

Proof. The generalized characters of CF_{0}(G) considered in Lemma 5 (resp. Lemma 6) form a free system of cardinality |Irr(B)| − 3 = rk_{O}CF_{0}(G) (resp. |Irr(B)| − 2 = rk_{O}CF_{0}(G)). Moreover I ∘ R̃ is the identity map on them, so R̃ coincides with R on CF_{0}(G) . Then R̃(f) − R(f) ∈ CF_{0}(H)^{⊥} = CF_{2′}(H, b_{z}; O) : if f_{0} ∈ CF_{0}(G), Rf_{0} = R̃f_{0} and (R̃f − Rf, Rf_{0})_{H} = (R̃f, R̃f_{0})_{H} −(Rf, Rf_{0})_{H} = (f, f_{0})_{G} −(f, f_{0})_{G} = 0 by Step 1 and the fact that R̃ is an isometry. So R̃(f)(h) = R(f)(h) if h ∈ H \ H_{2′}.

Let’s check Step 2. R̃ is well defined and provides a correspondence with signs which bijects Irr(B) and Irr(b_{z}) since they have the same cardinality. Now let us verify the inner products of I.5.(ii). If u ∈ D \ {1} = D^{#}, then C_{G}(u) ⊂ H. Thus we must check (d^{(u,b_{u})}R̃(ψ_{i}), d^{(u,b_{u})}R̃(ψ_{j}))_{C_{H}(u)} = (d^{(u,b_{u})}ψ_{i}, d^{(u,b_{u})}ψ_{j})_{C_{H}(u)}. This clearly follows from the above Lemma and 4 (iii). So R̃ satisfies (i) and (ii) of I.5. For all u ∈ D^{#}, R̃_{ℓ′}^{<u>} is the identity map on CF_{ℓ′}(C_{H}(u), b_{u}; K) by Lemmas 4 (ii) and 8. This implies that R̃ is an isotypie : take R̃^{<u>} to be the identity map on CF(C_{H}(u), b_{u}; K) ([Br2] 4.5, 4.6).

There remains Step 3. Let Σ = (∑_{g∈G} gzg^{−1})^{2} ∈ Z(O[G]). The main tool to prove Step 3 consists in computing f_{0}(Σ) for adequate f_{0}’s in CF_{0}(G).

Lemma 9. If χ ∈ Irr(B), then χ(Σ) = |G|^{2} \frac{χ(z)^{2}}{χ(1)} . If f_{0} ∈ CF_{0}(G), then f_{0}(Σ) = 0.

Proof. The equality χ(Σ) = |G|^{2} \frac{χ(z)^{2}}{χ(1)} is a consequence of Schur’s Lemma : ∑_{g∈G} gzg^{−1} ∈

Z(OG), so it acts by a scalar on the representation space of χ. This scalar equals \frac{χ(z)}{χ(1)}|G|.

So Σ acts by its square and χ(Σ) = |G|^{2} \frac{χ(z)^{2}}{χ(1)} as claimed.

It remains to check that, f_{0}(Σ) = 0. Let T_{G}(z, H) = {g ∈ G / gzg^{−1} ∈ H} and Σ′ = (∑_{g∈T_{G}(z,H)} gzg^{−1})^{2} ∈ Z(O[H]). We first show that f_{0}(Σ) = (G : H)Rf_{0}(Σ′). Let H ⊂ H^{#} be a system of representatives of the G-conjugacy classes of elements of G whose 2-parts are conjugate to some u in D^{#}. Note that any h, h′ in H^{#} which are G-conjugate are in fact H-conjugate as z is the only involution of <h> and <h′ >. So H is also a system of representatives of H^{#} mod. H-conjugacy. As f_{0} ∈ CF_{0}(G) and Rf_{0} ∈ CF_{0}(H), we have f_{0}(Σ) = ∑_{h∈H} f_{0}(h)|{(g, g′) ∈ G × G / gzg^{−1}g′zg′^{−1} =_{G} h}| and Rf_{0}(Σ′) = ∑_{h∈H} Rf_{0}(h)|{(g, g′) ∈ T_{G}(z, H) × T_{G}(z, H) / gzg^{−1}g′zg′^{−1} =_{H} h}|. For h ∈ H, f_{0}(h) = Rf_{0}(h) by Lemma 4 (iii) ; moreover G (respectively H)

acts by translation on the set {(g, g′) ∈ G × G / gzg^{−1}g′zg′^{−1} =_{G} h} (respectively {(g, g′) ∈ T_{G}(z, H) × T_{G}(z, H) / gzg^{−1}g′zg′^{−1} =_{H} h}) so its cardinality is (G : C_{G}(h))|{(g, g′) ∈ G × G / gzg^{−1}g′zg′^{−1} = h}| (respectively (H : C_{H}(h))|{(g, g′) ∈ T_{G}(z, H) × T_{G}(z, H) / gzg^{−1}g′zg′^{−1} = h}|). But if h = (gzg^{−1})(g′zg′^{−1}) with g, g′ ∈ G, these two involutions normalize <h> so they centralize z ; thus g, g′ are in fact in T_{G}(z, H) and, recalling C_{G}(h) = C_{H}(h), |{(g, g′) ∈ G × G / gzg^{−1}g′zg′^{−1} =_{G} h}| = (G : H)|{(g, g′) ∈ T_{G}(z, H) × T_{G}(z, H) / gzg^{−1}g′zg′^{−1} =_{H} h}|. So f_{0}(Σ) = (G : H)Rf_{0}(Σ′) as claimed.

Now it remains to show that Rf_{0}(Σ′) = 0. If χ ∈ Irr(b_{z}), χ(gzg^{−1}) = 0 when g ∉ H since such a gzg^{−1} cannot be H-conjugate to any u in D^{#}. So χ(Σ′) = ^{\frac{χ(∑_{g∈T_{G}(z,H)} gzg^{−1})^{2}}{χ(1)}} =

\frac{|H|^{2}χ(z)^{2}}{χ(1)} = |H|^{2}χ(z^{2}) = |H|^{2}χ(1). This implies R(f_{0})(Σ′) = |H|^{2}R(f_{0})(1) = 0.

Before we give the proof of Step 3, we need the following elementary argument :

Lemma 10. If a, a′, a′′, b, b′, b′′ ∈ K are such that bb′b′′ ≠ 0 and a + a′ + a′′ = b + b′ + b′′ = \frac{a^{2}}{b} + \frac{a′^{2}}{b′} + \frac{a′′^{2}}{b′′} = 0, then \frac{a}{b} = \frac{a′}{b′} = \frac{a′′}{b′′}.

Proof. (ab′ − a′b)^{2} = (b + b′)(a^{2}b′ + a′^{2}b) − (a + a′)^{2}bb′ = (−b′′)(\frac{−a′′^{2}bb′}{b′′}) − a′′^{2}bb′ = 0, so \frac{a}{b} = \frac{a′}{b′}. Then apply symmetry.

Asumme now that a sum of three characters with signs αχ_{i} + α′χ_{i′} + α′′χ_{i′′} is in CF_{0}(H) and that I(αχ_{i} + α′χ_{i′} + α′′χ_{i′′}) = ϵψ_{i} + ϵ′ψ_{i′} + ϵ′′ψ_{i′′} is one of the equalities in Lemma 5 or 6. Then αχ_{i}(1) + α′χ_{i′}(1) + α′′χ_{i′′}(1) = 0 and ϵψ_{i}(1) + ϵ′ψ_{i′}(1) + ϵ′′ψ_{i′′}(1) = 0. On the other hand, Lemma 9 tells us that ϵψ_{i} + ϵ′ψ_{i′} + ϵ′′ψ_{i′′}(Σ) = \frac{(ϵψ_{i}(z))^{2}}{ϵψ_{i}(1)} + \frac{(ϵ′ψ_{i′} (z))^{2}}{ϵ′ψ_{i′} (1)} + \frac{(ϵ′′ψ_{i′′} (z))^{2}}{ϵ′′ψ_{i′′} (1)} =

0. Then, by Lemma 8 and Proposition 4, \frac{(ϵψ_{i}(z))^{2}}{ϵψ_{i}(1)} + \frac{(ϵ′ψ_{i′} (z))^{2}}{ϵ′ψ_{i′} (1)} + \frac{(ϵ′′ψ_{i′′} (z))^{2}}{ϵ′′ψ_{i′′} (1)} = \frac{(ϵχ_{i}(z))^{2}}{ϵψ_{i}(1)} +

\frac{(ϵ′χ_{i′} (z))^{2}}{ϵ′ψ_{i′} (1)} + \frac{(ϵ′′χ_{i′′} (z))^{2}}{ϵ′′ψ_{i′′} (1)} = \frac{(αχ_{i}(1))^{2}}{ϵψ_{i}(1)} + \frac{(α′χ_{i′} (1))^{2}}{ϵ′ψ_{i′} (1)} + \frac{(α′′χ_{i′′} (1))^{2}}{ϵ′′ψ_{i′′} (1)} = 0. One then applies Lemma 10 to αχ_{i}(1), α′χ_{i′}(1), α′′χ_{i′′}(1), and ϵψ_{i}(1), ϵ′ψ_{i′}(1), ϵ′′ψ_{i′′}(1), it tells us that αϵψ_{i}(1), α′ϵ′ψ_{i′}(1), α′′ϵ′′ψ_{i′′}(1) have the same sign.

In case (aa), this provides the result we seek (Lemma 5).

In case (ab), the last two equations of Lemma 6 give the desired relation between δ_{1}ψ_{1}, δ_{2}ψ_{2}, ϵ_{1}κψ_{5} and (ψ_{μ})_{μ∈Irr(D)\{λ_{1},λ_{2},λ_{3},λ_{4}}}. On the other hand I(χ_{μ} − δ_{2}χ_{2} + δ_{4}χ_{4}) = I(χ_{μ} − δ_{1}χ_{1} − δ_{2}χ_{2}) + I(δ_{1}χ_{1} + δ_{4}χ_{4}) = ψ_{μ} − ψ_{2} + ψ_{4} adds δ_{4}ψ_{4} to the relation. The sum of the second and the third relations of Lemma 6 adds δ_{3}ψ_{3}. This completes the proof of Step 3.

Remark. At this point, we can conclude that χ(z) = χ(1) for any χ in Irr(B) of height zero, when B is the principal block, this essentially proves the theorem of Brauer-Suzuki. In the general case, we obtain : There exists μ invertible in O (precisely μ = \frac{(H:D)χ′(1)}{(G:D)χ(1)} where χ ∈ Irr(b_{z}) is of height zero and χ′ ∈ Irr(B) satisfies ±R̃(χ′) = χ) such that μBTr_{H}^{G}(z) is an involution in Z(OGB). The following is an isomorphism (see [Br2]1.5) :

Z(OHb_{z})  →  Z(OGB)
b_{z}Tr_{C_{H}(h)}^{H}(h)  ↦  \{ \begin{matrix}μBTr_{C_{G}(h)}^{G}(h) & if h_{2} ∈_{H} D^{#} \\ μ^{2}BTr_{H}^{G}(z)Tr_{C_{G}(h)}^{G}(hz) & if h_{2} ∉_{H} D^{#} \end{matrix}

Remark. In the other case of control, that is when D is quasidihedral and case (ba) occurs (see Proposition 0 (iii)), the same result can be obtained when the additional hypothesis is satisfied : z and y are not G-conjugate. The proof is similar using D^{#} = {u ∈ D / z ∈<u>}.

References.

[A-Br] J.L. Alperin, M. Broué, Local methods in block theory, Ann. of Math. 110 (1979),

143-157.

[B1] R. Brauer, Some applications of the theory of blocks of characters of finite groups, II, J. Algebra 1(1964), 307-334.

[B2] R. Brauer, On blocks and sections in finite groups II, Amer. J. Math. 90(1968), 895-925.

[B3] R. Brauer, On 2-blocks with dihedral defect groups, Symposia Mathematica,13 (1974), 367-393.

[Br1] M. Broué, Théorie locale des blocs d’un groupe fini, Proceedings of the International Congress of Mathematicians, Berkeley, 1986, 360-368.

[Br2] M. Broué, Isométries parfaites, types de blocs, catégories dérivées, Astérisque,181-182

(1990). [Br-P1] M. Broué, Ll. Puig, Characters and local structure in G-algebras, J. Algebra, 63 (1980),

306-317. [Br-P2] M. Broué, Ll. Puig, A Frobenius theorem for blocks, Invent. Math. 56(1980), 117-128.

[D] E.C. Dade, Character theory pertaining to finite simple groups, in Finite Simple Groups ed. by M. Powell and G. Higman, Academic, 1971.

[E] K. Erdmann, Blocks of tame representation type and related algebras, Springer Lecture Notes 1428, 1990.

[L] M. Lechuga, Contribution à l’étude locale des blocs, Thesis, 1989, Université Paris 7.

[Li] M. Linckelmann, A derived equivalence for blocks with dihedral defect groups, (manuscript).

[O] J.B. Olsson, On 2-blocks with quaternion and quasidihedral defect groups, J. Algebra, 36-2(1975), 212-241.

[P] Ll. Puig, Nilpotent blocks and their source algebras, Invent. Math.93(1988), 77-116.