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Publication# Varieties of modules for Z/2Z x Z/2Z

Article

Résumé

Let k be an algebraically closed field of characteristic 2. We prove that the restricted nilpotent commuting variety C, that is the set of pairs of (n x n)-matrices (A, B) such that A(2) = B-2 = [A, B] = 0, is equidimensional. C can be identified with the 'variety of n-dimensional modules' for Z/2Z x Z/2Z, or equivalently, for k[X, Y]/(X-2, Y-2). On the other hand, we provide an example showing that the restricted nilpotent commuting variety is not equidimensional for fields of characteristic > 2. We also prove that if e(2) = 0 then the set of elements of the centralizer of e whose square is zero is equidimensional. Finally, we express each irreducible component of C as a direct sum of indecomposable components of varieties of Z/2Z x Z/2Z-modules. (c) 2007 Elsevier Inc. All rights reserved.

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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.