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MOP – algorithmic modality analysis for parabolic actions. (English) Zbl 1050.20033

Summary: Let \(G\) be a simple algebraic group and \(P\) a parabolic subgroup of \(G\). The group \(P\) acts on the Lie algebra \({\mathfrak p}_u\) of its unipotent radical \(P_u\) via the adjoint action. The modality of this action, \(\text{mod}(P:{\mathfrak p}_u)\), is the maximal number of parameters upon which a family of \(P\)-orbits on \({\mathfrak p}_u\) depends. More generally, we also consider the modality of the action of \(P\) on an invariant subspace \(\mathfrak n\) of \({\mathfrak p}_u\), that is \(\text{mod}(P:{\mathfrak n})\). In this note we describe an algorithmic procedure, called MOP, which allows one to determine upper bounds for \(\text{mod}(P:{\mathfrak n})\).
The classification of the parabolic subgroups \(P\) of exceptional groups with a finite number of orbits on \({\mathfrak p}_u\) was achieved with the aid of MOP. We describe the results of this classification in detail in this paper. In view of the results of L. Hille and G. Röhrle [Transform. Groups 4, No. 1, 35-52 (1999; Zbl 0924.20035)], this completes the classification of parabolic subgroups of all reductive algebraic groups with this finiteness property.
Besides this result we present other applications of MOP, and illustrate an example.

MSC:

20G15 Linear algebraic groups over arbitrary fields
17B45 Lie algebras of linear algebraic groups
20G05 Representation theory for linear algebraic groups
14L30 Group actions on varieties or schemes (quotients)
68W30 Symbolic computation and algebraic computation

Citations:

Zbl 0924.20035

Software:

GAP
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References:

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