2-Blocks with minimal nonabelian defect groups III

2-Blocks with minimal nonabelian defect groups III

Benjamin Sambale

June 6, 2015

Abstract

We prove that two2-blocks of (possibly different) finite groups with a common minimal nonabelian defect group and the same fusion system are isotypic (and therefore perfectly isometric) in the sense of Broué. This continues former work by [Cabanes-Picaronny, 1992], [Sambale, 2011] and [Eaton-Külshammer-Sambale, 2012].

Keywords:minimal nonabelian defect groups, perfect isometries, isotypies AMS classification:20C15, 20C20

1 Introduction

Since its appearance in 1990, Broué’s Abelian Defect Conjecture gained much attention among representation theorists. On the level of characters it predicts the existence of a perfect isometry between a block with abelian defect group and its Brauer correspondent. These blocks have a common defect group and the same fusion system. Although Broué’s Conjecture is false for nonabelian defect groups (see [4]), one can still ask if perfect isometries or even isotypies exist. We affirmatively answer this question for p = 2 and minimal nonabelian defect groups (see Theorem 9 below). These are the nonabelian defect groups such that any proper subgroup is abelian. Doing so, we verify the character-theoretic version of Rouquier’s Conjecture [17, A.2] in this special case (see Corollary 10 below). At the same time we provide a new infinite family of defect groups supporting a blockwiseZ^∗-Theorem.

By Rédei’s classification of minimal nonabelian p-groups, one has to consider three distinct families of defect groups. For two of these families the result already appeared in the literature (see [3, 19, 5]). Hence, it suffices to handle the remaining family which we will do in the next section. The proof of the main result is an application of Horimoto-Watanabe [10, Theorem 2]. The last section of the present paper also contains a related result for the nonabelian defect group of order27and exponent9.

Our notation is fairly standard. We consider blocks B of finite groups with respect to a p-modular system (K,O, F) where O is a complete discrete valuation ring with quotient fieldK of characteristic0 and field of fractions F of characteristic p. As usual, we assume that K is “large” enough and F is algebraically closed.

The number of irreducible ordinary characters (resp. Brauer characters) ofB is denoted byk(B)(resp.l(B)).

Moreover,k_i(B)is the number of those irreducible characters of B which have height i≥0. For other results on block invariants and fusion systems we often refer to [20]. Moreover, for the definition and construction of perfect isometries we follow [1, 3]. A cyclic group of ordern∈Nis denoted byC_n.

∗Institut für Mathematik, Friedrich-Schiller-Universität, 07743 Jena, Germany, benjamin.sambale@uni-jena.de

2 A class of minimal nonabelian defect groups

LetB be a non-nilpotent2-block of a finite group Gwith defect group

D=hx, y|x^2r =y²= [x, y]²= [x, x, y] = [y, x, y] = 1i ∼=C₂²oC2^r (1) wherer≥2,[x, y] :=xyx⁻¹y⁻¹and[x, x, y] := [x,[x, y]].

We have already investigated some properties ofB in [19], and later gave simplified proofs in [20, Chapter 12].

For the convenience of the reader we restate some of these results.

Lemma 1([20, Lemma 12.3]). Let z:= [x, y]. Then the following holds:

(i) Φ(D) = Z(D) =hx², zi ∼=C2^r−1×C2. (ii) D⁰=hzi ∼=C₂.

(iii) |Irr(D)|= 5·2^r−1.

Recall that a (saturated) fusion systemF on a p-group P determines the following subgroups:

Z(F) :={x∈P :xis fixed by every morphism inF }, foc(F) :=hf(x)x⁻¹:x∈Q≤P, f ∈Aut_F(Q)i, hyp(F) :=hf(x)x⁻¹:x∈Q≤P, f ∈O^p(AutF(Q))i.

Lemma 2. The fusion systemF of B is the constrained fusion system of the finite group A₄oC₂r whereC₂r

acts as a transposition inAut(A₄)∼=S₄. In particular,B has inertial index1 andQ:=hx², y, zi ∼=C₂r−1×C₂² is the only F-essential subgroup of D. Moreover, Aut_F(Q)∼=S₃. Without loss of generality,Z(F) = hx²iand hyp(B) =foc(B) =foc(F) =hy, zi.

Proof. We have seen in [20, Proposition 12.7] that F is constrained and coincides with the fusion system of A4 oC2^r. The construction of the semidirect product A4 oC2^r is slightly different in [20], but it is easy to see that both constructions give isomorphic groups. The remaining claims follow from the proof of [20, Proposition 12.7].

By a result of Watanabe [25, Theorem 3 and Lemma 3], the hyperfocal subgroup of a 2-block is trivial or non-cyclic. Hence, our situation with a Klein-four (hyper)focal subgroup represents the first non-trivial example in some sense. Recall that aB-subsection is a pair(u, bu)such thatu∈D andbu is a Brauer correspondent of B in CG(u).

Lemma 3. The setR:= Z(D)∪ {xⁱy^j:i, j∈Z, iodd} is a set of representatives for theF-conjugacy classes of D with |R| = 2^r+1. For u ∈ R let (u, b_u) be a B-subsection. Then b_u has defect group C_D(u). Moreover, l(b_u) = 1wheneveru∈ R \ hx²i.

Proof. By Lemma 2, it is easy to see thatRis in fact a set of representatives for theF-conjugacy classes ofD.

Observe that huiis fully F-normalized for all u∈ R. Hence, by [20, Lemma 1.34], bu has defect group CD(u) and fusion systemC_F(hui). It is easy to see thatC_F(hui)is trivial unlessu∈Z(F) =hx²i. This showsl(bu) = 1 foru∈ R \ hx²i.

Theorem 4([20, Theorem 12.4]). We have k(B) = 5·2^r−1,k0(B) = 2^r+1,k1(B) = 2^r−1 andl(B) = 2.

Proof. By Lemma 2, we have |D : foc(B)| = 2^r. In particular,2^r | k0(B) by [16, Theorem 1]. Moreover, [11, Theorem 1.1] implies2^r+1≤k0(B). By Lemma 3 we havel(bx) = 1. Thus, we obtain k0(B) = 2^r+1 by a result of Robinson (see [20, Theorem 4.12]). In order to determine l(B), we use induction on r. Let u:= x². Then bu dominates a block bu of CG(u)/hui with defect group D := D/hui ∼= D8 and fusion system F := F/hui.

By [13, Theorem 6.3], hx², y, zi/hui ∼=C₂² is the only F-essential subgroup ofD. Therefore, a result of Brauer (see [20, Theorem 8.1]) shows that l(b_u) = l(b_u) = 2. By Lemma 3 and [20, Theorem 1.35] it follows that

k(B)> k0(B). Since |Z(D) : Z(D)∩foc(B)|= 2^r−1, we have 2^r−1 |ki(B)for i≥1 by [16, Theorem 2]. Thus, by [15, Theorem 3.4] we obtain

2^r+2≤k0(B) + 4(k(B)−k0(B))≤

∞

X

i=0

ki(B)2²ⁱ≤ |D|= 2^r+2.

This givesk1(B) = 2^r−1andk(B) =k0(B) +k1(B) = 5·2^r−1. In case r= 2, [20, Theorem 1.35] implies l(B) =k(B)− X

16=u∈R

l(bu) = 10−8 = 2.

Now letr≥3 and1 6=hui<hx²i. Thenbu as above has the same type of defect group as B except thatr is smaller. Hence, induction givesl(bu) =l(bu) = 2. Now the claiml(B) = 2 follows again by [20, Theorem 1.35].

In the following results we denote the set of irreducible characters ofB of height ibyIrri(B).

Proposition 5 ([20, Proposition 12.9]). The set Irr₀(B)contains four 2-rational characters and two families of2-conjugate characters of size2ⁱfor every i= 1, . . . , r−1. The characters of height1split into two2-rational characters and one family of2-conjugate characters of size2ⁱ for every i= 2, . . . , r−2.

Proposition 6. There are2-rational charactersχ_i∈Irr(B)fori= 1,2,3 such that Irr0(B) ={χi∗λ:i= 1,2, λ∈Irr(D/foc(B))}, Irr₁(B) ={χ3∗λ:λ∈Irr(Z(D)foc(B)/foc(B))}.

In particular, the characters of height 1have the same degree and|{χ(1) :χ∈Irr₀(B)}| ≤2.

Proof. We have already seen in the proof of Theorem 4 that the action of D/foc(B) on Irr₀(B) via the ∗- construction has two orbits, and the action of Z(D)foc(B)/foc(B) on Irr1(B) is regular. By Proposition 5 we can choose 2-rational representatives for these orbits. Notice that we identify the sets Irr(D/foc(B)) and Irr(Z(D)foc(B)/foc(B))with subsets ofIrr(D)in an obvious manner.

In the situation of Proposition 6 it is conjectured thatχ1(1)6=χ2(1)(see [14]).

Proposition 7 ([20, Proposition 12.8]). The Cartan matrix of B is given by 2^r−1

3 1 1 3

up to basic sets.

Observe that Proposition 7 also gives the Cartan matrix for the defect groupD₈and the corresponding fusion system (this would be the caser= 1).

Now we are in a position to obtain the generalized decomposition matrix of B. This completes partial results in [19, Section 3.3].

Proposition 8. Let R andχi be as in Lemma 3 and Proposition 6 respectively. Then there are basic sets for bu (u∈ R)and signs, σ∈ {±1} such that the generalized decomposition numbers ofB have the following form

u x²ⁱ x²ⁱz x²ⁱ⁺¹ x²ⁱ⁺¹y d^u_χ

1ϕ (1,0) 1 1 1

d^u_χ

2ϕ (0, ) −

d^u_χ3ϕ (σ, σ) −2σ 0 0 .

Proof. Since the Galois group of Q(e^2πi/2r) over Q acts on the columns of the generalized decomposition matrix (cf. Proposition 5), we only need to determine the numbers d^u_χ

iϕ foru∈ {x, xy, x^2j, x^2jz} (i= 1,2,3, j= 1, . . . , r). First letu=x. Then the orthogonality relations show that

2^r|d^x_χ1ϕ|²+ 2^r|d^x_χ2ϕ|²+ 2^r−1|d^x_χ3ϕ|²= 2^r+1.

Sinceχ1 andχ2 have height0, we haved^x_χ1ϕ6= 06=d^x_χ2ϕ (see [20, Proposition 1.36]). It follows thatd^x_χiϕ=±1 fori= 1,2andd^x_χ3ϕ= 0, becauseχi is2-rational. By replacingϕwith−ϕif necessary (i. e. changing the basic set for bx), we may assume thatd^x_χ1ϕ = 1. We set d^x_χ2ϕ =:0. Similarly, we obtain d^xy_χ1ϕ = 1, d^xy_χ2ϕ =±1 and d^xy_χ

3ϕ= 0. Now since the columnsd^xandd^xy of the generalized decomposition matrix are orthogonal, we obtain d^xy_χ

2ϕ=−0.

Now letu:=x^2j for somej∈ {1, . . . , r}. LetIBr(b_u) ={ϕ1, ϕ₂}(see proof of Theorem 4). Then by Proposition 7 we get

2^r|d^u_χ1ϕ1|²+ 2^r|d^u_χ2ϕ1|²+ 2^r−1|d^u_χ3ϕ1|²= 3·2^r−1, 2^r|d^u_χ1ϕ2|²+ 2^r|d^u_χ2ϕ2|²+ 2^r−1|d^u_χ3ϕ2|²= 3·2^r−1, 2^rd^u_χ1ϕ1d^u_χ1ϕ2+ 2^rd^u_χ2ϕ1d^u_χ2ϕ2+ 2^r−1d^u_χ3ϕ1d^u_χ3ϕ2= 2^r−1.

Obviously, d^u_χ1ϕ1d^u_χ2ϕ1 = 0 and we may assume that (d^u_χ1ϕ1, d^u_χ1ϕ2) = (1,0) and (d^u_χ2ϕ1, d^u_χ2ϕ2) = (0, j) for a signj∈ {±1}. Moreover,d^u_χ3ϕ1 =d^u_χ3ϕ2 =:σj∈ {±1}. Now letu:=x^2jz. Then we have

2^r|d^u_χ1ϕ|²+ 2^r|d^u_χ2ϕ|²+ 2^r−1|d^u_χ3ϕ|²= 2^r+2. It is known that2|d^u_χ

3ϕ6= 0, sinceb_u is major (see [20, Proposition 1.36]). This givesd^u_χ

1ϕ= 1,d^u_χ

2ϕ=±1and d^u_χ

3ϕ=±2. By the orthogonality tod^x2j we obtain thatd^u_χ

3ϕ=−2σj andd^u_χ

2ϕ=_j.

It remains to show that the signs_j and σ_j do not depend onj. For this we consider charactersλ, ψ∈Irr(D) whose values are given as follows

x^2j x^2jz x xy

λ 1 1 1 −1

ψ 2 −2 0 0

.

Observe that ψ is the inflation of the irreducible character of D/hx²i ∼= D8 of degree 2. It is easy to see that (λ+ψ)(x^2ky) = −1 = 1−2 = (λ+ψ)(x^2kz) for every k ∈ Z. It follows that λ+ψ is F-stable, i. e.

(λ+ψ)(u) = (λ+ψ)(v)wheneveru andv areF-conjugate. By Broué-Puig [1],χ1∗(λ+ψ) is a generalized character ofB. In particular, the scalar product(χ1∗(λ+ψ), χ3)G is an integer. This number can be computed by using the so-called contribution numbers m^u_χ1χ3 :=d^u_χ1C_u⁻¹d^u_χ3^T where Cu is the Cartan matrix of bu and d^u_χi is the row of the generalized decomposition matrix corresponding to(u, bu)andχi. In caseu=x^2j we have

C_u⁻¹= 2^−r−2

3 −1

−1 3

by Proposition 7. This givesm^u_χ

1χ₃ = 2^−r−1σ_j. Similarly,m^u_χ

1χ₃ =−2^−r−1σ_j foru=x^2jz. Thus, we obtain (χ1∗(λ+ψ), χ3)G =X

u∈R

(λ+ψ)(u)m^u_χ1χ3 = X

u∈Z(D)

(λ+ψ)(u)m^u_χ1χ3

= (3 + 1)

2^−r−1σ_r+ 2^−r−1

r−1

X

j=1

σ_j2^r−j−1

= 2^−r+1σ_r+

r−1

X

j=1

σ_j2^−j.

Ifσ1=σj for somej6= 1, then it follows immediately thatσ1=. . .=σr(otherwise the scalar product above is not an integer). Now suppose that−σ1=σ2=. . .=σr. In this case we replaceχ3 by the2-rational character χ3∗τ where τ ∈ Irr(Z(D)foc(B)/foc(B))such that τ(x²) = −1. This changes σ1, but does not affect σj for j >1.

A similar argument with the scalar product(χ2∗(λ+ψ), χ3)G implies that 1 =. . .=r. In case0 =−1, we replaceχ2 byχ2∗τ whereτ ∈Irr(D/foc(B))such thatτ(x) =−1. Observe again that this changes0, but keepsj forj >0. This completes the proof.

3 The main result

Theorem 9. LetB andBe be2-blocks of (possibly different)finite groups with a common minimal nonabelian defect group and the same fusion system. Then B andBe are isotypic (and therefore perfectly isometric).

Proof. We may assume thatBis not nilpotent by Broué-Puig [2]. LetDbe a defect group ofBandB. Ife |D|= 8, then the claim follows from [3]. Now suppose that D is given as in (1). We will attach a tilde to everything associated withB. By Proposition 8 and [10, Theorem 2] there is a perfect isometrye I: CF(G, B)→CF(G,e B)e whereCF(G, B)denotes the space of class functions with basisIrr(B)overK. It remains to show thatIis also an isotypy. In order to do so, we follow [3, Section V.2]. For each u∈ D let CF(CG(u)2⁰, bu)be the space of class functions onCG(u)which vanish on thep-singular classes and are spanned byIBr(bu). The decomposition mapd^u_G: CF(G, B)→CF(CG(u)2⁰, bu)is defined by

d^u_G(χ)(s) :=χ(eb_uus) = X

ϕ∈IBr(bu)

d^u_χϕϕ(s)

forχ∈Irr(B)ands∈CG(u)2⁰ where eb_u is the block idempotent ofbu overO. ThenI determines isometries I^u: CF(CG(u)2⁰, bu)→CF(C

Ge(u)2⁰,beu) by the equation d^u

Ge ◦I = I^u ◦d^u_G. Note that I¹ is the restriction of I. We need to show that I^u can be extended to a perfect isometrycI^u : CF(CG(u), bu)→CF(C

Ge(u),beu). Suppose first that bu is nilpotent. Then by Proposition 8, d^u_G(χ1) = ϕand d^u

Ge(I(χ1)) =eϕewhere IBr(bu) = {ϕ} and IBr(beu) = {ϕ}e for some signs ,e∈ {±1}. It follows thatI^u(ϕ) =eϕ. Lete ψ∈Irr₀(b_u)and ψe∈Irr₀(be_u)be2-rational characters. Then it is well-known that ϕ=d¹_C

G(u)(ψ) and Irr(bu) ={ψ∗λ :λ∈ Irr(D)} (see [2]). Therefore, we may define cI^u by cI^u(ψ∗λ) :=eψe∗λforλ∈Irr(D). ThencI^u is a perfect isometry and

Ic^u(ϕ) =Ic^u(d¹_C

G(u)(ψ)) =d¹_C

Ge(u)(cI^u(ψ)) =ed¹_C

Ge(u)(ψ) =e eϕe=I^u(ϕ).

Hence,cI^uextendsI^u. Moreover,cI^udoes not depend on the generator ofhui, since the signsandewere defined by means of2-rational characters.

Assume next that b_u is non-nilpotent. Then u ∈ hx²i and b_u has defect group D. By Proposition 8, we can choose basic setsϕ₁,ϕ₂ (resp.fϕ₁,ϕf₂) forb_u (resp.be_u) such thatϕ_i=d^u_G(χ_i)andϕe_i=d^u

Ge(I(χ_i))fori= 1,2.

Then I^u(ϕi) = ϕei fori = 1,2. Since the Cartan matrix of bu with respect to the basic setϕ1, ϕ2 is already fixed (and given by Proposition 7), we find2-rational charactersψi∈Irr0(bu)such thatd¹_C

G(u)(ψi) =iϕi with i∈ {±1}fori= 1,2 (see proof of Proposition 8). Similarly, one hasψei∈Irr0(beu)such thatd¹_C

Ge(u)(ψei) =eiϕei. Then, by what we have already shown, there exists a perfect isometry cI^u : CF(CG(u), bu) → CF(C

Ge(u),beu) sendingψi toieiψei fori= 1,2. We have

cI^u(ϕi) =icI^u(d¹_C

G(u)(ψi)) =id¹_C

Ge(u)(cI^u(ψi)) =eid¹_C

Ge(u)(ψei) =ϕei=I^u(ϕi)

fori= 1,2. This shows thatcI^uextendsI^u. Moreover, it is easy to see thatcI^udoes not depend on the generator ofhui.

Altogether we have proved the theorem if Dis given as in (1). By [20, Theorem 12.4] it remains to handle the case

D∼=hx, y|x^2r =y^2r = [x, y]²= [x, x, y] = [y, x, y] = 1i

wherer≥2. HereB andBe are Morita equivalent and therefore perfectly isometric. However, a Morita equiv- alence does not automatically provide an isotypy. Nevertheless, in this special case the Morita equivalence is a composition of various “natural” equivalences (namely Fong reductions, Külshammer-Puig reduction and Külshammer’s reduction for blocks with normal defect groups, see [5, proof of Theorem 1]). In particular, the generalized decomposition matrices ofBandBecoincide up to signs (see [24]). Now we can use the same methods as above in order to construct an isotypy. In fact, for everyB-subsection(u, bu)one has that bu is nilpotent or u= [x, y]andbu Morita equivalent toB (see proof of [19, Proposition 4.3]). We omit the details.

Corollary 10. Let B be a 2-block of a finite group Gwith minimal nonabelian defect groupD 6∼=D8. ThenB is isotypic to a Brauer correspondent inNG(hyp(B)).

Proof. Let bD be a Brauer correspondent of B in DCG(D). Since DCG(D)⊆ NG(hyp(B)), the Brauer cor- respondent b := b^N_D^G(hyp(B)) of B has defect group D. By Theorem 9, it suffices to show that B and b have the same fusion system. Observe thatNG(D, bD)⊆NG(hyp(B)). In particular,B andb have the same inertial quotient. If there is only the trivial fusion system onD, then we are done (this applies ifDis metacyclic of order at least16). In caseD∼=Q8,Bis a controlled block (see e. g. [3]). SinceBandbhave the same inertial quotient, it follows that these blocks also have the same fusion system. It remains to consider the two other families of defect groups (see [20, Theorem 12.4]). For one of these families the fusion system is again controlled (see [20, Proposition 12.7]). Finally, ifDis given as in (1), then the fusion system is constrained and the automorphisms of the essential subgroup (if it exists) also act on hyp(B). Hence, B is nilpotent if and only ifb is nilpotent.

Again the claim follows from Theorem 9.

We remark that Corollary 10 would be false in case D ∼=D8. The principal 2-block of GL(3,2) gives a coun- terexample. IfB is a block of a finite group Gwith defect group as given in (1), then B is also isotypic to a Brauer correspondent inCG(u)whereu∈Z(F). This resembles Glauberman’sZ^∗-Theorem.

In the situation of Theorem 9 (or Corollary 10) it is desirable to extend the isotypies to Morita equivalences (as we did in [5]). This is not always possible if|D|= 8, since for example the principal2-blocks of the symmetric groupsS4andS5are not Morita equivalent. Nevertheless, the possible Morita equivalence classes in case|D|= 8 are known by Erdmann’s classification of tame algebra [6] (at least over F, cf. [9]). In view of [5] one may still ask if two non-nilpotent2-blocks with isomorphic defect groups as in Section 2 are Morita equivalent. We will see that the answer is again negative.

Consider the groupsG₁:=A₄oC₂randG₂:=A₅oC₂r constructed similarly as in Lemma 2. ThenG₁/Z(G₁)∼= S₄ andG₂/Z(G₂)∼=S₅. LetB_ibe the principal2-block ofG_i, and letB_i be the principal2-block ofG_i/Z(G_i) fori= 1,2. Then the Cartan matrix ofB_i is just the Cartan matrix of B_i multiplied by |Z(G_i)|= 2^r−1. It is known that the Cartan matrices ofB₁andB₂do not coincide (regardless of the labeling of the simple modules).

Therefore,B1 andB2 are not Morita equivalent.

Nevertheless, the structure of a finite groupGwith a minimal nonabelian Sylow2-subgroupP as given in (1) is fairly restricted. More precisely, Glauberman’sZ^∗-Theorem impliesx²∈Z^∗(G), and the structure ofG/Z^∗(G) follows from the Gorenstein-Walter Theorem [7]. In particular,Ghas at most one nonabelian composition factor by Feit-Thompson.

We use the opportunity to present a related result forp= 3which extends [20, Theorem 8.15].

Theorem 11. Let B andBe be non-nilpotent blocks of (possibly different)finite groups both with defect group C₉oC₃. ThenB andBe are isotypic.

Proof. As in the proof of Theorem 9, we will make use of [10, Theorem 2]. Let D:=hx, y|x⁹=y³= 1, yxy⁻¹=x⁴i

be a defect group of B, and let F be the fusion system of B. By Stancu [21], B is controlled with inertial index 2, and we may assume that x and x⁻¹ are F-conjugate (see proof of [20, Theorem 8.8]). Then R :=

{1, x, x³, y, y², xy, xy²} is a set of representatives for theF-conjugacy classes ofD (see proof of [20, Theorem 8.15]). It suffices to show that the generalized decomposition numbers ofB are essentially unique (up to basic sets and signs and permutations of rows). Since the Galois group ofQ(e^2πi/9)overQacts on the columns of the generalized decomposition matrix, we only need to determine the numbers d^u_χϕ foru∈ {x, x³, y, xy}. By [20, Theorem 8.15] there are four3-rational charactersχ_i∈Irr(B)(i= 1, . . . ,4) such thatχ₁,χ₂,χ₃ have height0 andχ₄ has height1. Since foc(B) =hxi, we see that

Irr(B) ={χ_i∗λ:i= 1,2,3, λ∈Irr(D/foc(B))} ∪ {χ₄}.

Let u := x³. Then IBr(b_u) = {ϕ} and d^u_χ

iϕ are non-zero (rational) integers. Moreover, d^u_χ

4ϕ ≡ 0 (mod 3).

After permutingχ1,χ2 andχ3 and changing the basic set forbu if necessary, we may assume that d^u_χ1ϕ = 2,

d^u_χ

2ϕ=:1∈ {±1}, d^u_χ

3ϕ=:2 ∈ {±1}and d^u_χ

4ϕ= 33∈ {±3}. Now letu:=x. Then d^u_χ

iϕ=±1fori= 1,2,3 and d^u_χ4ϕ = 0. We may choose a basic set for bu such that d^u_χ1ϕ = 1. Then by the orthogonality relations, d^u_χ2ϕ=−1 andd^u_χ3ϕ=−2. Next letu:=y. Thenbu dominates a block ofCG(u)/huiwith cyclic defect group CD(u)/hui ∼=C3 and inertial index2. This yieldsIBr(bu) ={ϕ1, ϕ2}and the Cartan matrix of bu is given by

3 2 1

1 2

(not only up to basic sets, but this is not important here). We can choose a basic set such that(d^u_χ1ϕ1, d^u_χ1ϕ2) = (1,1),(d^u_χ2ϕ1, d^u_χ2ϕ2) = (σ1,0),(d^u_χ3ϕ1, d^u_χ3ϕ2) = (0, σ2)and(d^u_χ4ϕ1, d^u_χ4ϕ2) = (0,0)for some signsσ1, σ2∈ {±1}.

Finally foru:=xywe obtaind^u_χ

1ϕ= 1,d^u_χ

iϕ =−σi−1 fori= 2,3 andd^u_χ

4ϕ= 0after changing the basic set if necessary. The following table summarizes the results

u x³ x y xy

d^u_χ1ϕ 2 1 (1,1) 1 d^u_χ2ϕ 1 −1 (σ1,0) −σ1

d^u_χ

3ϕ ₂ −2 (0, σ₂) −σ2

d^u_χ4ϕ 33 0 (0,0) 0 .

It suffices to show that i=σi fori= 1,2 (observe that we do not need the ordinary decomposition numbers in order to apply [10, Theorem 2]). For this, letλ∈Irr(D/hx³i)such that λ(x) =e^2πi/3 and λ(y) = 1. Then the generalized characterψ:=λ+λ−2·1D ofD is constant onhxi \ hx³iand thusF-stable. By [1],χ1∗ψis a generalized character ofB and(χ1∗ψ, χ2)G∈Z. As in the proof of Theorem 9, we compute

(χ1∗ψ, χ2)G =X

u∈R

ψ(u)m^u_χ1χ2=ψ(x)m^x_χ1χ2+ψ(xy)m^xy_χ1χ2+ψ(xy²)m^xy_χ1²_χ2 =1 31+2

3σ1.

This shows 1 = σ1. Similarly, one gets 2 = σ2 by computing (χ1∗ψ, χ3)G. Hence, [10, Theorem 2] gives a perfect isometry I : CF(G, B)→ CF(G,e B). In order to show thate I is also an isotypy, we make use of the notation introduced in the proof of Theorem 9. Letu∈Dsuch thatb_uis nilpotent. Then by the table above, we haveIBr(b_u) ={±d^u_G(χ₂)}. Thus, one can extendI^ujust as in Theorem 9. Now suppose thatb_uis non-nilpotent and thus u=y (up to inversion). We choose a basic setϕ1,ϕ2 forbu as above such that d^u_G(χi) =ϕ_i−1 for i = 2,3. Now we have to determine the ordinary decomposition numbers of bu with respect to ϕ1, ϕ2. The defect group ofbuisCD(y) =hx³, yi ∼=C3×C3andfoc(bu) =hx³i. By Kiyota [12],k(bu) = 9. Therefore, there are3-rational charactersψi∈Irr(bu)such that

Irr(bu) ={ψi∗λ:i= 1,2,3, λ∈Irr(hx³, yi/hx³i)}.

By the Cartan matrix ofbu given above (with respect toϕ1,ϕ2), it follows immediately thatd¹_C

G(u)(ψi) =iϕi

with_i∈ {±1} fori= 1,2after a suitable permutation ofψ₁,ψ₂,ψ₃. Similarly,d¹_C

Ge(u)(ψe_i) =e_iϕe_i. By a result of Usami [22], there is a perfect isometry CF(CG(u), bu) → CF(C

Ge(u),beu). However, we need the additional information that ψ_i is mapped to ±ψe_i. In order to show this, we use [10, Theorem 2] again. Observe that d^u_C

G(u)(ψ_i) =ζ_id¹_C

G(u)(ψ_i) =ζ_iiϕ_i for a cube root of unityζ_i. But sinced^u_ψ

iϕi is rational, we haveζ_i= 1. Now an elementary application of the orthogonality relations shows that the generalized decomposition matrix ofb_u (inC_G(u)) is determined by

v 1 y x³ x³y

d^v_ψ

1ϕ (₁,0) (₁,0) _{1 1}

d^v_ψ

2ϕ (0, ₂) (0, ₂) _{2 2}

d^v_ψ

3ϕ (3, 3) (3, 3) −3 −3

.

It follows that there is a perfect isometryIc^u : CF(CG(u), bu)→ CF(C

Ge(u),beu) such thatcI^u(ψi) =ieiψei for i= 1,2. Therefore Ic^u extends I^u. As in the proof of Theorem 9, it is also clear that cI^u is independent of the choice of the generator ofhui. This finishes the proof.

The proof method of Theorem 11 also works for other defect groups. In fact, Watanabe [23] showed inde- pendently (using more complicated methods) that two p-blocks (p > 2) with a common metacyclic, minimal nonabelian defect group and the same fusion system are perfectly isometric. Again, this gives evidence for the character-theoretic version of Rouquier’s Conjecture (see [25, Theorem 2]). As another remark, Holloway- Koshitani-Kunugi [8, Example 4.3] constructed a perfect isometry between the principal 3-block of G :=

Aut(SL(2,8))∼=²G2(3)and its Brauer correspondent. SinceGhas a Sylow3-subgroup isomorphic toC9oC3, this is a special case of Theorem 11. Note that in the introduction of Ruengrot [18] it is erroneously stated that these blocks arenot perfectly isometric.

Acknowledgment

This work is supported by the German Research Foundation and the Daimler and Benz Foundation. The author thanks Atumi Watanabe for providing a copy of [23]. Moreover, the author thanks Burkhard Külshammer for answering some questions.

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