Radha Kessar

ZJ-THEOREMS FOR FUSION SYSTEMS

For p an odd prime, we generalise the Glauberman-Thompson p-nilpotency theorem [5, Ch. 8, Theorem 3.1] to arbitrary fusion systems. We define a notion of Qd(p)- free fusion systems and show that if F is a Qd(p)-free fusion system on some finite p-group P then F is controlled by W(P) for any Glauberman functor W, generalising Glauberman’s ZJ-theorem [3] to arbitrary fusion systems.

Local control in fusion systems of p-blocks of finite groups

Abstract If p is an odd prime, b a p -block of a finite group G such that SL (2, p ) is not involved in N G ( Q , e )/ C G ( Q ) for any b -subpair ( Q , e ), then N G ( Z ( J ( P ))) controls b -fusion, where P is a defect group of b . This is a block theoretic analogue of Glaubermans ZJ -Theorem. Several results of general interest about fusion and blocks are also proved.

SYMMETRIC GROUPS, WREATH PRODUCTS, MORITA EQUIVALENCES, AND BROUÉ'S ABELIAN DEFECT GROUP CONJECTURE

It is shown that for any prime p, and any non-negative integer w less than p, there exist p-blocks of symmetric groups of defect w, which are Morita equivalent to the principal p-block of the group Sp [rmoust ] Sw. Combined with work of J. Rickard, this proves that Broues abelian defect group conjecture holds for p-blocks of symmetric groups of defect at most 5.

Conjectures of Alperin and Broué for 2-blocks with elementary abelian defect groups of order 8

Abstract Using the classification of finite simple groups, we prove Alperins weight conjecture and the character theoretic version of Broués abelian defect conjecture for 2-blocks of finite groups with an elementary abelian defect group of order 8.

On Perfect Isometries for Tame Blocks

Any 2-block of a finite group G with a quaternion defect group Q8 is Morita equivalent to the corresponding block of the centraliser H of the unique involution of Q8 in G; this answers positively an earlier question raised by M. Broue.

Crossover morita equivalences for blocks of the covering groups of the symmetric and alternating groups

In [S], Joanna Scopes discovered a method for generating Morita equivalences between blocks of symmetric groups and thus for showing that Donavan’s conjecture, that there are only a finite number of Morita equivalence classes of blocks with a given defect group, holds for the blocks of the symmetric groups. This method has led in various different directions. It was generalized by Puig [P1] to demonstrate not only Morita equivalences but also the more restrictive Puig equivalences, thus establishing Puig’s conjecture, that there are only a finite number of Puig equivalence classes for a given defect group, for blocks of the symmetric group. A variant was adapted by the first author to prove Donovan’s conjecture for blocks of the Schur covers of the symmetric and alternating groups, [K]. A related technique was used in [Jo] for blocks of the general linear group, and an adaptation of the method was developed in [HK1], [HK2] to find Morita equivalences between blocks in various other algebraic groups. The method also lead Rickard to a way of demonstrating derived quivalences between blocks of symmetric groups, and this method was then taken up by Chuang and Rouquier [ChR] to show that for a given weight there is only one derived equivalence class of symmetric blocks which, along with [ChK], settled the Broue conjecture for symmetric blocks. In this paper we intend to return to [K] and show that, in fact, the results therein reflected only half of the picture. The results in [K] demonstrated the existence of Morita equivalences between blocks of the covering groups S̃n of Sn or between blocks of the covering groups Ãn of An. We will now reconsider the situation and show that we can equally well get “crossovers” between blocks of Ãn and S̃n. More specifically, the various characters are associated with strict partitions of n and the Morita eqivalences are obtained by an involution which is a variant of the Scopes involution used in Scopes’ original work. The cases treated in [K] were those in which the involution is parity-preserving, and in this paper we will be interested in cases where it is parity-reversing.

Equivalences for blocks of the Weyl groups

We describe the source algebras of the blocks of the Weyl groups of type B and type D in terms of the source algebras of the blocks of the symmetric groups. As a consequence, we show that Puigs conjecture on the finiteness of the number of isomorphism classes of source algebras for blocks of finite groups with a fixed defect group holds for these classes of groups. We also show how certain isomorphisms between subalgebras of block algebras of the symmetric groups can be lifted to block algebras of the Weyl groups of type B.

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 Broues 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.

On the endomorphism algebras of modular Gelfand–Graev representations

We study the endomorphism algebras of a modular Gelfand-Graev representation of a finite reductive group by investigating modular properties of homomorphisms constructed by Curtis and Curtis-Shoji.

A block theoretic analogue of a theorem of Glauberman and Thompson

If p is an odd prime, G a finite group and P a Sylow-p-subgroup of G, a theorem of Glauberman and Thompson states that G is p-nilpotent if and only if NG(Z(J(P))) is p-nilpotent, where J(P) is the Thompson subgroup of P generated by all abelian subgroups of P of maximal order. Following a suggestion of G. R. Robinson, we prove a block-theoretic analogue of this theorem.