Projection Maps for Tensor Products of <i>gl</i>(<i>r, C</i>)-Representations

Abstract

We investigate the tensor product T =V(λ_1)⊗ ⋯ ⊗_T =V(λm) of the finite dimensional irreducible G = gl(r, C) modules labelled by partitions _λ_1,⋯,λm of m not necessarily distinct numbers _n_1,⋯,nm respectively. We determine the centralizer algebra EndG(T) and the projection maps of T onto its irreducible G-summands and give an explicit construction of the corresponding maximal vectors. In the special case that ni = 1 for i = 1,⋯,m, the results reduce to the well-known results of Schur and Weyl.