On Weyl's construction and Schur-Weyl duality for the symplectic group

The object of this thesis is to study the Schur-Weyl duality between the Brauer algebra and the symplectic group over the complex field. We will first present some results of representation theory of finite groups and the classical Schur-Weyl duality for general linear groups. Our approach will be based on an important result on semisimple algebras over an algebraically closed field known as Double Centralizer Theorem. We will then describe the Weyl's construction for symplectic groups of a finite-dimensional complex representation. Finally, giving some important facts on invariant theory and introducing Brauer diagrams, we will give a proof of the symplectic Schur-Weyl duality.