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Concept# Representation theory

Summary

Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essence, a representation makes an abstract algebraic object more concrete by describing its elements by matrices and their algebraic operations (for example, matrix addition, matrix multiplication). The theory of matrices and linear operators is well-understood, so representations of more abstract objects in terms of familiar linear algebra objects helps glean properties and sometimes simplify calculations on more abstract theories.
The algebraic objects amenable to such a description include groups, associative algebras and Lie algebras. The most prominent of these (and historically the first) is the representation theory of groups, in which elements of a group are represented by invertible matrices such that the group operation is matrix multiplication

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Group (mathematics)

In mathematics, a group is a non-empty set with an operation that satisfies the following constraints: the operation is associative, has an identity element, and every element of the set has an inver

Group theory

In abstract algebra, group theory studies the algebraic structures known as groups.

The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fi

The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fi

Lie group

In mathematics, a Lie group (pronounced liː ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract conce

This thesis is in the context of representation theory of finite groups. More specifically, it studies biset functors. In this thesis, I focus on two biset functors: the Burnside functor and the functor of p-permutation modules. For the Burnside functor we first give a result that characterize some B-groups; B-groups being the essential ingredient in the classification of composition factors of the Burnside functor. The second result compares the Burnside functor and the functor of free modules. Note that the functor of free modules is not a biset functor since the inflation of a free module is not necessarily free. To compare those functors we will work on an adjunction between the category of biset functors and the category of functors that do not have inflation. An aspect of the work done on the functor of p-permutation module is to compare the functor of p-permutation modules and the functor of ordinary representations. On the other hand, because of the classification of p-permutation modules, we try to express the functor o p-permutation modules in terms of the functor of projective modules (which is not a biset functor). We will use an adjunction between the category of biset functors and a category that contains the functor of projective modules.

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Study the basics of representation theory of groups and associative algebras.

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This course consists of two parts. The first part covers basic concepts of molecular symmetry and the application of group theory to describe it. The second part introduces Laplace transforms and Fourier series and their use for solving ordinary and partial differential equations in chemistry & c.e.

For a finite group G, certain G-algebras associated to finite G-sets are studied, which generalize both the group algebra and Drinfeld's double. The modular representation theory of these algebras is considered and a striking reciprocity theorem is proved.

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.

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