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

AbstractFor 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 = Cr 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 C of all transformations on T commuting with the action of G, to construct the irreducible C-representations, and to identify C with a certain subalgebra B(r)p,q of the Brauer algebra B(r)p+q