Abelian Defect Group Conjecture Research Articles

Broué's abelian defect group conjecture for blocks with cyclic hyperfocal subgroups

AbstractIn this paper, we prove that the hyperfocal subalgebra of a block with an abelian defect group and a cyclic hyperfocal subgroup is Rickard equivalent to the group algebra of the semidirect of the hyperfocal subgroup by the inertial quotient of the block. In particular, the hyperfocal subalgebra is a Brauer tree algebra, which is analogous to the structure of blocks with cyclic defect groups. As a consequence, we show that Broué's abelian defect group conjecture holds for blocks with cyclic hyperfocal subgroups.

Descent of splendid Rickard equivalences in GLn(q)

Let n be a positive integer and q a prime power. We prove that a refined version of Broué's abelian defect group conjecture holds for unipotent ℓ-blocks of GLn(q), where ℓ∤q. We also give a sufficient condition on general ℓ-blocks of GLn(q) to satisfy the refined abelian defect group conjecture.

Modular representations of finite groups and Lie theory

This article discusses the modular representation theory of finite groups of Lie type from the viewpoint of Broué's abelian defect group conjecture. We discuss both the defining characteristic case, the inspiration for Alperin's weight conjecture, and the non-defining case, the inspiration for Broué's conjecture. The modular representation theory of general finite groups is conjectured to behave both like that of finite groups of Lie type in defining characteristic, and in non-defining characteristic, to a large extent.The expected behavior of modular representation theory of finite groups of Lie type in defining characteristic is particularly difficult to grasp along the lines of Broué's conjecture, and we raise a new question related to the change of central character.We introduce a degeneration method in the modular representation theory of finite groups of Lie type in non-defining characteristic. Combined with the rigidity property of perverse equivalences, this provides a setting for two-variable decomposition matrices, for large characteristic. This should help make progress towards finding decomposition matrices, an outstanding problem with few general results beyond the case of general linear groups. This last part is based on joint work with David Craven and Olivier Dudas.

The strengthened Broué abelian defect group conjecture for SL(2,pn) and GL(2,pn)

We show that each p-block of SL(2,pn) and GL(2,pn) over an arbitrary complete discrete valuation ring is splendidly Rickard equivalent to its Brauer correspondent, hence give new evidence for a refined version of Broué's abelian defect group conjecture proposed by Kessar and Linckelmann.

Mod ℓ cohomology of some Deligne–Lusztig varieties for GLn(q)

In this article, we study the mod ℓ cohomology of some Deligne–Lusztig varieties for GLn(q). We prove that the cohomology groups of these varieties are torsion-free under some conditions on the characteristic. Under the torsion-free assumption we can compute the cohomology groups explicitly and we prove that the cohomology complex satisfies a partial-tilting condition, which is one of the necessary conditions in the geometric version of Broué's abelian defect group conjecture.

Broué's conjecture for 2-blocks with elementary abelian defect groups of order 32

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 Broue's Abelian Defect Group Conjecture in this situation.

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).

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 \(\ell \).

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.

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.

Morita equivalence classes of $2$-blocks of defect three

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.

On Rouquier blocks for finite classical groups at linear primes

W. Turner has proved that Broué's abelian defect group conjecture holds for certain unipotent blocks of the finite general linear group, the so-called Rouquier blocks. This together with theorems of A. Marcus and of J. Chuang and R. Rouquier proves that the conjecture holds for all blocks of such groups. We prove that other finite classical groups also possess analogues of Rouquier blocks at linear primes.

Broué's abelian defect group conjecture and 3-decomposition numbers of the sporadic simple Conway group [formula omitted

In the representation theory of finite groups, Broué's abelian defect group conjecture says 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 B of the normaliser NG(P) of P in G are derived equivalent. We prove that Broué's conjecture, and even Rickard's splendid equivalence conjecture, are true for the unique 3-block A of defect 2 of the sporadic simple Conway group Co1, implying that both conjectures hold for all 3-blocks of Co1. To do so, we determine the 3-decomposition numbers of A, and we actually show that A is Puig equivalent to the principal 3-block of the symmetric group S6 of degree 6.

Perverse equivalences and Brouéʼs conjecture

We give a new approach to the construction of derived equivalences between blocks of finite groups, based on perverse equivalences, in the setting of Brouéʼs 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.

The McKay conjecture and Brauer's induction theorem

Let $G$ be an arbitrary finite group. The McKay conjecture asserts that $G$ and the normaliser $N_G (P)$ of a Sylow $p$-subgroup $P$ in $G$ have the same number of characters of degree not divisible by $p$ (that is, of $p'$-degree). We propose a new refinement of the McKay conjecture, which suggests that one may choose a correspondence between the characters of $p'$-degree of $G$ and $N_G (P)$ to be compatible with induction and restriction in a certain sense. This refinement implies, in particular, a conjecture of Isaacs and Navarro. We also state a corresponding refinement of the Brou\'e abelian defect group conjecture. We verify the proposed conjectures in several special cases.

Brouéʼs abelian defect group conjecture holds for the double cover of the Higman–Sims sporadic simple group

In the representation theory of finite groups, there is a well-known and important conjecture, due to Brouéʼ, saying 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 B of the normaliser NG(P) of P in G are derived equivalent. We prove in this paper, that Brouéʼs abelian defect group conjecture, and even Rickardʼs splendid equivalence conjecture are true for the faithful 3-block A with an elementary abelian defect group P of order 9 of the double cover 2.HS of the Higman–Sims sporadic simple group. It then turns out that both conjectures hold for all primes p and for all p-blocks of 2.HS.

On blocks stably equivalent to a quantum complete intersection of dimension 9 in characteristic 3 and a case of the Abelian defect group conjecture

Using a stable equivalence due to Rouquier, we prove that Broue's abelian defect group conjecture holds for 3-blocks of defect 2 whose Brauer correspondent has a unique isomorphism class of simple modules. The proof makes use of the fact, also due to Rouquier, that a stable equivalence of Morita type between self-injective algebras induces an isomorphism between the connected components of the outer automorphism groups of the algebras.

Brouéʼs abelian defect group conjecture holds for the sporadic simple Conway group [formula omitted

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 A N of the normaliser N G ( 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 Co 3 . 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 Co 3 .