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

• Published in: Journal of algebra Vol. 166; no. 3; pp. 529 - 567
• Main Authors: Benkart, G., Chakrabarti, M., Halverson, T., Leduc, R., Lee, C.Y., Stroomer, J.
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 15.06.1994
• ISSN: 0021-8693; 1090-266X
• DOI: 10.1006/jabr.1994.1166

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 = 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.