Two contributions to the representation theory of algebraic groups
Deux contributions a la théorie de représentations de groupes algébriques
Résumé
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.
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.
Une partie de cette thèse étudie des produits tensoriels de deux représentations irreductibles d'un groupes algébrique simple. Il s'agit de comprendre les tenseurs pures dans la componente de Cartan du produit.
Une partie étudie la propriété de séparation d'un sous-ensemble dans un espace vectoriel complex.
Une partie étudie la propriété de séparation d'un sous-ensemble dans un espace vectoriel complex.