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 mu(1)(R)(G). Let b be a block of RG with abelian defect group D. Let b' be its Brauer correspondant in N-G(D). It is conjectured by Broue 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 Broue'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.

Trace maps for Mackey algebras

Baptiste Thierry Pierre Rognerud

Let G be a finite group and R be a commutative ring. The Mackey algebra μR(G) shares a lot of properties with the group algebra RG however, there are some differences. For example, the group algebra is a symmetric algebra and this is not always the case for the Mackey algebra. In this paper we present a systematic approach to the question of the symmetry of the Mackey algebra, by producing symmetric associative bilinear forms for the Mackey algebra. Using the fact that the category of Mackey functors is a closed symmetric monoidal category, we prove that the Mackey algebra μR(G) is a symmetric algebra if and only if the family of Burnside algebras RB(H) for H≤G is a family of symmetric algebras with a compatibility condition. As a corollary, we recover the well known fact that over a field of characteristic zero, the Mackey algebra is always symmetric. Over the ring of integers the Mackey algebra of G is symmetric if and only if the order of G is square free. Finally, if (K, O, k) is a p-modular system for G, we show that the Mackey algebras μO(G) and μk(G) are symmetric if and only if the Sylow p-subgroups of G are of order 1 or p.

Elsevier2015

Relative Projectivity and Relative Endotrivial Modules

Caroline Lassueur

This dissertation is concerned with modular representation theory of finite groups, and more precisely, with the study of classes of representations, which we shall term relative endotrivial modules. Given a prime number p, a finite group G of order divisible by p, we shall say that a kG-module M is endotrivial relatively to the kG-module V if its endomorphism algebra Endk(M) is isomorphic, as a kG-module, to a direct sum of a trivial module and another module which is projective relatively to V , i.e. in short Endk(M) ≅ k ⊕ (V – projective). More accurately, in the first part of the text projectivity relative to kG-modules is used to define groups of relative endotrivial modules, which are obtained by replacing the notion of projectivity with that of relative projectivity in the definition of ordinary endotrivial modules. However, in order to achieve this goal we first need to develop the theory of projectivity relative to modules, in particular with respect to standard group operations such as induction, restriction and inflation. Then, for finite groups having a cyclic Sylow p-subgroup, using the structure of the group T(G) of endotrivial modules described in [MT07], we give a complete classification of the groups of relative endotrivial modules. We also study the case of groups that have a Sylow p-subgroup isomorphic to a Klein group C2 × C2, as well as the case of p-nilpotent groups. In a second part of the text, it is shown how our new groups of relative endotrivial modules provide a natural context to generalise the Dade group of a p-group P to an arbitrary finite group. The classification of endo-permutation modules and the complete description of the structure of the Dade group D(P) was completed in 2004 by S. Bouc with [Bou06]. This adventure had started about 25 years earlier with the first papers and results by E. Dade in [Dad78a] and [Dad78b] in 1978, and the final classification was in fact achieved through the non-effortless combined work of several (co)-authors between 1998 and 2004, including S. Bouc, J. Carlson, N. Mazza and J. Thévenaz. It is most interesting to note that crucial building pieces for this classification are indeed the endotrivial modules, which are particular cases of endo-permutation modules. Yet, for an arbitrary finite group G, no satisfying equivalent group structure to the Dade group on a class of kG-modules has been defined so far. With the goal to fill this gap, we turn the problem upside down, in some sense, and show how one can regard an endo-permutation module as an endotrivial module, of course not in the ordinary sense, but in the relative sense. This shall enable us to endow a set of isomorphism classes of endo-p-permutation modules with a group structure, similar to that of the Dade group. We shall call this new group, the generalised Dade group of the group G, explicitly compute its structure and show how it is closely related to that of the G-stable points of the Dade group of a Sylow p-subgroup of G.

EPFL2012

Equivalences between blocks of cohomological Mackey algebras

Baptiste Thierry Pierre Rognerud

Let G be a finite group and (K, O, k) be a p-modular system “large enough”. Let R = 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 coμR(G). Here, we prove that a so-called 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.

Springer Verlag2015

Relative Projectivity and Relative Endotrivial Modules

Caroline Lassueur

EPFL2012