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Algorithmic orbit classification for some Borel group actions. (English) Zbl 0612.17005

Let H be a reductive algebraic group over a base field K, let G be a Borel subgroup, and let U be the Lie algebra of the unipotent radical of G. The problem is to describe the orbit structures in the G-modules U and \(U^*\). To attack this problem the authors have developed an algorithm which uses the root system of G and the weight space of the G-module. The procedure, which is quite technical, is presented in detail in the paper and is illustrated by several examples.
The algorithm was implemented on a computer to obtain a complete classification of the orbits in U or \(U^*\) when H is \(A_ n\) (n\(\leq 6)\), \(B_ 2\), \(G_ 2\), \(B_ 3\) or \(C_ 3\). The complete classification is also obtained for the orbits in \(U^*\) when H is \(C_ 4\) or \(A_ 7\). In other cases, partial results are given.
Reviewer: Th.Farmer

MSC:

17B20 Simple, semisimple, reductive (super)algebras
20G05 Representation theory for linear algebraic groups
22E25 Nilpotent and solvable Lie groups
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References:

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