Abstract Let B be a block of a finite group with respect to an algebraically closed field F of characteristic p>0. In a recent paper, Otokita gave an upper bound for the Loewy length LL(ZB) of the center ZB of B in terms of a defect group D of B. We refine his methods in order to prove the optimal bound LL(ZB)≤LL(FD) whenever D is abelian. We also improve Otokita’s bound for non-abelian defect groups. As an application, we classify the blocks B such that LL(ZB)≥|D|/2. 1. Introduction We consider a block (algebra) B of FG where G is a finite group and F is an algebraically closed field of characteristic p>0. In general, the structure of B is quite complicated and can only be described in restrictive special cases (e.g. blocks of defect 0). For this reason, we are content here with the study of the center ZB of B. This is a local F-algebra in the sense that the Jacobson radical JZB has codimension 1. It is well known that the dimension of ZB itself equals the number k(B) of irreducible complex characters in B. In particular, this dimension is locally bounded by a theorem of Brauer and Feit [3]. Moreover, the number l(B) of irreducible Brauer characters in B is given by the dimension of the Reynolds ideal RB≔ZB∩SB, where SB is the socle of B. It follows that the dimension of the quotient ZB/RB is locally determined by Brauer’s theory of subsections. Here a B-subsection is a pair (u,b), where u∈G is a p-element and b is a Brauer correspondent of B in CG(u). In order to give better descriptions of ZB, we make use of the Loewy length LL(A) of a finite-dimensional F-algebra A which is the smallest positive integer l such that (JA)l=0. A result by Okuyama [22] states that LL(ZB)≤∣D∣, where ∣D∣ is the order of a defect group D of B. In fact, there is an open conjecture by Brauer [2, Problem 20] asserting that even dimZB≤∣D∣. In a previous paper [16] jointly with Shigeo Koshitani, we have shown conversely that LL(B) is bounded from below in terms of ∣D∣. There is no such bound for LL(ZB), but again an open question by Brauer [2, Problem 21] asks if dimZB can be bounded from below in terms of ∣D∣. Recently, Okuyama’s estimate has been improved by Otokita [23]. More precisely, if exp(D) is the exponent of D, he proved that LL(ZB)≤∣D∣−∣D∣exp(D)+1. (1.1) The present note is inspired by Otokita’s methods. Our first result gives a local bound on the Loewy length of ZB/RB. Since (JZB)(RB)⊆(JB)(SB)=0, we immediately obtain a bound for LL(ZB). In our main theorem we apply this bound to blocks with abelian defect groups as follows. Theorem 1. Let Bbe a block of FGwith abelian defect group D. Then LL(ZB/RB)<LL(FD)and LL(ZB)≤LL(FD). If, in the situation of Theorem 1, D has type (pa1,…,par), then LL(FD)=pa1+⋯+par−r+1 as is well known. For p-solvable groups G, the stronger assertion LL(B)=LL(FD) holds (see [19, Theorem K]). Similarly, if D is cyclic, one can show more precisely that LL(ZB)=LL(ZB/RB)+1=dimZB/RB+1=∣D∣−1l(B)+1 (see [16, Corollary 2.6]). For nilpotent blocks, Theorem 1 is best possible by Broué–Puig [5]. We conjecture conversely that the inequality is strict for non-nilpotent blocks (cf. Corollary 5 and Proposition 7 below). Arguing inductively, we also improve Otokita’s bound for blocks with non-abelian defect groups. More precisely, we show in Theorem 12 that LL(ZB)≤∣D∣p+∣D∣p2−∣D∣p3 (see also Proposition 15). Extending Otokita’s work again, we use our results to classify all blocks B with LL(ZB)≥∣D∣/2 in Corollary 16. We conjecture that in the non-abelian defect case, the inequality LL(ZB)≤LL(FD) is still satisfied. This holds for example if D⊴G (see [23, proof of Lemma 2.4]). We support this observation by computing the Loewy lengths of the centers of some blocks with small defect. Finally, we take the opportunity to improve [23, Proposition 3] (see Proposition 2.2). To do so, we recall that the inertial quotient I(B) of B is the group NG(D,bD)/DCG(D) where bD is a Brauer correspondent of B in CG(D). By the Schur–Zassenhaus Theorem, I(B) can be embedded in the automorphism group Aut(D). Then FDI(B)≔{x∈FD:a−1xa=xfora∈I(B)} is the algebra of fixed points. Moreover, for a subset U⊆G, we define U+≔∑u∈Uu∈FG. Then RFG has an F-basis consisting of the sums S+, where S runs through the p′-sections of G (see for example [17]). Note that the trivial p′-section is the set Gp of p-elements of G. 2. Abelian defect groups By the results mentioned in the introduction, we may certainly restrict ourselves to blocks with positive defect. Proposition 2. Let Bbe a block of FGwith defect group D≠1. Let (u1,b1),…,(uk,bk)be a set of representatives for the conjugacy classes of non-trivial B-subsections. Then the map ZB/RB→⨁i=1kZbi/Rbi,z+RB↦∑i=1kBr⟨ui⟩(z)1bi+Rbiis an embedding of F-algebras where Br⟨ui⟩:ZFG→ZFCG(ui)denotes the Brauer homomorphism. In particular, LL(ZB/RB)≤max{LL(Zbi/Rbi):i=1,…,k}. Proof First we consider the whole group algebra FG instead of B. For this, let v1,…,vr be a set of representatives for the conjugacy classes of non-trivial p-elements of G. Let z≔∑g∈Gαgg∈ZFG. Then z is constant on the conjugacy classes of G. It follows that z is constant on the p′-sections of G if and only if Br⟨vi⟩(z)=∑g∈CG(vi)αgg is constant on the p′-sections of CG(vi) for i=1,…,r. Therefore, the map ZFG/RFG→⨁i=1rZFCG(vi)/RFCG(vi), z+RFG↦∑i=1rBr⟨vi⟩(z)+RFCG(vi) is a well-defined embedding of F-algebras. Now the first claim follows easily by projecting onto B, i.e. replacing z by z1B. The last claim is an obvious consequence.□ Proof of Theorem 1 Let D≅Cpa1×⋯×Cpar. It is well known that LL(FD)=LL(FCpa1⊗⋯⊗FCpar)=pa1+⋯+par−r+1. Hence, it suffices to show that LL(ZB/RB)≤pa1+⋯+par−r. We argue by induction on r. If r=0, then we have D=1, ZB=RB and LL(ZB/RB)=0. Thus, we may assume that r≥1. Let I≔I(B) be the inertial quotient of B. In order to apply Proposition 2, we consider a B-subsection (u,b) with 1≠u∈D. Then b has defect group D and inertial quotient CI(u). Since I is a p′-group, we have D=Q×[D,CI(u)] with Q≔CD(CI(u))≠1. Let β be the Brauer correspondent of b in CG(Q)⊆CG(u). By Watanabe [37, Theorem 2], the Brauer homomorphism BrD induces an isomorphism between Zb and Zβ. Since the intersection of a p′-section of G with CG(D) is a union of p′-sections of CG(D), it follows that BrD(Rb)⊆Rβ. On the other hand, dimFRb=l(b)=l(β)=dimFRβ by [38, Theorem 1]. Thus, we obtain Zb/Rb≅Zβ/Rβ and it suffices to show that LL(Zβ/Rβ)≤pa1+⋯+par−r. Let β¯ be the unique block of CG(Q)/Q dominated by β. By [18, Theorem 7] (see also [8, Theorem 1.2]), it follows that a source algebra of β is isomorphic to a tensor product of FQ and a source algebra of β¯. Since β is Morita equivalent to its source algebra, we may assume in the following that β=FQ⊗β¯. Let Q≅Cpa1×⋯×Cpas with 1≤s≤r. Since the defect group D/Q of β¯ has rank r−s<r, induction implies that LL(Zβ¯/Rβ¯)≤pas+1+⋯+par−r+s≕l. In particular, (JZβ¯)l⊆Rβ¯. Since Q is an abelian p-group, we have RFQ=SFQ≅F. Consequently, LL(FQ/RFQ)=pa1+⋯+pas−s≕l′, i.e. (JFQ)l′⊆RFQ. Moreover, SFQ⊗Sβ¯⊆S(FQ⊗β¯). Hence, RFQ⊗Rβ¯⊆Z(FQ⊗β¯)∩S(FQ⊗β¯)=R(FQ⊗β¯)=Rβ. Since JZβ=J(FQ⊗Zβ¯)=JFQ⊗Zβ¯+FQ⊗JZβ¯, we see that (JZβ)l+l′ is a sum of terms of the form (JFQ)i⊗(JZβ¯)j with i+j=l+l′. If i>l′, then (JFQ)i=0. Similarly, if j>l, then (JZβ¯)j=0. It follows that (JZβ)l+l′=(JFQ)l′⊗(JZβ¯)l⊆RFQ⊗Rβ¯⊆Rβ. This proves the theorem, because l+l′=pa1+⋯+par−r.□ Our theorem shows that Otokita’s bound (1.1) is only optimal for nilpotent blocks with cyclic defect groups or defect group C2×C2 (see [23, Corollary 3.1]). The next result strengthens [23, Proposition 2.2]. Proposition 3. Let Bbe a block of FGwith defect group D. Moreover, let c≔dimFZ(D)I(B)and z≔LL(FZ(D)I(B)). Then LL(ZB/RB)≤k(B)−l(B)+z−cand in particular LL(ZB)≤k(B)−l(B)+z−c+1. Proof Let K≔Ker(BrD)∩ZB⊴ZB. Since ZB is local, we have K⊆JZB. Furthermore, RB+K/K annihilates the radical JZB/K of ZB/K. It follows that RB+K/K is contained in the socle of ZB/K. By Broué [4, Proposition (III)1.1], it is known that BrD induces an isomorphism between ZB/K and the symmetric F-algebra FZ(D)I(B). The socle of the latter algebra has dimension 1. Hence, dimRB+K/K≤1. On the other hand, Gp+∈RFG. Therefore, 1BGp+∈RB and BrD(1BGp+)=BrD(1B)BrD(Gp+)=BrD(1B)CG(D)p+. Here, BrD(1B) is the block idempotent of bDNG(D), where bD is a Brauer correspondent of B in CG(D). In particular, 1bDBrD(1B)=1bD and 0≠1bDCG(D)p+=1bDBrD(1B)CG(D)p+=1bDBrD(1BGp+). From that we obtain 1BGp+∉K and dimRB+K/K=1. This implies RB+K/K=S(ZB/K) and LL(ZB/RB+K)=z−1. Now we consider the lower section of ZB. Here, we have dimRB+K/RB=dimRB+K−dimRB=1+dimK−l(B)=1+dimZB−dimZB/K−l(B)=1+k(B)−c−l(B). The claim follows easily.□ The invariant c in Proposition 3 is just the number of orbits of I(B) on Z(D). Moreover, if D and I(B) are given, the number z can be calculated by computer. It happens frequently that I(B) acts trivially on Z(D). In this case, c=∣Z(D)∣ and z is determined by the isomorphism type of Z(D) as explained earlier. In particular, LL(ZB)≤k(B)−l(B) whenever additionally Z(D) is non-cyclic. Now we give a general upper bound on z. Lemma 4. Let Pbe a finite abelian p-group, and let Ibe a p′-subgroup of Aut(P). Then LL(FPI)≤LL(FCP(I))+LL(F[P,I])−12. Proof Since FPI=FCP(I)⊗F[P,I]I, we may assume that CP(I)=1. It suffices to show that JFPI⊆(JFP)2. It is well known that JFP is the augmentation ideal of FP and JFPI=JFP∩FPI. In particular, I acts naturally on JFP and on JFP/(JFP)2. We regard P/Φ(P) as a vector space over Fp. By [10, Remark VIII.2.11], there exists an isomorphism of Fp-spaces Γ:JFP/(JFP)2→F⊗FpP/Φ(P) sending 1−x+(JFP)2 to 1⊗xΦ(P) for x∈P. After choosing a basis, it is easy to see that Γ(wγ)=Γ(w)γ for w∈JFP/(JFP)2 and γ∈I. Let w∈JFPI⊆JFP. Then Γ(w+(JFP)2) is invariant under I. It follows that Γ(w+(JFP)2) is a linear combination of elements of the form λ⊗x, where λ∈F and x∈CP/Φ(P)(I). However, by hypothesis, CP/Φ(P)(I)=CP(I)Φ(P)/Φ(P)=Φ(P) and, therefore, Γ(w+(JFP)2)=0. This shows w∈(JFP)2 as desired.□ We describe a special case which extends Theorem 1. Here, the action of I(B) on D is called semiregular if all orbits on D⧹{1} have length ∣I(B)∣. Corollary 5. Let Bbe a block of FGwith abelian defect group Dsuch that I≔I(B)acts semiregularly on [D,I]. Then LL(ZB)=LL(ZF[D⋊I])=LL(FDI)≤LL(FCD(I))+LL(F[D,I])−12. Proof Let Q≔CD(I) and let b be a Brauer correspondent of B in CG(Q). By [37, Theorem 2], ZB≅Zb. Moreover, by [18, Theorem 7], we have ZB≅FQ⊗Zb¯, where b¯ is the block of CG(Q)/Q dominated by b. As usual, b¯ has defect group D/Q≅[D,I] and inertial quotient I(b¯)≅I(B). It follows that LL(ZB)=LL(FQ)+LL(Zb¯)−1. On the other hand, FDI≅FQ⊗F[D,I]I and F[D⋊I]≅FQ⊗F[[D,I]⋊I]. Hence, we may assume that Q=1 and [D,I]=D≠1. Let (u1,b1),…,(uk,bk) be a set of representatives for the G-conjugacy classes of non-trivial B-subsections. Since I acts semiregularly on D, every block bi has inertial quotient I(bi)≅CI(ui)=1. Hence, bi is nilpotent and l(bi)=1. With the notation of Proposition 3, it follows that k(B)−l(B)=∑i=1kl(bi)=∣D∣−1∣I∣=c−1 and LL(ZB)≤LL(FDI). By the proof of Proposition 3, we also have the opposite inequality LL(ZB)≥LL(FDI). It is easy to see that ZF[D⋊I]=FDI⊕Γ, where Γ is the subspace spanned by the non-trivial p′-class sums of D⋊I. By hypothesis, every non-trivial p′-conjugacy class is a p′-section of D⋊I. Hence, we obtain Γ⊆RF[D⋊I]. The claim LL(FDI)=LL(ZF[D⋊I]) follows. The last claim is a consequence of Lemma 4.□ Corollary 5 applies for instance whenever I has prime order. For example, if ∣I∣=2, we have equality LL(ZB)=LL(FCD(I))+LL(F[D,I])−12 by [16, Proposition 2.6]. However, in general for a block B with abelian defect group D, it may happen that LL(ZB)>LL(FDI). An example is given by the principal 3-block of G=(C3×C3)⋊SD16. Here LL(ZB)=3 and dimFDI=2. In the situation of Corollary 5, I is a complement in the Frobenius group [D,I]⋊I. In particular, the Sylow subgroups of I are cyclic or quaternion groups. It follows that I has trivial Schur multiplier. By a result of the first author [20], the Brauer correspondent of B in NG(D) is Morita equivalent to F[D⋊I]. In this way we see that Corollary 5 is in accordance with Broué’s Abelian Defect Group Conjecture. Moreover, Alperin’s Weight Conjecture predicts l(B)=k(I) in this situation. By a result of the second author (see [33, Lemma 9] and [31, Theorem 5]), we also have dimZB≤∣CD(I)∣(∣[D,I]∣−1l(B)+l(B))≤∣D∣. Further properties of this class of blocks have been obtained in Kessar–Linckelmann [14]. Nevertheless, it seems difficult to express LL(FDI) explicitly in terms of D and I. Some special cases have been considered in [35, Section 6.3]. Our next aim concerns the sharpness of Theorem 1. For this, we need to discuss twisted group algebras of the form Fα[D⋊I(B)]. Lemma 6. Let Pbe a finite abelian p-group, and let Ibe a non-trivial p′-subgroup of Aut(P). Then LL(ZFα[P⋊I])<LL(FP)for every α∈H2(I,F×). Proof For the sake of brevity, we write PI instead of P⋊I. We may normalize α such that x·y in Fα[PI] equals xy∈PI for all x∈P and y∈PI. By Passman [24, Theorem 1.6], JZFα[PI]=JFα[PI]∩ZFα[PI]=(JFP·Fα[PI])∩ZFα[PI]. An element x∈PI is called α-regular if x·y=y·x in Fα[PI] for all y∈CPI(x). A conjugacy class of PI is called α-regular if it consists of α-regular elements. It is known that ZFα[PI] has a basis consisting of the α-regular class sums (see for example [7, Remark 4 on p. 155]). Hence, let K be an α-regular conjugacy class of PI. If K⊆P, then clearly ∣K∣1−K+∈ZFα[PI]∩JFP⊆JZFα[PI], since JFP is the augmentation ideal of FP. Now assume that K⊆PI⧹P and x∈K. Then the P-orbit of x (under conjugation) is the coset x[x,P]. Hence, K is a disjoint union of cosets x1[x1,P],…,xm[xm,P]. Since I acts faithfully on P, we have [xi,P]≠1 and [xi,P]+∈JFP for i=1,…,m. It follows that K+∈(JFP·Fα[PI])∩ZFα[PI]=JZFα[PI]. In this way we obtain an F-basis of JZFα[PI]. Let l≔LL(ZFα[PI]). Then there exist conjugacy classes K1,…,Ks⊆P and elements x1,…,xt∈PI⧹P such that s+t=l−1 and (∣K1∣1−K1+)⋯(∣Ks∣1−Ks+)x1[x1,P]+⋯xt[xt,P]+≠0 in Fα[PI]. Since xi[xi,P]=[xi,P]xi, we conclude that 0≠(∣K1∣1−K1+)⋯(∣Ks∣1−Ks+)[x1,P]+⋯[xt,P]+∈FP. (2.1) Since (2.1) does not depend on α anymore, we may assume that α=1 in the following. Since ZF[PI]=FCP(I)⊗ZF[[P,I]⋊I] and FP=FCP(I)⊗F[P,I], we may assume that CP(I)=1. By Lemma 4 we have s≤LL(FPI)−1≤LL(FP)−12<LL(FP)−1. Thus, we may assume that t>0. Since x1 acts non-trivially on [x1,P], we obtain ∣[x1,P]∣≥3 and [x1,P]+∈(JF[x1,P])2⊆(JFP)2. Also, ∣Ki∣1−Ki+∈JFP for i=1,…,s. Therefore, (2.1) shows that (JFP)l≠0 and the claim follows.□ Proposition 7. Let Bbe a block of FGwith abelian defect group D. Suppose that the character-theoretic version of Broué’s Conjecture holds for B. Then LL(ZB)=LL(FD)if and only if Bis nilpotent. Proof A nilpotent block B satisfies LL(ZB)=LL(FD) by Broué–Puig [5]. Thus, we may assume conversely that LL(ZB)=LL(FD). Broué’s Conjecture implies ZB≅Zb where b is the Brauer correspondent of B in NG(D). By Külshammer [20], Zb≅ZFα[D⋊I(B)] for some α∈H2(I(B),F×). Now Lemma 6 shows that I(B)=1. Hence, B must be nilpotent.□ 3. Non-abelian defect groups We start with a result about nilpotent blocks which might be of independent interest. Proposition 8. For a non-abelian p-group Pwe have JZFP⊆JF[P′Z(P)]·FPand LL(ZFP)≤LL(FP′Z(P))<LL(FP). Proof We have already used that JFP is the augmentation ideal of FP and JZFP=ZFP∩JFP. Hence, JZFP is generated as an F-space by the elements 1−z and K+, where z∈Z(P) and K⊆P⧹Z(P) is a conjugacy class. Each such K has the form K=xU with x∈P and U⊆P′. Since ∣U∣=∣K∣ is a multiple of p, we have U+∈JFP′. On the other hand, 1−z∈JFZ(P) for z∈Z(P). Setting N≔P′Z(P) we obtain JZFP⊆FP·JFN. Since P acts on FN preserving the augmentation, we also have FP·JFN=JFN·FP. This shows LL(ZFP)≤LL(FN). For the second inequality, note that N≤Z(P)Φ(P)<P. Hence, FN+=(JFN)LL(FN)−1⊆(JFP)LL(FN)−1 and (JFP)LL(FP)−1=FP+≠FN+. Therefore, we must have LL(FN)<LL(FP).□ If P has class 2, we have P′≤Z(P) and JFZ(P)⊆JZFP. Hence, Proposition 8 implies LL(ZFP)=LL(FZ(P)) in this case. In the following we improve (1.1) for non-abelian defect groups. We make use of Otokita’s inductive method: LL(ZB)≤max{(∣⟨u⟩∣−1)LL(Zb¯):(u,b)B-subsection}+1 (3.1) (see [23, proof of Theorem 1.3]). Here b¯ denotes the block of CG(u)/⟨u⟩ dominated by b. By [29, Lemma 1.34], we may assume that b¯ has defect group CD(u)/⟨u⟩ where D is a defect group of B. We start with a detailed analysis of the defect groups of large exponent. Lemma 9. Let Pbe a p-group such that Z(P)is cyclic and ∣P:Z(P)∣=p2. Then one of the following holds: P≅⟨x,y∣xpd−1=yp=1,y−1xy=x1+pd−2⟩≕Mpdfor some d≥3. P≅⟨x,y,z∣xpd−2=yp=zp=[x,y]=[x,z]=1,[y,z]=xpd−3⟩≕Wpdfor some d≥3. P≅Q8. Proof Let ∣P∣=pd with d≥3. If exp(P)=pd−1, then the result is well known. Thus, we may assume that exp(P)=pd−2. Let Z(P)=⟨x⟩ and D=⟨x,y,z⟩. Since ⟨x,y⟩≅⟨x,z⟩≅Cpd−2×Cp, we may assume that yp=zp=1. Since P is non-abelian, we have 1≠[y,z]∈P′≤Z(P). In particular, P has nilpotency class 2. It follows that [y,z]p=[yp,z]=1 and, therefore, [y,z]=xpd−3. Consequently, the isomorphism type of P is uniquely determined. Conversely, one can construct such a group as a central product of Cpd−2 and an extraspecial group of order p3.□ Proposition 10. Let Bbe a block of FGwith defect group D≅Mpdor D≅Q8. Then one of the following holds: LL(ZB)=pd−2−1l(B)+1≤pd−2=LL(ZFD)≤LL(FD). ∣D∣=8and LL(ZB)≤3. Proof Suppose first that p=2. If ∣D∣=8, then there are in total five possible fusion systems for B and none of them is exotic (see [29, Theorem 8.1]). By [6], the fusion system of B determines the perfect isometry class of B. Since perfect isometries preserve the isomorphism type of ZB, we may assume that B is the principal block of FH, where H∈{D8,Q8,S4,SL(2,3),GL(3,2)}. A computation with GAP [9] reveals that LL(ZB)≤3 in all cases. Note that we may work over the field with two elements, since the natural structure constants of ZFH (and of ZB) lie in the prime field of F. (The fusion system corresponding to H=GL(3,2) can be handled alternatively with Proposition 3.) If D≅M2d with d≥4, then B is nilpotent ([29, Theorem 8.1]) and the result follows from the remark after Proposition 8. Now assume that p>2. By [36], B is perfectly isometric to its Brauer correspondent in NG(D). Hence, we may assume that D⊴G. It is known that B has cyclic inertial quotient I(B) of order dividing p−1 (see [29, proof of Theorem 8.8]). Hence, by [20] we may assume that G=D⋊I(B). Then G has only one block and ZB=ZFG. Moreover, l(B)=∣I(B)∣. After conjugation, we may assume that I(B)=⟨a⟩ acts non-trivially on ⟨x⟩ and trivially on ⟨y⟩ with the notation from Lemma 9. Since ∣D′∣=p, the conjugacy classes of D are either singletons in Z(D) or cosets of D′. Some of these classes are fused in G. The classes in G⧹D are cosets of ⟨x⟩. As usual, ZFG is generated by the class sums and JZFG is the augmentation ideal (intersected with ZFG). In particular, JZFG contains the class sums of conjugacy classes whose length is divisible by p. Let U1,…,Uk be the non-trivial orbits of I(B) on Z(D). Then JZFG also contains the sums l(B)1G−Ui+ for i=1,…,k. For u,v∈D we have u(D′)+·v(D′)+=uv((D′)+)2=0,u(D′)+·v⟨x⟩+=uv(D′)+⟨x⟩+=0,u⟨x⟩+·v⟨x⟩+=uv(⟨x⟩+)2=0,u(D′)+·(l(B)1G−Ui+)=l(B)u(D′)+−l(B)u(D′)+=0,u⟨x⟩+·(l(B)1G−Ui+)=l(B)u⟨x⟩+−l(B)u⟨x⟩+=0. It follows that (JZFG)2=(JZF⟨x,a⟩)2. Now the claim can be shown with [16, Corollary 2.8].□ Lemma 11. Let Bbe a block of FGwith defect group D≅Wpd. Then LL(ZB)≤pd−1−p+1. Proof If ∣D∣=8, then the claim holds by Proposition 10. Hence, we may exclude this case in the following. We consider B-subsections (u,b) with 1≠u∈D. As usual, we may assume that b has defect group CD(u). Suppose first that I(B) acts faithfully on Z(D). We apply Proposition 2. If u∉Z(D), then CD(u)≅Cpd−2×Cp. Thus, Theorem 1 implies LL(Zb/Rb)≤pd−2+p−2. Now assume that u∈Z(D). The centric subgroups in the fusion system of b are maximal subgroups of D. In particular, they are abelian of rank 2. Now by [29, Proposition 6.11], it follows that b is a controlled block. Since I(b)≅CI(B)(u)=1, b is nilpotent and Zb≅ZFD. By Proposition 8, we obtain LL(Zb/Rb)≤LL(Zb)=LL(ZFD)≤LL(FZ(D))=pd−2. Hence, Proposition 2 gives LL(ZB)≤LL(ZB/RB)+1≤pd−2+p−1≤pd−1−p+1. Now we deal with the case where I(B) is non-faithful on Z(D). We make use of (3.1). Let ∣⟨u⟩∣=ps. The dominated block b¯ has defect group CD(u)/⟨u⟩. If u∉Z(D), then ∣CD(u)/⟨u⟩∣=pd−s−1≥p and (ps−1)LL(Zb¯)≤(ps−1)pd−s−1≤pd−1−p. Next suppose that u∈Z(D). Then D′⊆⟨u⟩ and b¯ has defect group D/⟨u⟩≅Cpd−s−2×Cp×Cp. In case ⟨u⟩<Z(D), we have s≤d−3 and Theorem 1 implies (ps−1)LL(Zb¯)≤(ps−1)(pd−s−2+2p−2)≤pd−2+2pd−2−2pd−3−3p+2≤pd−1−p. Finally, assume that ⟨u⟩=Z(D). By [33, Lemma 3], we have I(b¯)≅I(b)≅CI(B)(u)≠1. We want to show that I(b¯) acts semiregularly on D/Z(D). Let D=⟨x,y,z⟩ as in Lemma 9, and let γ∈I(b¯). Then yγ≡yizj(modZ(D)) and zγ≡ykzl(modZ(D)) for some i,j,k,l∈Z. Since D has nilpotency class 2, we have [y,z]=[y,z]γ=[yγ,zγ]=[yizj,ykzl]=[y,z]il−jk. It follows that il−jk≡1(modp) and I(b¯)≤SL(2,p). As a p′-subgroup of SL(2,p), I(b¯) acts indeed semiregularly on D/Z(D). Thus, Corollary 5 shows that (ps−1)LL(Zb¯)≤(pd−2−1)p=pd−1−p. Therefore, the claim follows from (3.1).□ We do not expect that Lemma 11 is sharp. In fact, Jennings’s Theorem [11] shows that LL(FWp3)=4p−3. Even in this small case the perfect isometry classes are not known (see for example [27]). We are now in a position to deal with all non-abelian defect groups. Theorem 12. Let Bbe a block of FGwith non-abelian defect group of order pd. Then LL(ZB)≤pd−1+pd−2−pd−3. Proof We argue by induction on d. Let D be a defect group of B. Again we will use (3.1). Let (u,b) be a B-subsection with u∈D of order ps≠1. As before, we may assume that the dominated block b¯ has defect group CD(u)/⟨u⟩. If CD(u)/⟨u⟩ is cyclic, then CD(u) is abelian and, therefore, CD(u)<D. Hence, (ps−1)LL(Zb¯)≤(ps−1)pd−s−1≤pd−1−1≤pd−1+pd−2−pd−3−1. Suppose next that CD(u)/⟨u⟩ is abelian of type (pa1,…,par) with r≥2. If s=d−2, then D fulfills the assumption of Lemma 9. Hence, by Proposition 10 and Lemma 11, we conclude that LL(ZB)≤pd−1−p+1≤pd−1+pd−2−pd−3. Consequently, we can restrict ourselves to the case s≤d−3. Theorem 1 shows that LL(Zb¯)≤pa1+⋯+par−r+1≤pa1+⋯+ar−1+par−1≤∣CD(u)∣ps+1+p−1. Hence, one gets (ps−1)LL(Zb¯)≤(ps−1)(pd−s−1+p−1)≤pd−1+ps+1−ps−1≤pd−1+pd−2−pd−3−1. It remains to consider the case where CD(u)/⟨u⟩ is non-abelian. Here induction gives (ps−1)LL(Zb¯)≤(ps−1)(pd−s−1+pd−s−2−pd−s−3)≤pd−1+pd−2−pd−3−1. Now the claim follows with (3.1).□ In the situation of Theorem 12 we also have dimZFD≤∣Z(D)∣+pd−∣Z(D)∣p≤pd−1+pd−2−pd−3, but it is not clear if LL(ZB)≤dimZFD. Doing the analysis in the proof above more carefully, our bound can be slightly improved, but this does not affect the order of magnitude. Note also that Theorem 12 improves Equation (1.1) even in case p=2, because then exp(D)≥4. Nevertheless, we develop a stronger bound for p=2 in the following. We begin with the 2-blocks of defect 4. The definition of the minimal non-abelian group MNA(2,1) can be found in [29, Theorem 12.2]. The following proposition covers all non-abelian 2-groups of order 16. Proposition 13. Let Bbe a block of FGwith defect group D. Then LL(ZB)≤{3ifD≅C4⋊C4,4ifD∈{M16,D8×C2,Q8×C2,MNA(2,1)},5ifD∈{D16,Q16,SD16,W16}.In all cases, we have LL(ZB)≤LL(FD). Proof The case D≅M16 has already been done in Proposition 10. For the metacyclic group D≅C4⋊C4, B is nilpotent (see [29, Theorem 8.1]) and the result follows from Proposition 8. For the dihedral, quaternion, semidihedral and minimal non-abelian groups the perfect isometry class is uniquely determined by the fusion system of B (see [6, 32]). Moreover, all these fusion systems are non-exotic (see [29, Theorem 10.17]). In particular, LL(ZB)≤LL(ZFH) for some finite group H. More precisely, if B is non-nilpotent, we may consider the following groups H: PGL(2,7) and PSL(2,17) if D≅D16, SL(2,7) and 𝚂𝚖𝚊𝚕𝚕𝙶𝚛𝚘𝚞𝚙(240,89)≅2.S5 if D≅Q16, M10 (Mathieu group), GL(2,3) and PSL(3,3) if D≅SD16, 𝚂𝚖𝚊𝚕𝚕𝙶𝚛𝚘𝚞𝚙(48,30)≅A4⋊C4 if D≅MNA(2,1). For all these groups H, the number LL(ZFH) can be determined with GAP [9]. Finally, for D∈{D8×C2,Q8×C2,W16}, one can enumerate the possible generalized decomposition matrices of B up to basic sets (see [28, Propositions 3, 4 and 5]). In each case the isomorphism type of ZB can be determined with a result of Puig [26]. We omit the details. Observe that we improve Lemma 11 for D≅W16. Finally, the claim LL(ZB)≤LL(FD) can be shown with Jennings’s Theorem [11] or one consults [12, Corollary 4.2.4 and Table 4.2.6].□ Next we elaborate on Lemma 9. Lemma 14. Let Bbe a 2-block of FGwith non-abelian defect group Dsuch that there exists a z∈Z(D)with D/⟨z⟩≅C2n×C2where, n≥2. Then LL(ZB)<∣D∣/2. Proof By hypothesis, there exist two maximal subgroups M1 and M2 of D containing z such that M1/⟨z⟩≅M2/⟨z⟩≅C2n. It follows that M1 and M2 are abelian. Since D=M1M2, we obtain Z(D)=M1∩M2 and ∣D:Z(D)∣=4. This implies ∣D′∣=2 (see e.g. [1, Lemma 1.1]). Obviously, D′≤⟨z⟩. By Lemma 9, we may assume that Z(D) is abelian of rank 2. Suppose for the moment that B is nilpotent. Since Z(D) is not cyclic, D≇M2m for all m. Now a result of Koshitani–Motose [21, Theorems 4 and 5] shows that LL(ZB)=LL(ZFD)≤LL(FD)<∣D∣2. For the remainder of the proof, we may assume that B is not nilpotent. Suppose that Z(D)=Φ(D). Then D is minimal non-abelian and it follows from [29, Theorem 12.4] that D≅MNA(r,1) for some r≥2. By Proposition 13, we can assume that r≥3. By the main result of [32], B is isotypic to the principal block of H≔A4⋊C2r. In particular, LL(ZB)≤LL(FH). Note that H contains a normal subgroup N≅C2r−1×C2×C2 such that H/N≅S3 (see [32, Lemma 2]). By Passman [25, Theorem 1.6], (JFH)2⊆(JFN)(FH)=(FH)(JFN). It follows that LL(FH)≤2LL(FN)=2(2r−1+2)<2r+1=∣D∣2. Thus, we may assume ∣D:Φ(D)∣=8 in the following. Let F be the fusion system of B. Suppose that there exists an F-essential subgroup Q≤D (see [29, Definition 6.1]). Then z∈Z(D)≤CD(Q)≤Q and Q is abelian. Moreover, ∣D:Q∣=2. It is well known that AutF(Q) acts faithfully on Q/Φ(Q) (see [29, p. 64]). Since D/Q≤AutF(Q), we obtain D′⊈Φ(Q). On the other hand, z2∈Φ(Q). This shows that D′=⟨z⟩ and D/D′ has rank 2. However, this contradicts ∣D:Φ(D)∣=8. Therefore, B is a controlled block and Aut(D) is not a 2-group. Let 1≠α∈Aut(D) be of odd order. Then α acts trivially on D′ and on Ω(Z(D))/D′, since Z(D) has rank 2. Hence, α acts trivially on Ω(Z(D)) and also on Z(D). But then α acts non-trivially on D/⟨z⟩≅C2n×C2 which is impossible. This contradiction shows that there are no more blocks with the desired property.□ Proposition 15. Let Bbe a 2-block of FGwith non-abelian defect group of order 2d. Then LL(ZB)<2d−1. Proof We mimic the proof of Theorem 12. Let D be a defect group of B, and let (u,b) be a B-subsection such that u has order 2s>1. As usual, let b¯ be the block of CG(u)/⟨u⟩ dominated by b. It suffices to show that (2s−1)LL(Zb¯)≤2d−1−2. If CD(u)/⟨u⟩ is cyclic, then CD(u) is abelian and CD(u)<D. Then we obtain (2s−1)LL(Zb¯)≤(2s−1)2d−s−1=2d−1−2d−s−1. We may assume that s=d−1. Then by Proposition 10, we may assume that D is dihedral, semidihedral or quaternion. Moreover, by Proposition 13, we may assume that d≥5. Then [29, Theorem 8.1] implies LL(ZB)≤dimZB=k(B)≤2d−2+5<2d−1. Now suppose that CD(u)/⟨u⟩ is abelian of type (2a1,…,2ar) with r≥2. As in Theorem 12, we may assume that s≤d−3. If a1=1 and r=2, then by Lemma 14, we may assume that CD(u)<D. Hence, we obtain (2s−1)LL(Zb¯)≤(2s−1)(2d−s−2+1)≤2d−2+2d−3≤2d−1−2 in this case. Now suppose that r≥3 or ai>1 for i=1,2. If r=3 and a1=a2=a3=1, we have (2s−1)LL(Zb¯)≤2d−1−4. In the remaining cases, we have s≤d−4 and (2s−1)LL(Zb¯)≤(2s−1)(2d−s−2+3)≤2d−2+3·2d−4≤2d−1−2. Finally, suppose that CD(u)/⟨u⟩ is non-abelian. Then the claim follows by induction on d.□ Corollary 16. Let Bbe a block of FGwith defect group D. Then LL(ZB)≥∣D∣/2if and only if one of the following holds: Dis cyclic and l(B)≤2, D≅C2n×C2for some n≥1, D≅C2×C2×C2and Bis nilpotent, D≅C3×C3and Bis nilpotent. Proof Suppose that LL(ZB)≥∣D∣/2. Then by Theorem 12 and Proposition 15, D is abelian. If D is cyclic, we have LL(ZB)=∣D∣−1l(B)+1. If additionally l(B)≥3, then we get the contradiction ∣D∣≤4. Now suppose that D is not cyclic. Then ∣D∣2≤LL(ZB)≤∣D∣p+p−1 by Theorem 1 and we conclude that p2≤∣D∣≤2p(p−1)p−2. This yields p≤3. Suppose first that p=3. Then we have D≅C3×C3 and 5=LL(ZB)≤k(B)−l(B)+1 by Proposition 3. It follows from [15] that I(B)∉{C4,C8,Q8,SD16} (note that k(B)−l(B) is determined locally). The case I(B)≅C2 is excluded by Corollary 5. Hence, we may assume that I(B)∈{C2×C2,D8}. By [30, Theorem 3] and [34, Lemma 2], B is isotypic to its Brauer correspondent in NG(D). This gives the contradiction LL(ZB)≤3. Therefore, B must be nilpotent and LL(ZB)=5. Now let p=2. Then D has rank at most 3 by Theorem 1. If the rank is 3, we obtain LL(ZB)≤2d−2+2 and d=3. In this case, I(B)∈{1,C3,C7,C7⋊C3} acts semiregularly on [D,I(B)] except if I(B)≅C7⋊C3. By Corollary 5, we see that B is nilpotent or I(B)≅C7⋊C3. By [13], B is isotypic to its Brauer correspondent in NG(D) and this gives LL(ZB)=2 if I(B)≅C7⋊C3. Therefore, B is nilpotent and LL(ZB)=4. It remains to handle defect groups of rank 2. Here, D≅C2n×C2 for some n≥1. If n≥2, then B is always nilpotent and LL(ZB)=2n+1. If n=1, then both possibilities l(B)∈{1,3} give LL(ZB)≥2. Conversely, we have seen that all our examples actually satisfy LL(ZB)≥∣D∣/2.□ The following approach gives more accurate results for a given arbitrary defect group. For a finite p-group P we define a recursive function L as follows: L(P)≔{pa1+⋯+par−r+1ifP≅Cpa1×⋯×Cpar,pd−2ifP≅Mpdwithpd≠8,pd−1−p+1ifP≅Wpdwithpd≠16,3ifP∈{D8,Q8,C4⋊C4},4ifP∈{D8×C2,Q8×C2,MNA(2,1)},5ifP∈{D16,Q16,SD16,W16},max{(∣⟨u⟩∣−1)L(CP(u)/⟨u⟩):1≠u∈P}+1otherwise. Then, by the results above, every block B of FG with defect group D satisfies LL(ZB)≤L(D). For example, there are only three non-abelian defect groups of order 36 giving the worst case estimate LL(ZB)≤287. 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The Quarterly Journal of Mathematics – Oxford University Press
Published: Sep 1, 2018
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