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<h1>1 Preface</h1><p>
<P>
<H3>Sections</H3>
<oL>
<li> <A HREF="CHAP001.htm#SECT001">Root Systems</a>
<li> <A HREF="CHAP001.htm#SECT002">Citing Unipot</a>
</ol><p>
<p>
<a name = "I0"></a>

<font face="Gill Sans,Helvetica,Arial">Unipot</font> is a package for <font face="Gill Sans,Helvetica,Arial">GAP</font>4 <a href="biblio.htm#GAP4"><cite>GAP4</cite></a>. The  version 1.0
of   this  package  was   the  content  of  my  diploma   thesis
<a href="biblio.htm#SH2000"><cite>SH2000</cite></a>.
<p>
Let <var>U</var> be a unipotent  subgroup of  a  Chevalley  group of Type
<var>L(K)</var>.  Then it is generated by  the  elements <var>x<sub>r</sub>(t)</var> for all
<var>rinPhi<sup>+</sup>,tinK</var>. The roots of the underlying  root  system
<var>Phi</var>  are  ordered  according  to  the height  function.  Each
element of  <var>U</var> is  a product of the  root elements <var>x<sub>r</sub>(t)</var>. By
Theorem 5.3.3 from  <a href="biblio.htm#Carter72"><cite>Carter72</cite></a>  each element of <var>U</var> can  be
uniquely  written  as a product  of root  elements with roots in
increasing order. This unique form is called the canonical form.
<p>
The main  purpose  of this  package is to compute the  canonical
form of an element of the group <var>U</var>. For we have implemented the
unipotent subgroups  of Chevalley groups and  their elements  as
<font face="Gill Sans,Helvetica,Arial">GAP</font> objects and  installed  some  operations  for  them.  One
method for the operation <code>Comm</code>  uses the Chevalley's commutator
formula, which we have implemented, too.
<p>
<p>
<h2><a name="SECT001">1.1 Root Systems</a></h2>
<p><p>
We  are  using  the  root  systems and  the  structure constants
available in <font face="Gill Sans,Helvetica,Arial">GAP</font> from  the simple  Lie algebras. We  also are
using the same ordering of roots available in <font face="Gill Sans,Helvetica,Arial">GAP</font>.
<p>
Note that the structure constants in <font face="Gill Sans,Helvetica,Arial">GAP</font>4.1 are not generated
corresponding  to  a Chevalley  basis, so  computations  in  the
groups of  type <var>B<sub>l</sub></var>  may produce an  error and computations in
groups  of  types  <var>B<sub>l</sub></var>,  <var>C<sub>l</sub></var>  and <var>F<sub>4</sub></var> 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.
<p>
Since the revision  4.2  of  <font face="Gill Sans,Helvetica,Arial">GAP</font> the  structure constants are
generated corresponding to a  Chevalley basis, so that they meet
all our assumptions.
<p>
Therefore  the package  requires at least  the  revision  4.2 of
<font face="Gill Sans,Helvetica,Arial">GAP</font>.
<p>
Beginning with version 1.2 of <font face="Gill Sans,Helvetica,Arial">Unipot</font>, the new package loading
mechanism of <font face="Gill Sans,Helvetica,Arial">GAP</font>4.4 is used and therefore, <font face="Gill Sans,Helvetica,Arial">GAP</font>4.4 is required.
<p>
<p>
<h2><a name="SECT002">1.2 Citing Unipot</a></h2>
<p><p>
If you  use <font face="Gill Sans,Helvetica,Arial">Unipot</font> to solve a problem or publish some  result
that was partly obtained  using <font face="Gill Sans,Helvetica,Arial">Unipot</font>, I would appreciate it
if you would  cite <font face="Gill Sans,Helvetica,Arial">Unipot</font>,  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.
<p>
Specifically, please refer to:
<p>
<pre>
[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://...)
</pre>
<p>
(Should the reference style require full addresses please use:
``Arbeitsgruppe Algebra,
  Mathematisches Institut,
  Justus-Liebig-Universit&auml;t Gie&szlig;en,
  Arndtstr. 2,
  35392 Gie&szlig;en, Germany'')
<p>
<p>
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<P>
<address>Unipot manual<br>Oktober 2004
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