Tensor Product Representations of General Linear Groups and Their Connections with Brauer Algebras Original Research Article

For the complex general linear group G = GL(r, image) we investigate the tensor product module T= (circle times operatorp V)circle times operator(circle times operatorq V) of p copies of its natural representation V = imager and q copies of the dual spare V* of V. We describe the maximal vectors of T and from that obtain an explicit decomposition of T into its irreducible G-summands. Knowledge of the maximal vectors allows us to determine the centralizer algebra image of all transformations on T commuting with the action of G, to construct the irreducible image-representations, and to identify image with a certain subalgebra image(r)p,q of the Brauer algebra image(r)p+q.