Abelian Defect Group Research Articles

On Blocks with Abelian Defect Groups of Small Rank

Let B be a p-block of a finite group with abelian defect group D. Suppose that D has no elementary abelian direct summand of order p4. Then, we show that B satisfies Brauer’s k(B)-Conjecture (i.e. \({k(B)\le|D|}\)). Together with former results, it follows that Brauer’s k(B)-Conjecture holds for all blocks of defect at most 3. We also obtain some related results.

Splendid and perverse equivalences

Inspired by the works of Rickard on splendid equivalences [9] and of Chuang and Rouquier on perverse equivalences [6], we are here interested in the combination of both, i.e. a splendid perverse equivalence. This is naturally the right framework to understand the relations between global and local perverse equivalences between blocks of finite groups, as a splendid equivalence induces local derived equivalences via the Brauer functor. We prove that under certain conditions, we have an equivalence between a perverse equivalence between the homotopy category of p-permutation modules and local derived perverse equivalences, in the case of abelian defect groups.

Extensions of Rickard equivalences

Broué conjectures that a block of a finite group G with abelian defect group P is Rickard equivalent to its Brauer correspondent in the normalizer NG(P). We find a character-theoretic criterion on extensions of Rickard equivalences between blocks. As an application, we prove that the conjecture holds for the principal block of a finite group with abelian defect group and cyclic hyperfocal subgroup.

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.

Broué's isotypy conjecture for the sporadic groups and their covers and automorphism groups

Let B be a p-block of a finite group G with abelian defect group D such that S ≤ G ≤ 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 Broué. This has been done by Rouquier for principal blocks and it remains to deal with the non-principal blocks.

On stable equivalences with endopermutation source

We show that a bimodule between block algebras which has a fusion stable endopermutation module as a source and which induces Morita equivalences between centralisers of nontrivial subgroups of a defect group induces a stable equivalence of Morita type; this is the converse to a theorem of Puig. The special case where the source is trivial has long been known by many authors. The earliest instance of a result deducing a stable equivalence of Morita type from local Morita equivalences with possibly nontrivial endopermutation source is due to Puig, in the context of blocks with abelian defect groups with a Frobenius inertial quotient. The present note is motivated by an application, due to Biland, to blocks of finite groups with structural properties known to hold for hypothetical minimal counterexamples to the Zp⁎-Theorem.

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.

Equivalences between blocks of cohomological Mackey algebras

Let \(G\) be a finite group and \((K,\mathcal {O},k)\) be a \(p\)-modular system which is large enough. Let \(R=\mathcal {O}\) or \(k\). There is a bijection between the blocks of the group algebra \(RG\) and the central primitive idempotents (the blocks) of the so-called cohomological Mackey algebra \(co\mu _{R}(G)\). Here, we introduce the notion of permeable derived equivalence and we prove that a permeable derived equivalence between two blocks of group algebras implies the existence of a derived equivalence between the corresponding blocks of cohomological Mackey algebras. In particular, in the context of Broué’s abelian defect group conjecture, if two blocks are splendidly derived equivalent, then the corresponding blocks of cohomological Mackey algebras are derived equivalent.

Equivalences between blocks of p-local Mackey algebras

Let G be a finite group and (K,O,k) be a p-modular system. Let R=O or k. There is a bijection between the blocks of the group algebra and the blocks of the so-called p-local Mackey algebra μR1(G). Let b be a block of RG with abelian defect group D. Let b′ be its Brauer correspondant in NG(D). It is conjectured by Broué that the blocks RGb and RNG(D)b′ are derived equivalent. Here we look at equivalences between the corresponding blocks of p-local Mackey algebras. We prove that an analogue of the Broué's conjecture is true for the p-local Mackey algebras in the following cases: for the principal blocks of p-nilpotent groups and for blocks with defect 1.

On the p-extensions of inertial blocks

Assume that a block is inertial and has abelian defect groups. We prove that a bimodule inducing a basic Morita equivalence between this block and its Brauer correspondence admits an abelian equivariant p-extension.

Cartan matrices and Brauer’s k(B)-conjecture III

For a block B of a finite group we prove that \({k(B) \le ({\rm det} C - 1)/l(B)+l(B) \le {\rm det} C}\) where k(B) [respectively l(B)] is the number of irreducible ordinary (respectively Brauer) characters of B, and C is the Cartan matrix of B. As an application, we show that Brauer’s k(B)-Conjecture holds for every block with abelian defect group D and inertial quotient T provided there exists an element u ∈ D such that C T (u) acts freely on \({D/\langle u\rangle}\). This gives a new proof of Brauer’s Conjecture for abelian defect groups of rank at most 2. We also prove the conjecture in case \({l(B) \le 3}\).

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.

Zero character heights and Abelian defect groups of blocks in finite groups

Let G be a group, Irr(G) be the set of all the irreducible ordinary characters of G, B ∊ Bl(G) be a block of G and Irr(B) be the set of all the irreducible ordinary characters of G which are in the block B. In this paper the object is to study the ordinary irreducible characters of height zero in relation to the Abelian defect groups of the blocks in which such characters sit. We prove that if B ∊ Bl(G) is a block of G with a defect group D such that every χ ∊ Irr(B) has height zero, then D is Abelian.

Parabolic Deligne–Lusztig varieties

Motivated by the Broué conjecture on blocks with abelian defect groups for finite reductive groups, we study “parabolic” Deligne–Lusztig varieties and construct on those which occur in the Broué conjecture an action of a braid monoid, whose action on their ℓ-adic cohomology will conjecturally factor through a cyclotomic Hecke algebra. In order to construct this action, we need to enlarge the set of varieties we consider to varieties attached to a “ribbon category”; this category has a Garside family, which plays an important role in our constructions, so we devote the first part of our paper to the necessary background on categories with Garside families.

A characterization of saturated fusion systems over abelian 2-groups

Given a saturated fusion system F over a 2-group S, we prove that S is abelian provided any element of S is F-conjugate to an element of Z(S). This generalizes a Theorem of Camina–Herzog, leading to a significant simplification of its proof. More importantly, it follows that any 2-block B of a finite group has abelian defect groups if all B-subsections are major. Furthermore, every 2-block with a symmetric stable center has abelian defect groups.

On the Harris–Knörr correspondence in p-solvable groups

Fix a prime p and let N be a normal subgroup of a finite p-solvable group G. Suppose that b is a p-block of N with abelian defect group Q and B is a p-block of G covering b. Let b⁎ be the Brauer correspondent of b in NN(Q) and let B⁎ be the unique p-block of NG(Q) that covers b⁎ and induces B. We show that there exist height preserving bijections Ψ of Irr(B) onto Irr(B⁎) and Ω of IBr(B) onto IBr(B⁎) such that the decomposition numbers dχφ and dΨ(χ)Ω(φ) are equal for all χ∈Irr(B) and all φ∈IBr(B).

2-blocks with abelian defect groups

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 Donovanʼs 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×C2m for m⩾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 Donovanʼs 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.

3-Blocks with Abelian defect groups isomorphic to $Z_{3^m } \times Z_{3^n } $

Let G be a finite group and let p be a fixed prime number. Let B be a p-block of G with defect group D. In this paper, we give results on 3-blocks with abelian defect groups isomorphic to \(Z_{3^m } \times Z_{3^n } \). We are particularly interested in the number of irreducible ordinary characters and the number of irreducible Brauer characters in the block. We calculate two important block invariants k(B) and l(B) in this case.

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.