1 Preface

Sections

1. Root Systems
2. Future of Unipot
3. Citing Unipot

Unipot is a package for GAP4 GAP4. This package is the content of my diploma thesis SH2000.

Let U be a unipotent subgroup of a Chevalley group of Type L(K). Then it is generated by the elements x_r(t) for all r Î F⁺,t Î K. The roots of the underlying root system F are ordered according to the height function. Each element of U is a product of the root elements x_r(t). By the Theorem 5.3.3 from Carter72 each element of U can be uniquely written as a product of root elements with roots in increasing order. This unique form is called the canonical form.

The main purpose of this package is to compute the canonical form of an element of the group U. For we have implemented the unipotent subgroups of Chevalley groups and their elements as GAP objects and installed some operations for them. One method for the operation Comm uses the Chevalley's commutator formula, which we have implemented, too.

1.1 Root Systems

We are using the root systems and the structure constants available in GAP via the simple Lie algebras. We also are using the ordering of roots available in GAP.

Note that the structure constants in GAP4.1 are not generated corresponding to a Chevalley basis, so computations in the groups of type B_l may produce an error and computations in groups of types B_l, C_l and F₄ may lead to wrong results. In the groups of other types we haven't seen any wrong results but can not guarantee that all results are correct.

In the revision 4.2 of GAP the structure constants are generated corresponding to a Chevalley basis, so that they meet all our assumptions.

Therefore the package requires the revision 4.2 of GAP.

1.2 Future of Unipot

In one of the future versions of the package Unipot we plan to implement some other features. Here is a small list of them:

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GAP4.2 provides special root system objects. We should use them.

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Provide some root systems in common notations (like Carter or Bourbaki).

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Allow the user to provide his own table of structure constants.

--

Provide whole Chevalley groups as GAP objects

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Provide root subgroups

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The elements of Chevalley groups should act on the underlying simple Lie algebra as automorphisms

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There are many known properties of the Chevalley groups and their unipotent subgroups like simplicity, central series, etc. Implement them.

1.3 Citing Unipot

If you use Unipot to solve a problem or publish some result that was partly obtained using Unipot, I would appreciate it if you would cite Unipot, just as you would cite another paper that you used. (Below is a sample citation.) Again I would appreciate if you could inform me about such a paper.

Specifically, please refer to:

[Hal00] Sergei Haller. Unipot --- a system for computing with elements
        of unipotent subgroups of Chevalley groups, Version 1.1.
        Justus-Liebig Universitaet Giessen, Germany, July 2000. 
        (ftp://ftp-pclabor.hrz.uni-giessen.de/SHadow/unipot/)

(Should the reference style require full addresses please use: ``Arbeitsgruppe Algebra, Mathematisches Institut, Justus-Liebig Universität Gießen, Arndtstr. 2, 35392 Gießen, Germany'')

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