Let $V$ be a finite dimensional complex vector space. A subset $X$ in $V$ has the separation property if the following holds: For any pair $l$, $m$ of linearly independent linear functions on $V$ there is a point $x$ in $X$ such that $l(x)=0$ and $m(x)$ is nonzero. We study the the case where $V=\C[x,y]_n$ is an irreducible representation of $\SL_2$. The subsets we are interested in are the closures of $\SL_2$-orbits $O_f$ of forms in $\C[x,y]_n$. We give an explicit description of those orbits that have the separation property: The closure of $O_f$ has the separation property if and only if the form $f$ contains a linear factor of multiplicity one. In the second part of this thesis we study tensor products $V_{\lambda}\otimes V_{\mu}$ of irreducible $G$-representations (where $G$ is a reductive complex algebraic group). 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 {\it small}. We show that in general, Cartan components are small. We present what happens for $G=\SL_2$ and $G=\SL_3$ and discuss the representations of the special linear group in detail.
Two contributions to the representation theory of algebraic groups