Tensor product representations of general linear groups and their connections with brauer algebras

For the complex general linear group G = GL(r, C) we investigate the tensor product module T= (⨂^p V)⨂(⨂^q V) of p copies of its natural representation V = C^r 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 퓎 of all transformations on T commuting with the action of G, to construct the irreducible C-representations, and to identify 퓎 with a certain subalgebra ⨂^(r) _{p, q} of the Brauer algebra ⨂^(r)_p+q.