Simple module | EPFL Graph Search

In mathematics, specifically in ring theory, the simple modules over a ring R are the (left or right) modules over R that are non-zero and have no non-zero proper submodules. Equivalently, a module M is simple if and only if every cyclic submodule generated by a non-zero element of M equals M. Simple modules form building blocks for the modules of finite length, and they are analogous to the simple groups in group theory. In this article, all modules will be assumed to be right unital modules over a ring R. Examples Z-modules are the same as abelian groups, so a simple Z-module is an abelian group which has no non-zero proper subgroups. These are the cyclic groups of prime order. If I is a right ideal of R, then I is simple as a right module if and only if I is a minimal non-zero right ideal: If M is a non-zero proper submodule of I, then it is also a right ideal, so I is not minimal. Conversely, if I is not minimal, then there is a non-zer

Modules d'endo-p-permutation

Jean-Marie Urfer

This dissertation is concerned with the study of a new family of representations of finite groups, the endo-p-permutation modules. Given a prime number p, a finite group G of order divisible by p and an algebraically closed field k of characteristic p, we say that a kG-module M is an endo-p-permutation module if its endomorphism algebra Endk(M) is a p-permutation kG-module, that is a direct summand of a permutation kG-module. This generalizes the notion, first introduced by E. Dade in 1978, of endo-permutation modules for p-groups . For P a p-group, E. Dade defined an abelian group structure on the set of isomorphism classes of indecomposable endo-permutation kP-modules with vertex P and he proved that the complete description of the structure of this group is equivalent to the classification of endo-permutation kP-modules. This group of isomorphism classes is now called the Dade group of the p-group P. The problem of describing the Dade group for an arbitrary p-group was recently solved by S. Bouc. This opens the question of studying endo-p-permutation modules, which are the natural generalization to arbitrary finite groups of endo-permutation modules. In the following text, we present the basic properties of endo-p-permutation modules and give a characterization of indecomposable endo-p-permutation modules with vertex P via properties of their sources modules. In particular, when the normalizer of P controls p-fusion, we are able to give a complete classification of sources of indecomposable endo-p-permutation modules with vertex P, using Bouc's description of the Dade group. When p is odd, we also give an alternative proof of a theorem of Dade concerned with extensions of endo-permutation modules, using our previous results. We present a consequence of this theorem of Dade on the structure of the multiplicity module associated to an indecomposable endo-p-permutation module. Finally we study some concrete examples of endo-p-permutation modules such as relative syzygies and relative Heller translates. We prove also that the Green correspondent of an indecomposable kNG(P)- endo-p-permutation module with vertex P is not in general an endo-p-permutation kG-module. The study of such representations is motivated by the important role they play in certain areas of representations theory. For instance, endo-permutation modules, and more generally endo-p-permutation modules (as is proved here), appear in the study of simple modules for p-solvable groups.

EPFL2006

Generic direct summands of tensor products for simple algebraic groups and quantum groups at roots of unity

Jonathan Gruber

Let G be either a simple linear algebraic group over an algebraically closed field of characteristic l>0 or a quantum group at an l-th root of unity. The category Rep(G) of finite-dimensional G-modules is non-semisimple. In this thesis, we develop new techniques for studying Krull-Schmidt decompositions of tensor products of G-modules.More specifically, we use minimal complexes of tilting modules to define a tensor ideal of singular G-modules, and we show that, up to singular direct summands, taking tensor products of G-modules respects the decomposition of Rep(G) into linkage classes. In analogy with the classical translation principle, this allows us to reduce questions about tensor products of G-modules in arbitrary l-regular linkage classes to questions about tensor products of G-modules in the principal block of G. We then identify a particular non-singular indecomposable direct summand of the tensor product of two simple G-modules in the principal block (with highest weights in two given l-alcoves), which we call the generic direct summand because it appears generically in Krull-Schmidt decompositions of tensor products of simple G-modules (with highest weights in the given l-alcoves). We initiate the study of generic direct summands, and we use them to prove a necessary condition for the complete reducibility of tensor products of simple G-modules, when G is a simple algebraic group of type A_n.

EPFL2022

Modules d'endo-p-permutation

Jean-Marie Urfer

EPFL2006