Cartan components and decomposable tensors

Abstract

We study tensor products Vλ ⊗ Vμ of irreducible representations of a connected, simply-connected, complex reductive group G. In general, such a tensor product is no longer irreducible. A fundamental question is how the irreducible components are embedded in the tensor product. A special component of the tensor product is the so-called Cartan component Vλ+μ which is the component with maximal highest weight. It appears exactly once in the decomposition. Another interesting subset of Vλ ⊗ Vμ is the set of decomposable tensors. The following question arises in this context: Is the set of decomposable tensors in the Cartan component of such a tensor product given as the closure of the G-orbit of a highest weight vector? If this is the case, we say that the Cartan component is small. We give a characterization and a combinatorial description of representations with small Cartan components. Our results show that in general Cartan components are small.