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<div class="ChapSects"><a href="chap1.html#X7DFB63A97E67C0A1">1 <span class="Heading">Introduction</span></a>
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<h3>1 <span class="Heading">Introduction</span></h3>
<p>This manual describes the <strong class="pkg">Repsn</strong> package for computing matrix representations in characteristic zero of finite groups. Most of the functions in <strong class="pkg">Repsn</strong> have been written according to the algorithm described in the author's Ph.D thesis <a href="chapBib.html#biBDab-03">[Dab-03]</a> and <a href="chapBib.html#biBDD-08">[DD-08]</a> (see <a href="chapBib.html#biBDab-05">[Dab-05]</a>).</p>
<p>For constructing representations of simple groups and their covers we use the algorithm described in <a href="chapBib.html#biBDix-93">[Dix-93]</a>. To use this algorithm for constructing a representation of a group G affording an irreducible character chi of G, we need to have a subgroup H of G such that the restriction of chi to H has a linear constituent with multiplicity one. In this case we say H is a <em>character subgroup</em> relative to chi (or a chi-subgroup). A chi-subgroup for each irreducible character chi of degree less than 100 of simple groups and their covers are listed in <a href="chapBib.html#biBDab-06">[Dab-06]</a> and <a href="chapBib.html#biBDab-07">[Dab-07]</a>.</p>
<p>All <strong class="pkg">Repsn</strong> functions are written entirely in the <strong class="pkg">GAP</strong> language. It is proved in <a href="chapBib.html#biBDab-05">[Dab-05]</a> and <a href="chapBib.html#biBDD-08">[DD-08]</a> that the algorithm is correct for any group with a character of degree less than 100. Indeed, if the group is solvable, there is no restriction on the character degree. In practice the program is quite fast when the degree is small, but can be very slow when it is necessary to call one of the subprograms which extend irreducible representations. In the latter case the number of element wise operations required to extend a representation of degree d is proportional to d^6.</p>
<p><strong class="pkg">Repsn</strong> is implemented in the <strong class="pkg">GAP</strong> language, and runs on any system supporting <strong class="pkg">GAP</strong>4. The <strong class="pkg">Repsn</strong> package is loaded into the current <strong class="pkg">GAP</strong> session with the command</p>
<pre class="normal">
gap> LoadPackage( "repsn" );
</pre>
<p>(see section <em>Loading a GAP Package</em> in the <strong class="pkg">GAP</strong> Reference Manual). One could install the <strong class="pkg">Repsn</strong> package on <strong class="pkg">GAP</strong>4.3. In this case it is loaded with the command</p>
<pre class="normal">
gap> RequirePackage( "repsn" );
</pre>
<p><strong class="pkg">Repsn</strong> has been developed by<br /></p>
<p>Vahid Dabbaghian<br /> Department of Mathematics<br /> Simon Fraser University<br /> Burnaby, British Columbia,<br /> V5A 1S6 Canada.<br /> e-mail: vdabbagh@sfu.ca<br /></p>
<p>Please send bug reports, suggestions and other comments to this e-mail address.</p>
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