Abelian Defect Group Research Articles

Translation by the full twist and Deligne–Lusztig varieties

We prove several conjectures about the cohomology of Deligne–Lusztig varieties: invariance under conjugation in the braid group, behaviour with respect to translation by the full twist, parity vanishing of the cohomology for the variety associated with the full twist. In the case of split groups of type A, and using previous results of the second author, this implies Broué–Michel's conjecture on the disjointness of the cohomology for the variety associated to any good regular element. That conjecture was inspired by Broué's abelian defect group conjecture and the specific form Broué conjectured for finite groups of Lie type [4, Rêves 1 et 2].

An algorithmic approach to perverse derived equivalences Broué's conjecture for [formula omitted

Following Craven and Rouquier's method [8] to tackle Broué's abelian defect group conjecture, we present two algorithms implementing that procedure in the case of principal blocks of defect D≅Cℓ×Cℓ for a prime ℓ; the first describes a stable equivalence between B0(G) and B0(NG(D)), and the second tries to lift such a stable equivalence to a perverse derived equivalence. As an application, we show that Broué's conjecture holds for the principal 5-block of the simple group Ω8+(2).

Some examples of Picard groups of blocks

We calculate examples of Picard groups for 2-blocks with abelian defect groups with respect to a complete discrete valuation ring. These include all blocks with abelian 2-groups of 2-rank at most three with the exception of the principal block of J1. In particular this shows directly that all such Picard groups are finite and Picent, the group of Morita auto-equivalences fixing the centre, is trivial. These are amongst the first calculations of this kind. Further we prove some general results concerning Picard groups of blocks with normal defect groups as well as some other cases.

Donovan\u2019s conjecture, blocks with abelian defect groups and discrete valuation rings

We give a reduction to quasisimple groups for Donovan’s conjecture for blocks with abelian defect groups defined with respect to a suitable discrete valuation ring {mathcal {O}}. Consequences are that Donovan’s conjecture holds for {mathcal {O}}-blocks with abelian defect groups for the prime two, and that, using recent work of Farrell and Kessar, for arbitrary primes Donovan’s conjecture for {mathcal {O}}-blocks with abelian defect groups reduces to bounding the Cartan invariants of blocks of quasisimple groups in terms of the defect. A result of independent interest is that in general (i.e. for arbitrary defect groups) Donovan’s conjecture for {mathcal {O}}-blocks is a consequence of conjectures predicting bounds on the {mathcal {O}}-Frobenius number and on the Cartan invariants, as was proved by Kessar for blocks defined over an algebraically closed field.

On Picard groups of blocks of finite groups

We show that the subgroup of the Picard group of a p-block of a finite group given by bimodules with endopermutation sources modulo the automorphism group of a source algebra is determined locally in terms of the fusion system on a defect group. We show that the Picard group of a block over a complete discrete valuation ring O of characteristic zero with an algebraic closure k of Fp as residue field is a colimit of finite Picard groups of blocks over p-adic subrings of O. We apply the results to blocks with an abelian defect group and Frobenius inertial quotient, and specialise this further to blocks with cyclic or Klein four defect groups.

Categorical actions on unipotent representations of finite unitary groups

Using Harish-Chandra induction and restriction, we construct a categorical action of a Kac-Moody algebra on the category of unipotent representations of finite unitary groups in non-defining characteristic. We show that the decategorified representation is naturally isomorphic to a direct sum of level 2 Fock spaces. From our construction we deduce that the Harish-Chandra branching graph coincides with the crystal graph of these Fock spaces, solving a recent conjecture of Gerber-Hiss-Jacon. We also obtain derived equivalences between blocks, yielding Broue’s abelian defect group conjecture for unipotent \(\ell \)-blocks at linear primes \(\ell \).

Towards Donovan's conjecture for abelian defect groups

We define a new invariant for a p-block of a finite group, 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.

Donovan’s conjecture and blocks with abelian defect groups

We give a reduction of Donovan's conjecture for abelian groups to a similar statement for quasisimple groups. Consequently we show that Donovan's conjecture holds for abelian $2$-groups.

Blocks of symmetric groups, semicuspidal KLR algebras and zigzag Schur-Weyl duality

We prove Turner's conjecture, which describes the blocks of the Hecke algebras of the symmetric groups up to derived equivalence as certain explicit Turner double algebras. Turner doubles are Schur-algebra-like `local' objects, which replace wreath products of Brauer tree algebras in the context of the Broué abelian defect group conjecture for blocks of symmetric groups with non-abelian defect groups. The main tools used in the proof are generalized Schur algebras corresponding to wreath products of zigzag algebras and imaginary semicuspidal quotients of affine KLR algebras.

Classifying blocks with abelian defect groups of rank 3 for the prime 2

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.

Block algebras with HH1 a simple Lie algebra

The purpose of this note is to add to the evidence that the algebra structure of a p-block of a finite group is closely related to the Lie algebra structure of its first Hochschild cohomology group. We show that if B is a block of a finite group algebra kG over an algebraically closed field k of prime characteristic p such that HH1(B) is a simple Lie algebra and such that B has a unique isomorphism class of simple modules, then B is nilpotent with an elementary abelian defect group P of order at least 3, and HH1(B) is in that case isomorphic to the Witt algebra HH1(kP)⁠. In particular, no other simple modular Lie algebras arise as HH1(B) of a block B with a single isomorphism class of simple modules.

Derived equivalences and stable equivalences of Morita type, II

We consider the question of lifting stable equivalences of Morita type to derived equivalences. One motivation comes from an approach to Broué’s abelian defect group conjecture. Another motivation is a conjecture by Auslander and Reiten on stable equivalences preserving the number of non-projective simple modules. A machinery is presented to construct lifts for a large class of algebras, including Frobenius-finite algebras introduced in this paper. In particular, every stable equivalence of Morita type between Frobenius-finite algebras over an algebraically closed field can be lifted to a derived equivalence. Consequently, the Auslander–Reiten conjecture is true for stable equivalences of Morita type between Frobenius-finite algebras. Examples of Frobenius-finite algebras are abundant, including representation-finite algebras, Auslander algebras, cluster-tilted algebras and certain Frobenius extensions. As a byproduct of our methods, we show that, for a Nakayama-stable idempotent element e in an algebra A over an algebraically closed field, each tilting complex over eAe can be extended to a tilting complex over A that induces an almost ν-stable derived equivalence studied in the first paper of this series. Moreover, the machinery is applicable to verify Broué’s abelian defect group conjecture for several cases mentioned in the literature.

Loewy lengths of blocks with abelian defect groups

We consider p p -blocks with abelian defect groups and in the first part prove a relationship between its Loewy length and that for blocks of normal subgroups of index p p . Using this, we show that if B B is a 2 2 -block of a finite group with abelian defect group D ≅ C 2 a 1 × ⋯ × C 2 a r × ( C 2 ) s D \cong C_{2^{a_1}} \times \cdots \times C_{2^{a_r}} \times (C_2)^s , where a i > 1 a_i > 1 for all i i and r ≥ 0 r \geq 0 , then d > LL ⁡ ( B ) ≤ 2 a 1 + ⋯ + 2 a r + 2 s − r + 1 d > \operatorname {LL}(B) \leq 2^{a_1}+\cdots +2^{a_r}+2s-r+1 , where | D | = 2 d |D|=2^d . When s = 1 s=1 the upper bound can be improved to 2 a 1 + ⋯ + 2 a r + 2 − r 2^{a_1}+\cdots +2^{a_r}+2-r . Together these give sharp upper bounds for every isomorphism type of D D . A consequence is that when D D is an abelian 2 2 -group the Loewy length is bounded above by | D | |D| except when D D is a Klein-four group and B B is Morita equivalent to the principal block of A 5 A_5 . We conjecture similar bounds for arbitrary primes and give evidence that it holds for principal 3 3 -blocks.

On the Loewy length of the center of a block with elementary abelian defect groups

ABSTRACTLet G be a finite group and k an algebraically closed field of characteristic p. In this paper we investigate the Loewy structure of centers of indecomposable group algebras kG, for groups G with a normal elementary abelian Sylow p-subgroup. Furthermore, we show a reduction result for the case that a normal abelian Sylow p-subgroup is acted upon by a subgroup of its automorphism group; this is fundamental in providing generic formulae for the Loewy lengths considered.

REGULAR ORBITS OF COPRIME LINEAR GROUPS IN LARGE CHARACTERISTIC

We prove that a finite coprime linear group$G$in characteristic$p\geq \frac{1}{2}(|G|-1)$has a regular orbit. This bound on$p$is best possible. We also give an application to blocks with abelian defect groups.

Isotypies for the quasisimple groups with exceptional Schur multiplier

Let [Formula: see text] be a block with abelian defect group [Formula: see text] of a quasisimple group [Formula: see text], such that [Formula: see text] has exceptional Schur multiplier. We show that, [Formula: see text] is isotypic to its Brauer correspondent in [Formula: see text] in the sense of Broué. The proof uses methods from a previous paper [B. Sambale, Broué’s isotypy conjecture for the sporadic groups and their covers and automorphism groups, Internat. J. Algebra Comput. 25 (2015) 951–976], and relies ultimately on computer calculations. Moreover, we verify the Alperin–McKay conjecture for all blocks of [Formula: see text].

Cartan matrices and Brauer's $k(B)$-conjecture IV

In this note we give applications of recent results coming mostly from the third paper of this series. It is shown that the number of irreducible characters in a $p$-block of a finite group with abelian defect group $D$ is bounded by $|D|$ (Brauer's $k(B)$-conjecture) provided $D$ has no large elementary abelian direct summands. Moreover, we verify Brauer's $k(B)$-conjecture for all blocks with minimal non-abelian defect groups. This extends previous results by various authors.

Refinements of the Orthogonality Relations for Blocks

For a block B of a finite group G there are well-known orthogonality relations for the generalized decomposition numbers. We refine these relations by expressing the generalized decomposition numbers with respect to an integral basis of a certain cyclotomic field. After that, we use the refinements in order to give upper bounds for the number of irreducible characters (of height 0) in B. In this way we generalize results from [Hethelyi-Kulshammer-Sambale, 2014]. These ideas are applied to blocks with abelian defect groups of rank 2. Finally, we address a recent conjecture by Navarro.

On centers of blocks with one simple module

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) [15].

RoCK blocks, wreath products and KLR algebras

We consider RoCK (or Rouquier) blocks of symmetric groups and Hecke algebras at roots of unity. We prove a conjecture of Turner asserting that a certain idempotent truncation of a RoCK block of weight d of a symmetric group $${\mathfrak {S}}_n$$ defined over a field F of characteristic e is Morita equivalent to the principal block of the wreath product $$\mathfrak S_e \wr \mathfrak S_d$$ . This generalises a theorem of Chuang and Kessar that applies to RoCK blocks with abelian defect groups. Our proof relies crucially on an isomorphism between $$F{\mathfrak {S}}_n$$ and a cyclotomic Khovanov–Lauda–Rouquier algebra, and the Morita equivalence we produce is that of graded algebras. We also prove the analogous result for an Iwahori–Hecke algebra at a root of unity defined over an arbitrary field.