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Computational and Combinatorial Methods for Algebraic Group Actions

Research Project
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01.04.2010
 - 31.03.2012

In our research in algebraic transformation groups we encountered many combinatorial and computational problems, in particular in questions related to representation theory and affine algebraic geometry. Some of them came up in a natural way when we were studying specific examples. Moreover, the development of Computer Algebra Programs like SINGULAR, LiE or GAP provide us with powerful tools which allow to push many computations much further than a few years ago. We are now able to calculate a number of new examples which were out of reach until recently and thus get new insight into the basic problems and a much better understanding of the algebraic and geometric background. This also changes the way how to look at these problems, namely from a computational point of view. In particular, the question of existence of a priori bounds and of estimates and the study of complexity of the computations became important. On the other hand, there are a lot of classical results, in particular calculations of invariants and covariants from the last century which have been used in the literature, but were never verified as far as we know. In this project we have among others the following aspects in mind. - Finite generation and finding generators; - Explicit calculation of invariants and covariants in representation theory; - Construction and study of examples with specific properties. In this context we plan to work on the following five specific problems. - Generators for the Group of Polynomial Automorphism of Affine n-Space: This is part of Pierre-Marie Poloni's research project; it is related to the thesis of Immanuel Stampfli, and there will also be some collaboration with Stefane Maubach. - Essential Dimension of Algebraic Groups: This is part of Roland Loetscher's thesis; here the method of multihomogeneous covariants plays a fundamental role. - G-equivariant multiplication on homogeneous spaces: This is a new project which came up in the study of equivariant Hilbert schmes. - Computation of Invariants and Covariants of Binary Forms: This is the subject of Mihaela Popoviciu's thesis. - On the Monoid of Degrees on an Algebra: This is part of the research work of Stephane Venereau, with the aim to get a new and more combinatorial and computational insight into the classical Epimorphism Theorem and the new results about tame automorphisms of affine 3-space A^3.

Members (1)

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Hanspeter Kraft

Principal Investigator