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Morita equivalence classes of 2-blocks of defect three

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Charles W. Eaton
Charles W. Eaton
Charles W. Eaton
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Abstract

We give a complete description of the Morita equivalence classes of blocks with elementary abelian defect groups of order 8 and of the derived equivalences between them. A consequence is the verification of Brou\'e's abelian defect group conjecture for these blocks. It also completes the classification of Morita and derived equivalence classes of 2-blocks of defect at most three defined over a suitable field.

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... Then Given two finite groups N ✁ G and a block b of ON , we define G[b] as the group of elements of G acting as inner automorphisms on b ⊗ O k. We will use the following result, extracted from [23], when dealing with automorphisms of products of quasisimple groups, similarly to what has been done in [13]. ...
... Corollary 2.9 ( [13]). Let G be a finite group and let N ✁ G be a normal subgroup with prime index ℓ = p. ...
... For the convenience of the reader, we start by writing the classification of 2-blocks with a smaller elementary abelian defect group . [19], [32], [7], [13], [14]). Let G be a finite group and let B be a block of OG with defect group D = (C 2 ) n where n ≤ 4. Then: ...
Morita equivalence classes of blocks with elementary abelian defect groups of order 32
Preprint
  • Aug 2019
We classify the Morita equivalence classes of blocks with elementary abelian defect groups of order 32 with respect to a complete discrete valuation ring with an algebraically closed residue field of characteristic two. As a consequence we prove Harada's conjecture for all these blocks, and we prove Brou\'e's abelian defect group conjecture for 30 of the 34 determined equivalence classes of blocks.
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... If G 1 = L, then [G 1 : L] = 2 and G/G 1 is a 2 ′-group (and so G = G 1 since (G, B) is a reduced pair). By [7] b L is Morita equivalent to a principal block, and so we are done in this case by Theorem 3.16. Hence G 1 = L. Let G 2 be the primage of G ∩ (H 2 × L 2 ) in G and G 3 the preimage of G ∩ (H × J 1 ). ...
... L 2 ) and there is a block B E of OE Morita equivalent to B 2 and with isomorphic defect groups. Now B E is a tensor product of blocks with elementary abelian defect groups of order 4 or 8, so B E is Morita equivalent to a principal block by [7] ...
... Suppose that D/O 2 (G) ∼ = (C 2 ) 4. Then by [5] B is source algebra equivalent to the principal block of O(A 4 × A 4 ), O(A 4 × A 5 ) or O(A 5 × A 5 ). Using the same argument as in the proof of [7, Theorem 1.1] (where further details may be found), by [24, Corollary 1.14] B is Morita equivalent to the principal block of a central extension of A 4 × A 4 , A 4 × A 5 or A 5 × A 5 , so sf O (B) ≤ (|P [7] and B E2 is is also Morita equivalent to a principal block by Theorem 5.7 (again noting that if it is nilpotent covered, then it is inertial and so Morita equivalent to the principal block of A 4 × C 4 or A 5 × C 4 ). Finally suppose that ...
Towards Donovan's conjecture for abelian defect groups
Article
  • Nov 2017
We define a new invariant for a p-block, the strong Frobenius number, which we use to address the problem of reducing Donovan's conjecture to normal subgroups of index p. As an application we use the strong Frobenius number to complete the proof of Donovan's conjecture for 2-blocks with abelian defect groups of rank at most 4 and for 2-blocks with abelian defect groups of order at most 64.
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... The block B of G is a crossed product of c with G/H, and from [9] the principal blocks of 2 G 2 (q) with q as specified above are in the same Morita equivalence class. Hence the possible Morita equivalence classes for B can be determined simply by applying Method 4.2. ...
... If H 1 = 2 G 2 (q) and H 2 = J 1 then T = 1, so G = H. Note that all the possibilities for c are Morita equivalent, since c = c 1 ⊗ OJ 1 and all the possibilities for c 1 are Morita equivalent, still from[9]. ...
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... A classification of all Morita equivalence classes of blocks with a smaller elementary abelian defect 2-group (C 2 ) n has been done by Alperin for n = 1 [1], various authors for n = 2 [16] [22] [6], Eaton for n = 3, 4 [9] [10] and the author for n = 5 [2]. We classify principal blocks with defect group (C 2 ) 6 . ...
... So in cases 6-8 we can limit our analysis to the subgroup Out(H 2 ), and, in case 9, G = H. The block B of G is a crossed product of c with G/H, and from [9] the principal blocks of 2 G 2 (q) with q as specified above are in the same Morita equivalence class. Hence the possible Morita equivalence classes for B can be determined simply by applying Method 4.2. ...
Morita equivalence classes of principal blocks with elementary abelian defect groups of order 64
Preprint
  • Oct 2020
We classify the Morita equivalence classes of principal blocks with elementary abelian defect groups of order 64 with respect to a complete discrete valuation ring with an algebraically closed residue field of characteristic two.
View
... The number of irreducible ordinary and Brauer characters of blocks with defect group (C 2 ) 4 has already been determined in [24] and [9]. Our work continues [8] in which a classification is given for blocks with elementary abelian defect groups of order 8. The Morita equivalence classes of block with Klein four defect groups are known by [11] (and over an appropriate discrete valuation ring by [27]). ...
... (iii) In [8] we were able to make use of the existence (shown in [17]) of a perfect isometry with the Brauer correspondent block to achieve a classification of an appropriate discrete valuation ring. In the current paper we are forced to work just over a field since the crossed products method of [22] and the split extension methods result of [18] are only known in this context. ...
Morita equivalence classes of blocks with elementary abelian defect groups of order 16
Article
  • Dec 2016
We classify the Morita equivalence classes of blocks with elementary abelian defect groups of order 16 with respect to an algebraically closed field of characteristic two. As a consequence, blocks with this defect group are derived equivalent to their Brauer correspondent in the normalizer of a defect group as predicted in Brou\'e's Conjecture. The principal innovation is the precise analysis of some crossed products as introduced by K\"ulshammer.
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... In fact, we may compute its structure constants as in Section 2 (these are integral). Charles Eaton has communicated privately that he determined the Morita equivalence class of B by relying heavily on the classification of the finite simple groups (his methods are described in [10] where he handles the elementary abelian defect group of order 8). We believe that the methods of the present paper are of independent interest. ...
On centers of blocks with one simple module
Preprint
  • May 2016
Let G be a finite group, and let B be a non-nilpotent block of G with respect to an algebraically closed field of characteristic 2. Suppose that B has an elementary abelian defect group of order 16 and only one simple module. The main result of this paper describes the algebra structure of the center of B. This is motivated by a similar analysis of a certain 3-block of defect 2 in [Kessar, 2012].
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... then B is tame and the claim follows from Theorem 2.2 (note that l(B) = 2 is only possible for D ∼ = D 8 ). Finally, if D ∼ = C 3 2 , then the claim follows from Eaton [6]. ...
Finite dimensional algebras not arising as blocks of group algebras
Preprint
  • Jan 2023
We develop new techniques to classify basic algebras of blocks of finite groups over algebraically closed fields of prime characteristic. We apply these techniques to simplify and extend previous classifications by Linckelmann, Murphy and Sambale. In particular, we fully classify blocks with 16-dimensional basic algebra.
View
... Motivated by Donovan's Conjecture in modular representation theory, there has been some interest in determining the possible Morita equivalence classes of p-blocks B of finite groups over a complete discrete valuation ring O with a given defect group D. While progress in the case p > 2 seems out of reach at the moment, quite a few papers appeared recently addressing the situation where D is an abelian 2-group. For instance, in [5,6,7,8,16] a full classification was obtained whenever D is an abelian 2-group of rank at most 3 or D ∼ = C 4 2 . Building on that, the first author determined in [1] the Morita equivalence classes of blocks with defect group D ∼ = C 5 2 . ...
Brou\'e's Conjecture for 2-blocks with elementary abelian defect groups of order 32
Preprint
  • Nov 2020
The first author has recently classified the Morita equivalence classes of 2-blocks B of finite groups with elementary abelian defect group of order 32. In all but three cases he proved that the Morita equivalence class determines the inertial quotient of B. We finish the remaining cases by utilizing the theory of lower defect groups. As a corollary, we verify Brou\'e's Abelian Defect Group Conjecture in this situation.
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... Donovan's conjecture has been verified in several cases: for example, the main result of [13] tells us there are only finitely many Morita equivalence classes of blocks with defect group (C 2 ) n , for n ∈ N. This inspired a program of explicitly describing these classes, which has been achieved for n = 2, 3, 4, and 5, in [11], [12], [14], and [4], respectively. These results and more can be found on the online block library [1]. ...
2$-Blocks whose defect group is homocyclic and whose inertial quotient contains a Singer cycle
Preprint
  • Dec 2019
In this paper we consider a block B with defect group D≅(C2m)nD \cong (C_{2^m})^n and inertial quotient E\mathbb{E} containing a Singer cycle (an element of order 2n−12^n-1). This implies E=E⋊F\mathbb{E} = E \rtimes F where E≅C2n−1E \cong C_{2^n-1} and F≤CnF \leq C_n. In particular, E acts freely on the non-trivial elements of D, and transitively on the elements of order 2. We will classify the possible Morita equivalence classes of B over O\mathcal{O}: when m=1, D is elementary abelian and B is Morita equivalent to the principal block of one of SL2(2n)⋊FSL_2(2^n) \rtimes F, D⋊ED \rtimes \mathbb{E}, or J1J_1 (where J1J_1 occurs only when n=3). When m>1m>1, B is Morita equivalent to D⋊ED \rtimes \mathbb{E}. We will also show that, with the exception of J1J_1, the Morita equivalences are basic.
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... Denote by Z n the residue class group modulo n. Recently, some authors proved the conjecture for some 2-blocks with abelian defect groups, such as Z 2 × Z 2 (see [7] and [29]), Z 2 × Z 2 × Z 2 (see [9]), and Z 2 n × Z 2 n (see [10] for n ≥ 2). The conjecture also has been verified in many other cases. ...
Blocks with defect group $\mathbb Z_{2^n}\times \mathbb Z_{2^n}\times \mathbb Z_{2^m}
Article
  • Mar 2018
In this paper, we prove that a block with defect group Z2n×Z2n×Z2m\mathbb Z_{2^n}\times \mathbb Z_{2^n}\times \mathbb Z_{2^m}, where n≥2n\geq 2 and m is arbitrary, is Morita equivalent to its Brauer correspondent.
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... Proof. If D is elementary abelian, then this is by [19], [7] and [8]. If D ∼ = C 4 × C 4 , then see [9] where it is shown that there are only two Morita equivalence classes. ...
Classifying blocks with defect group $C_{2^n}\times C_2\times C_2
Article
  • Sep 2017
In this paper we classify all blocks with defect group C2n×C2×C2C_{2^n}\times C_2\times C_2 up to Morita equivalence. The classification holds for blocks over a suitable discrete valuation ring as well as for those over an algebraically closed field. This case is significant because it involves comparison of Morita equivalence classes between a group and a normal subgroup of index 2, so requires novel reduction techniques which we hope will be of wider interest. We note that this completes the classification of 2-blocks with abelian defect at most 4 up to Morita equivalence. A consequence is that Broue's abelian defect group conjecture holds for such blocks.
View
... In fact, we may compute its structure constants as in Section 2 (these are integral). Charles Eaton has communicated privately that he determined the Morita equivalence class of B by relying heavily on the classification of the finite simple groups (his methods are described in [10] where he handles the elementary abelian defect group of order 8). We believe that the methods of the present paper are of independent interest. ...
On centers of blocks with one simple module
Article
  • May 2016
Let G be a finite group, and let B be a non-nilpotent block of G with respect to an algebraically closed field of characteristic 2. Suppose that B has an elementary abelian defect group of order 16 and only one simple module. The main result of this paper describes the algebra structure of the center of B. This is motivated by a similar analysis of a certain 3-block of defect 2 in [Kessar, 2012].
View
... This has been verified in only very few special situations. For example, the conjecture holds if D is cyclic or |D| ≤ 8 (see [29,37,40,30,18,10]). In case p = 2 or |D| = 9, the conjecture is known to hold for principal blocks (see [12,9,20]). ...
Brou\'e's isotypy conjecture for the sporadic groups and their covers and automorphism groups
Article
  • Apr 2015
  • INT J ALGEBR COMPUT
Let B be a p-block of a finite group G with abelian defect group D such that S\le G\le Aut(S), S'=S and S/Z(S) is a sporadic simple group. We show that B is isotypic to its Brauer correspondent in N_G(D) in the sense of Brou\'e. This has been done by [Rouquier, 1994] for principal blocks and it remains to deal with the non-principal blocks.
View
Towards Donovan's conjecture for abelian defect groups
Preprint
  • Nov 2017
We define a new invariant for a p-block, the strong Frobenius number, which we use to address the problem of reducing Donovan's conjecture to normal subgroups of index p. As an application we use the strong Frobenius number to complete the proof of Donovan's conjecture for 2-blocks with abelian defect groups of rank at most 4 and for 2-blocks with abelian defect groups of order at most 64.
View
Morita equivalence classes of 2‐blocks with abelian defect groups of rank 4
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Full-text available
We classify all 2‐blocks with abelian defect groups of rank 4 up to Morita equivalence. The classification holds for blocks over a suitable discrete valuation ring as well as for those over an algebraically closed field. An application is that Broué's abelian defect group conjecture holds for all blocks under consideration here.
View
Finite-dimensional algebras not arising as blocks of group algebras
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  • Jun 2023
We develop new techniques to classify basic algebras of blocks of finite groups over algebraically closed fields of prime characteristic. We apply these techniques to simplify and extend previous classifications by Linckelmann, Murphy and Sambale. In particular, we fully classify blocks with 16-dimensional basic algebra.
View
Morita equivalence classes of blocks with elementary abelian defect groups of order 32
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  • Jan 2021
We describe a general technique to classify blocks of finite groups, and we apply it to determine Morita equivalence classes of blocks with elementary abelian defect groups of order 32 with respect to a complete discrete valuation ring with an algebraically closed residue field of characteristic two. As a consequence we verify that a conjecture of Harada holds on these blocks.
View
2-blocks with an abelian defect group and a freely acting cyclic inertial quotient
Preprint
  • Oct 2020
We prove that when p = 2 a block with an abelian defect group, and a cyclic inertial quotient which acts freely on the defect group, is inertial. That is, such a block is basic Morita equivalent to its Brauer correspondent. This generalises a result of the second author on Singer cycle actions on homocyclic defect groups to all free actions on abelian defect groups.
View
2-Blocks whose defect group is homocyclic and whose inertial quotient contains a Singer cycle
Article
  • Jul 2020
We consider a block B of a finite group with defect group D≅(C2m)n and inertial quotient E containing a Singer cycle (an element of order 2n−1). This implies E=E⋊F, where E≅C2n−1, F≤Cn, and E acts transitively on the elements in D of order 2. We classify the basic Morita equivalence classes of B over a complete discrete valuation ring O: when m=1, B is basic Morita equivalent to the principal block of one of SL2(2n)⋊F, D⋊E, or J1 (where J1 occurs only when n=3). When m>1, B is basic Morita equivalent to D⋊E.
View
2$-Blocks whose defect group is homocyclic and whose inertial quotient contains a Singer cycle II
Preprint
Full-text available
  • Apr 2020
We consider 2-blocks of finite groups with defect group D=Q×RD=Q \times R and inertial quotient E\mathbb{E} where Q≅(C2m)nQ \cong (C_{2^m})^n, R≅C2rR \cong C_{2^r}, and E\mathbb{E} contains a Singer cycle of Aut⁡(Q)\operatorname{Aut}(Q) (an element of order 2n−12^n-1). We classify such blocks up to Morita equivalence when either E\mathbb{E} is cyclic or r=1. We achieve a partial classification when r>1r>1 and E is non-cyclic.
View
2-blocks with abelian hyperfocal subgroup of rank 3
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  • Mar 2019
In this paper, we calculate the numbers of irreducible ordinary characters and irreducible Brauer characters in a block of a finite group G, whose associated fusion system over a 2-subgroup P of G (which is a defect group of the block) has an abelian hyperfocal subgroup of rank 3.
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Classifying blocks with abelian defect groups of rank 3 for the prime 2
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  • Aug 2018
In this paper we classify all blocks with defect group C2n×C2×C2 up to Morita equivalence. Together with a recent paper of Wu, Zhang and Zhou, this completes the classification of Morita equivalence classes of 2-blocks with abelian defect groups of rank at most 3. The classification holds for blocks over a suitable discrete valuation ring as well as for those over an algebraically closed field. The case considered in this paper is significant because it involves comparison of Morita equivalence classes between a group and a normal subgroup of index 2, so requires novel reduction techniques which we hope will be of wider interest. We note that this also completes the classification of blocks with abelian defect groups of order dividing 16 up to Morita equivalence. A consequence is that Broue's abelian defect group conjecture holds for all blocks mentioned above.
View
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  • Jun 2018
In this paper, we prove that a block algebra with defect group Z2n×Z2n×Z2m, where n≥2 and m is arbitrary, is Morita equivalent to its Brauer correspondent.
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  • Sep 2011
We determine the structure of 2-blocks with minimal nonabelian defect groups, by making use of the classification of finite simple groups.
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Conjectures of Alperin and Broue for 2-blocks with elementary abelian defect groups of order 8
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Using the classification of finite simple groups we prove Alperin's weight conjecture and the character theoretic version of Broue's abelian defect group conjecture for 2-blocks of finite groups with an elementary abelian defect group of order 8. Comment: 40 pages
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2-Blocks with minimal nonabelian defect groups III
Article
  • Jan 2016
  • PAC J MATH
We prove that two 2-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 and Picaronny (J. Fac. Sci. Univ. Tokyo Sect. IA Math. 39:1 (1992), 141-161), Sambale (J. Algebra 337 (2011), 261-284) and Eaton et al. (J. Group Theory 15:3 (2012), 311-321).
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Morita equivalent blocks in Clifford theory of finite groups. - In: Astérisque. 181-182. 1990. S. 209-215
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Perverse equivalences and Broué’s conjecture
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  • Nov 2013
We give a new approach to the construction of derived equivalences between blocks of finite groups, based on perverse equivalences, in the setting of Broué’s abelian defect group conjecture. We provide in particular local and global perversity data describing the principal blocks and the derived equivalences for a number of finite simple groups with Sylow subgroups elementary abelian of order 9. We also examine extensions to automorphism groups in a general setting.
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Broué's abelian defect group conjecture holds for the sporadic simple Conway group Co3
Article
  • Dec 2011
In the representation theory of finite groups, there is a well-known and important conjecture due to M. Broué. He conjectures that, for any prime p, if a p-block A of a finite group G has an abelian defect group P, then A and its Brauer corresponding block AN of the normaliser NG(P) of P in G are derived equivalent (Rickard equivalent). This conjecture is called Strong Version of Brouéʼs Abelian Defect Group Conjecture. In this paper, we prove that the strong version of Brouéʼs abelian defect group conjecture is true for the non-principal 2-block A with an elementary abelian defect group P of order 8 of the sporadic simple Conway group Co3. This result completes the verification of the strong version of Brouéʼs abelian defect group conjecture for all primes p and for all p-blocks of Co3.
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2Blocks with minimal nonabelian defect groups
Article
  • Jul 2011
We study numerical invariants of 2-blocks with minimal nonabelian defect groups. These groups were classified by Rédei (see Rédei, 1947 [41]). If the defect group is also metacyclic, then the block invariants are known (see Sambale [43]). In the remaining cases there are only two (infinite) families of ‘interesting’ defect groups. In all other cases the blocks are nilpotent. We prove Brauerʼs k(B)-conjecture and Olssonʼs conjecture for all 2-blocks with minimal nonabelian defect groups. For one of the two families we also show that Alperinʼs weight conjecture and Dadeʼs conjecture are satisfied. This paper is a part of the authorʼs PhD thesis.
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The decomposition numbers in characteristic 2 of the groups of Ree type are determined, as well as the Loewy and socle series of the indecomposable projective modules. Moreover, we describe the Green correspondents of the simple modules. As an application of this and similar works on other simple groups with an abelian Sylow 2-subgroup, all of which have been classified apart from those considered in the present paper, we show that the Loewy length of an indecomposable projective module in the principal block of any finite group with an abelian Sylow 2-subgroup of order 2n is bounded by max{2n + 1, 2n}. This bound is the best possible.
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2-blocks with abelian defect groups
Article
  • May 2013
We give a classification, up to Morita equivalence, of 2-blocks of quasi-simple groups with abelian defect groups. As a consequence, we show that Donovan's conjecture holds for elementary abelian 2-groups, and that the entries of the Cartan matrices are bounded in terms of the defect for arbitrary abelian 2-groups. We also show that a block with defect groups of the form C2m×C2mC_{2^m} \times C_{2^m} for m≥2m \geq 2 has one of two Morita equivalence types and hence is Morita equivalent to the Brauer correspondent block of the normaliser of a defect group. This completes the analysis of the Morita equivalence types of 2-blocks with abelian defect groups of rank 2, from which we conclude that Donovan's conjecture holds for such 2-groups. A further application is the completion of the determination of the number of irreducible characters in a block with abelian defect groups of order 16. The proof uses the classification of finite simple groups.
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Donovan conjecture and Loewy length for principal 3-blocks of finite groups with elementary abelian Sylow 3-subgroup of order 9
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  • Aug 2006
In modular representation theory of finite groups there has been a well-known conjecture due to P. Donovan. Donovan conjecture is on blocks of group algebras of finite groups over an algebraically closed field k of prime characteristic p, which says that, for any given finite p-group P, up to Morita equivalence, there are only finitely many block algebras with defect group P. We prove that Donovan conjecture holds for principal block algebras in the case where P is elementary abelian 3-group of order 9. Moreover, under the same assumption, namely, if G is a finite group with elementary abelian Sylow 3-subgroup P of order 9, then the Loewy length of the principal block algebra B 0 (kG) of the group algebra kG is 5 or 7. The results here depend on the classification of finite simple groups.
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There are normal sub-blocks of nilpotent blocks which are NOT nilpotent or, equivalently, nilpotent extensions of non-nilpotent blocks. In this paper we determine the source algebra structure of the non-nilpotent blocks involved in these situations. Actually, we introduce a new type of blocks - called the inertial blocks - which include the nilpotent blocks and is closed by taking normal sub-blocks.
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The structure of blocks with a Klein four defect group
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We prove Erdmann’s conjecture (J Algebra 76:505–518, 1982) stating that every block with a Klein four defect group has a simple module with trivial source, and deduce from this that Puig’s finiteness conjecture holds for source algebras of blocks with a Klein four defect group. The proof uses the classification of finite simple groups.
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