# Deux contributions a la théorie de représentations de groupes algébriques

Abstract : 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.
Keywords :
Document type :
Theses
Domain :

https://tel.archives-ouvertes.fr/tel-00012189
Contributor : Karin Baur <>
Submitted on : Friday, April 28, 2006 - 4:08:14 PM
Last modification on : Tuesday, February 5, 2019 - 3:50:10 PM
Long-term archiving on: : Saturday, April 3, 2010 - 11:17:02 PM

### Identifiers

• HAL Id : tel-00012189, version 1

### Citation

Karin Baur. Deux contributions a la théorie de représentations de groupes algébriques. Mathématiques [math]. Universität Basel, 2002. Français. ⟨tel-00012189⟩

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