The goal of representation theory is to represent abstract algebraic objects like groups as subobjects of matrix groups. In this course we shall study linear representations of finite groups, which are homomorphisms G → GL(V) from a finite group G to the group of linear automorphisms of a vector space V. Interestingly, this does not only enable us to use tools from linear algebra to solve abstract algebraic problems, but to ultimately also use tools from algebraic geometry, module theory, etc.
When more structure (e.g. a topology on G and/or V) is imposed representation theory becomes very powerful and is used in harmonic analysis, and is deeply connected to the Erlangen and Langlands programs in geometry and number theory, respectively. There will not be enough time to
explore these topics, but this first course dealing with finite groups and complex vector spaces should at least open the door for the student interested in those areas.
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