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# 1 Preface

### Sections

- Root Systems
- 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 `x`_{r}(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 `x`_{r}(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.

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 `B`_{l} may produce an error and computations in
groups of types `B`_{l}, `C`_{l} and `F`_{4} 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.

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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Unipot manual

Oktober 2004