[Unipot] 1 Preface

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# 1 Preface

### Sections

1. Root Systems
2. Citing Unipot

Unipot is a package for GAP4 GAP4. The version 1.0 of this package was 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 xr(t) for all rinPhi+,tinK. The roots of the underlying root system Phi are ordered according to the height function. Each element of U is a product of the root elements xr(t). By 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 from the simple Lie algebras. We also are using the same 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 Bl may produce an error and computations in groups of types Bl, Cl 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.

Since 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 at least the revision 4.2 of GAP.

Beginning with version 1.2 of Unipot, the new package loading mechanism of GAP4.4 is used and therefore, GAP4.4 is required.

## 1.2 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:

    [Hal02] Sergei Haller. Unipot --- a system for computing with elements
            of unipotent subgroups of Chevalley groups, Version 1.2.
            Justus-Liebig-Universitaet Giessen, Germany, July 2002.
            (http://...)

(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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