Cartan components and decomposable tensors
We study tensor products $V_{\lambda}\otimes V_{\mu}$
of irreducible representations of a connected,
simply-connected, complex reductive group $G$.
In general, such a tensor product is not irreducible
anymore.
It is a fundamental question how the irreducible
components are embedded in the tensor product.
A special component of the tensor product is the
so-called Cartan component $V_{\lambda+\mu}$
which is the component with the maximal highest weight.
It appears exactly once in the decomposition.
Another interesting subset of $V_{\lambda}\otimes V_{\mu}$
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
{\itshape 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.
pdf