## Experimental Mathematics

- Experiment. Math.
- Volume 11, Issue 1 (2002), 57-67.

### MOP---Algorithmic Modality Analysis for Parabolic Group Actions

Ulf Jürgens and Gerhard Röhrle

#### Abstract

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, 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
mod ($P :\mathfrak n)$.
In this note we describe an algorithmic procedure,
called MOP, which allows one to
determine upper bounds for 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 from Hille and Röhrle (1999), 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.

#### Article information

**Source**

Experiment. Math., Volume 11, Issue 1 (2002), 57-67.

**Dates**

First available in Project Euclid: 10 July 2003

**Permanent link to this document**

https://projecteuclid.org/euclid.em/1057860314

**Mathematical Reviews number (MathSciNet)**

MR1960300

**Zentralblatt MATH identifier**

1050.20033

**Subjects**

Primary: 20G15: Linear algebraic groups over arbitrary fields

Secondary: 17B45: Lie algebras of linear algebraic groups [See also 14Lxx and 20Gxx]

**Keywords**

Linear algebraic groups Lie algebras modality of parabolic groups

#### Citation

Jürgens, Ulf; Röhrle, Gerhard. MOP---Algorithmic Modality Analysis for Parabolic Group Actions. Experiment. Math. 11 (2002), no. 1, 57--67. https://projecteuclid.org/euclid.em/1057860314