Brauer Correspondence Research Articles

Descent of equivalences for blocks with Klein four defect groups

We show that the splendid Rickard complexes for blocks with Klein four defect groups constructed by Rickard and Linckelmann descend to non splitting coefficient rings. As a corollary, Navarro's refinement of the Alperin-McKay conjecture holds for blocks with a Klein four defect group. We also prove that splendid Morita equivalences between blocks and their Brauer correspondents (if there exist) descend to non-split situations.

On the inductive Alperin–McKay conditions in the maximally split case

The Alperin–McKay conjecture relates height zero characters of an \(\ell \)-block with the ones of its Brauer correspondent. This conjecture has been reduced to the so-called inductive Alperin–McKay conditions about quasi-simple groups by the third author. The validity of those conditions is still open for groups of Lie type. The present paper describes characters of height zero in \(\ell \)-blocks of groups of Lie type over a field with q elements when \(\ell \) divides \(q-1\). We also give information about \(\ell \)-blocks and Brauer correspondents. As an application we show that quasi-simple groups of type \(\mathsf C\) over \(\mathbb {F}_q\) satisfy the inductive Alperin–McKay conditions for primes \(\ell \ge 5\) and dividing \(q-1\). Some methods to that end are adapted from Malle and Späth (Ann. Math. (2) 184:869–908, 2016).

Isotypies for p-blocks with non-abelian metacyclic defect groups, p odd

Let p be an odd prime, G be a finite group and b be a p-block of G with non-abelian metacyclic defect group P. Then it is known that a hyperfocal subgroup Q of b is cyclic. In this study motivated by Rouquier's conjecture on blocks with abelian hyperfocal subgroups, we show that b is isotypic to its Brauer correspondents in NG(P) and NG(Q).

Vertices for Iwahori–Hecke algebras and the Dipper–Du conjecture

Let $\mathscr{H}_n$ denote the Iwahori-Hecke algebra corresponding to the symmetric group $\mathfrak{S}_n$. We set up a Green correspondence for bimodules of these Hecke algebras, and a Brauer correspondence between their blocks. We examine Specht modules for $\mathscr{H}_n$ and compute the vertices of certain Specht modules, before using this to give a classification of the vertices of blocks of $\mathscr{H}_n$ in any characteristic. Finally, we apply this classification to resolve the Dipper-Du conjecture about the structure of vertices of indecomposable $\mathscr{H}_n$-modules.

Broué’s Perfect Isometry Conjecture Holds for the Double Covers of the Symmetric and Alternating Groups

In O. Brunat and J. Gramain (2014) recently proved that any two blocks of double covers of symmetric groups are Broue perfectly isometric provided they have the same weight and sign. They also proved a corresponding statement for double covers of alternating groups and Broue perfect isometries between double covers of symmetric and alternating groups when the blocks have opposite signs. Using both the results and methods of O. Brunat and J. Gramain in this paper we prove that when the weight of a block of a double cover of a symmetric or alternating group is less than p then the block is Broue perfectly isometric to its Brauer correspondent. This means that Broue’s perfect isometry conjecture holds for the double covers of the symmetric and alternating groups.We also explicitly construct the characters of these Brauer correspondents which may be of independent interest to the reader.

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.

Vertices of Specht modules and blocks of the symmetric group

This paper studies the vertices, in the sense defined by J.A. Green, of Specht modules for symmetric groups. The main theorem gives, for each indecomposable non-projective Specht module, a large subgroup contained in one of its vertices. A corollary of this theorem is a new way to determine the defect groups of symmetric groups. The main theorem is also used to find the Green correspondents of a particular family of simple Specht modules; as a corollary, this gives a new proof of the Brauer correspondence for blocks of the symmetric group. The proof of the main theorem uses the Brauer homomorphism on modules, as developed by M. Broué, together with combinatorial arguments using Young tableaux.

Vertices, sources and Green correspondents of the simple modules for the large Mathieu groups

We investigate the simple modules for the sporadic simple Mathieu groups M 22 , M 23 and M 24 as well as those of the automorphism group, the covering groups and the bicyclic extensions of M 22 in characteristics 2 and 3. We determine the vertices and sources as well as the Green correspondents of these simple modules. We also find two 3-blocks with elementary abelian defect groups of order 9 in these groups which are Morita equivalent to their Brauer correspondents.

Cohomology Algebras of Blocks of Finite Groups and Brauer Correspondence II

M. Linckelmann defined the cohomology algebras of blocks of finite groups. This note is an attempt to analyze an inclusion of cohomology algebras of blocks that corresponds under Brauer correspondence through transfer maps between the Hochschild cohomology algebras of the blocks.

Lifting Mckay Graphs and Relations to Prime Extensions

We give an algorithmic procedure for constructing the quivers and homogeneous relations of the Brauer correspondents of blocks of abelian noncyclic defect group in the almost simple groups in the ATLAS. This is one step in a program to compute the structure of the indecomposable projectives for these blocks. As an illustration, we determine an explicit tilting complex for the non-principal 3-block of the central extension of PSL(3,4) by C2.

On the Blocks of the Centralizer of the G-Fixed Points Subalgebra of the Interior G-Algebra

Let OG be a p-modular group algebra and H be a subgroup of G containing the normalizer of a p-subgroup P of G; in Section 2, for an interior G-algebra A, we associate a certain block of with each block of C A (A G ), and for an interior H-algebra B, we associate a certain block of with each block of C B (B H ); we have shown that these associate relationships respect Brauer Correspondence for block algebras and the tensor product of interior G-algebras, we have also characterized further Extended Green Correspondence by these associate relationships. Indeed, we attain on the points with the multiplicity 1 of G-algebras the generalized forms of these associate relationships in Section 1.

Cohomology algebras of 2-blocks of finite groups with defect groups of rank two

We shall calculate the cohomology algebras of 2-blocks of tame type and with wreathed defect groups; the cohomology algebra of B under consideration, in the sense of M. Linckelmann, is determined by particular B-subpairs, which we shall specify. In particular, depending our description of the cohomology algebras, we shall define “corestriction maps” between cohomology algebras of blocks which correspond under Brauer correspondence.

Cohomology Algebras of Blocks of Finite Groups and Brauer Correspondence

Cohomology Algebras of Blocks of Finite Groups and Brauer Correspondence

On Relatively Projective Modules and the Külshammer–Robinson Basis

In this paper, we rewrite the relations between the Külshammer–Robinson basis and Navarro basis in the local blocks, under Brauer correspondence. We define N-numbers and KR-matrix, then prove that they can be decided by decomposition numbers and Cartan matrix of local blocks. We give explicit connections between Külshammer–Robinson basis and projective modules. For N-blocks, we give other ways to describe the connections.

Glauberman Correspondence of p-Blocks of Finite Groups

Let G and A be finite groups with coprime orders. Suppose that A is solvable and that it acts on G by automorphisms. Let C=CG(A). By Irr(G) and IrrA(G) we denote the set of all irreducible and all A-invariant irreducible characters of G, respectively. Let D≤C be a fixed p-subgroup of G for a prime p. Using the Glauberman correspondenceπG,A:IrrAG→IrrC,A. Watanabe (J. Algebra216 (1999), 548–565) recently established a Glauberman correspondence between A-invariant p-blocks B of G with defect group D and p-blocks B1 of C with defect group D. Let Br(B) and Br(B1) be the Brauer correspondents of B and B1 in NG(D) and NC(D), respectively. The main result of this article asserts that the block algebras Br(B) and Br(B1) are Morita equivalent. Furthermore, if G is p-solvable and D is abelian, then the block algebras B and B1 are Morita equivalent.

Source Algebras and Source Modules

The aim of this article is to give a self-contained approach in module theoretic terms to two fundamental results in block theory, due to Puig 6, 14.6 : first, there is an embedding of the source algebra of the Brauer correspondent of a block of some finite group into a source algebra of that block, and second, the source algebras of the Brauer correspondence can be described explicitly. Our proof of the first result Theorem 5 and its Corollary 6 below is essentially the translation to our terminology of the proof in 1, 4.10 . The second result, describing the source algebras of the Brauer correspondent, follows also from 3 , but the proofs in 3 use both the main result on the structure of nilpotent blocks in 5 as well as techniques from 6, Sects. 4, 6 , while our approach from Proposition 9 onwards requires only some classical results on the structure of blocks with a central defect group. An account of the results in 6 on source algebras of blocks with a normal defect group can also be found in 7, Sect. 45 , except for the ˆ Ž . explicit description of the central extension E see Section 10 below , as being defined in terms of the multiplicity algebra of the block, which is Ž .Ž . quoted in 7, 45.10 b without proof, referring back to 6 at the end of 7, Sect. 45 .

On perfect isometries and isotypies in alternating groups

Perfect isometries and isotypies are constructed for alternating groups between blocks with abelian defect groups and the Brauer correspondents of these blocks. These perfect isometries and isotypies satisfy additional compatibility conditions which imply that an extended Broue conjecture holds for the principal block of an almost simple group with an abelian Sylow psubgroup and a generalized Fitting subgroup isomorphic to an alternating group. Let G be a finite group and let 0 be a complete discrete valuation ring with field of quotients K of characteristic 0 and residue class field k of characteristic p > 0. We suppose that K contains a primitive IGI-th root of unity. In [4, (6.1)] Michel Broue posed the following Isotypy Conjecture. Let e be a block of OG with abelian defect group D and let f be the Brauer correspondent of e in (9NG (D). Then e and f are isotypic blocks. If G has an abelian Sylow p-subgroup, then the conjecture can be posed for the principal block of 0G. In this case the authors have shown that the conjecture holds provided an extended conjecture holds for the principal block of almost simple groups with an abelian Sylow p-subgroup (see [10, (5E)]). In this paper the isotypy conjecture for an arbitrary block with abelian defect group is proved for alternating groups. In addition, the extended conjecture is proved for the principal p-block of almost simple groups with abelian Sylow p-subgroups and generalized Fitting subgroup isomorphic to an alternating group. We recall the basic definitions. Let -: 0 k be the canonical quotient mapping and let -: OG kG be the induced 0-algebra homomorphism of the group algebras. In particular, -: e -*e induces a bijection between central idempotents of OG and kG. If e is a block idempotent of OG, let KGeMod be the category of left KGe-modules of finite type and let ZK (G, e) be the Grothendieck group of KGeMod. Let Gv be the set of irreducible characters of G over K. We identify R.K (G, e) with the free abelian group on (G, e)v = {X E Gv X(ge) = X(g) for all g E G}. Let CF(G, K) be the K-space of K-valued class functions on G, and let CF(G, e, K) be the K-subspace of class functions ae in CF(G, K) such that ao(ge) = a(g). The Received by the editors March 26, 1996. 1991 Mathematics Subject Classification. Primary 20C15, 20C20; Secondary 20C30. The first author was supported in part by NSF grant DMS 9100310. The second author was supported in part by NSA grant MDA 904 92-H-3027. ?1997 American Mathematical Society

On Blocks with One Simple Module in Any Brauer Correspondent

We prove that a block of a finite group such that the defect groups are abelian and all the Brauer correspondents (the block itself included) have a unique isomorphism class of simple modules is nilpotent.

Virtually Irreducible Lattices and Brauer Correspondence of Blocks

Virtually Irreducible Lattices and Brauer Correspondence of Blocks

Block multiplicities and the Brauer correspondence

Block multiplicities and the Brauer correspondence