<html><head><title>[Unipot] 1 Preface</title></head> <body text="#000000" bgcolor="#ffffff"> [<a href = "chapters.htm">Up</a>] [<a href ="CHAP002.htm">Next</a>] [<a href = "theindex.htm">Index</a>] <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-Universität Gießen, Arndtstr. 2, 35392 Gießen, Germany'') <p> <p> [<a href = "chapters.htm">Up</a>] [<a href ="CHAP002.htm">Next</a>] [<a href = "theindex.htm">Index</a>] <P> <address>Unipot manual<br>Oktober 2004 </address></body></html>