Cartan components and decomposable tensors. (Q1423599)

The author studies tensor products \(V_\lambda\otimes V_\mu\) of irreducible representations of a connected, simply-connected, complex reductive group \(G\). A special component of the tensor product is the so-called Cartan component \(V_{\lambda+\mu}\) which is the component with maximal highest weight, and appears exactly once in the decomposition of \(V_\lambda\otimes V_\mu\). The author defines that the Cartan component \(V_{\lambda+\mu}\) of the tensor product \(V_\lambda\otimes V_{\mu}\) is small if the decomposable tensors of \(V_{\lambda+\mu}\) are given by \(\overline{G(v_\lambda\otimes v_\mu)}\). He gives a characterization and a combinatorial description of representations with small Cartan component, and shows that in general Cartan components are small.