Two contributions to the representation theory of algebraic groups

Baur, Karin. Two contributions to the representation theory of algebraic groups. 2002, Doctoral Thesis, University of Basel, Faculty of Science.


Official URL: http://edoc.unibas.ch/diss/DissB_6153

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Let V be a �nite 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) 6= 0. We study the the case where
V = C[x; y]n is an irreducible representation of SL2. The subsets we
are interested in are the closures of SL2{orbits Of of forms in C[x; y]n.
We give an explicit description of those orbits that have the separation
The closure of Of 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�
V� 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�+� which is the
component with the maximal highest weight. It appears exactly once
in the decomposition.
Another interesting subset of V�
V� 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
show that in general, Cartan components are small. We present what
happens for G = SL2 and G = SL3 and discuss the representations of
the special linear group in detail.
Advisors:Kraft, Hanspeter
Committee Members:Wallach, Nolan R.
Faculties and Departments:05 Faculty of Science > Departement Mathematik und Informatik > Ehemalige Einheiten Mathematik & Informatik > Algebra (Kraft)
UniBasel Contributors:Kraft, Hanspeter
Item Type:Thesis
Thesis Subtype:Doctoral Thesis
Thesis no:6153
Thesis status:Complete
Number of Pages:67
Identification Number:
edoc DOI:
Last Modified:22 Jan 2018 15:50
Deposited On:13 Feb 2009 15:43

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