Projection Maps for Tensor Products of $gl(r,\mathbf{C})$-Representations

Abstract

We investigate the tensor product $\mathcal{T}=V({\lambda }^1)\otimes \cdots \otimes V({\lambda }^m)$ of the finite dimensional irreducible $\mathcal{G}=gl(r,\mathbf{C})$ modules labelled by partitions ${\lambda }^1,\cdots ,{\lambda }^m$ of $m$ not necessarily distinct numbers $n_1,\cdots ,n_m$ respectively. We determine the centralizer algebra ${\text{End}}_{\mathcal{G}}(\mathcal{T})$ and the projection maps of $\mathcal{T}$ onto its irreducible $\mathcal{G}$-summands and give an explicit construction of the corresponding maximal vectors. In the special case that $n_i=1$ for $i=1,\cdots ,m$, the results reduce to the well-known results of Schur and Weyl.