DescriptionRepresentation Theory is a central field of modern mathematics with connections to many different braches such as analysis, algebra, number theory and even physics. The key idea of the whole theory is the study of (complicated) groups by representing them as matrices.We will start by reading Part I of the beautiful book of Serre "Linear Representations of Finite Groups" [1], where we will mainly learn the theory of characters attached to representations of finite groups. Then, there are various interesting directions that the project may take. Below we list a few of them:
Possible Directions:Further aspects of the representation theory of finite groups: Here the aim is to further study the representation theory of finite groups beyond the character theory, such as the theorems of Artin and Brauer, as for example in Part II of Serre's book.Representation Theory of Compact Groups : Here the aim will be to study the representation theory of compact groups. Representation Theory and L functions: The main aim of this direction is to study the link between the theory of Artin L-functions and representations of Galois groups. For this direction the module, Galois Theory III is a important corequisite. Young Diagrams, Frobenius's Character formula and Schur factors: In this direction the character theory of the symmetric group will be developed and we will also see how ideas of it can be used to study other representations by using tensor products. (see for example Lectures 4 and 6 of [2])
ResourcesThe following books will be used as references
[2] W. Fulton and J. Harris Representation Theory, A first course , GTM 129, Springer Prerequisites
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