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<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> <html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en"> <head> <script type="text/javascript" src="https://cdn.jsdelivr.net/npm/mathjax@2/MathJax.js?config=TeX-AMS-MML_HTMLor </script> <title>GAP (Polenta) - Chapter 3: An example application</title> <meta http-equiv="content-type" content="text/html; charset=UTF-8" /> <meta name="generator" content="GAPDoc2HTML" /> <link rel="stylesheet" type="text/css" href="manual.css" /> <script src="manual.js" type="text/javascript"></script> <script type="text/javascript">overwriteStyle();</script> </head> <body class="chap3" onload="jscontent()"> <div class="chlinktop"><span class="chlink1">Goto Chapter: </span><a href="chap0_mj.html">Top< <div class="chlinkprevnexttop"> <a href="chap0_mj.html">[Top of Book]</a> <a href="chap0_mj.ht <p id="mathjaxlink" class="pcenter"><a href="chap3.html">[MathJax off]</a></p> <p><a id="X81CAD2F27B2066C4" name="X81CAD2F27B2066C4"></a></p> <div class="ChapSects"><a href="chap3_mj.html#X81CAD2F27B2066C4">3 <span class="Heading">A <div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html </span> </div> <div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html </span> </div> <div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html </span> </div> </div> <h3>3 <span class="Heading">An example application</span></h3> <p>In this section we outline three example computations with functions from the previous chapt <p><a id="X7DAC33E37B977087" name="X7DAC33E37B977087"></a></p> <h4>3.1 <span class="Heading">Presentation for rational matrix groups</span></h4> <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">mats :=</span> [ [ [ 1, 0, -1/2, 0 ], [ 0, 1, 0, 1 ], [ 0, 0, 1, 0 ], [ 0, 0, 0, 1 ] ], [ [ 1, 1/2, 0, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, 1 ], [ 0, 0, 0, 1 ] ], [ [ 1, 0, 0, 1 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 0, 0, 1 ] ], [ [ 1, -1/2, -3, 7/6 ], [ 0, 1, -1, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 0, 1 ] ], [ [ -1, 3, 3, 0 ], [ 0, 0, 1, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 0, 1 ] ] ]; <span class="GAPprompt">gap></span> <span class="GAPinput">G := Group( mats );</span> <matrix group with 5 generators> # calculate an isomorphism from G to a pcp-group <span class="GAPprompt">gap></span> <span class="GAPinput">nat := IsomorphismPcpGroup( G );;</s <span class="GAPprompt">gap></span> <span class="GAPinput">H := Image( nat );</span> Pcp-group with orders [ 2, 2, 3, 5, 5, 5, 0, 0, 0 ] <span class="GAPprompt">gap></span> <span class="GAPinput">h := GeneratorsOfGroup( H );</span> [ g1, g2, g3, g4, g5, g6, g7, g8, g9] <span class="GAPprompt">gap></span> <span class="GAPinput">mats2 := List( h, x -> PreImage( nat, x ) # take a random element of G <span class="GAPprompt">gap></span> <span class="GAPinput">exp := [ 1, 1, 1, 1, 0, 0, 0, 0, 1 ];;</span> <span class="GAPprompt">gap></span> <span class="GAPinput">g := MappedVector( exp, mats2 );</spa [ [ -1, 17/2, -1, 233/6 ], [ 0, 1, 0, -2 ], [ 0, 1, -1, 2 ], [ 0, 0, 0, 1 ] ] # map g into the image of nat <span class="GAPprompt">gap></span> <span class="GAPinput">i := ImageElm( nat, g );</span> g1*g2*g3*g4*g9 # exponent vector <span class="GAPprompt">gap></span> <span class="GAPinput">Exponents( i );</span> [ 1, 1, 1, 1, 0, 0, 0, 0, 1 ] # compare the preimage with g <span class="GAPprompt">gap></span> <span class="GAPinput">PreImagesRepresentative( nat, i );< [ [ -1, 17/2, -1, 233/6 ], [ 0, 1, 0, -2 ], [ 0, 1, -1, 2 ], [ 0, 0, 0, 1 ] ] <span class="GAPprompt">gap></span> <span class="GAPinput">last = g;</span> true </pre></div> <p><a id="X79CF643081B3FB26" name="X79CF643081B3FB26"></a></p> <h4>3.2 <span class="Heading">Modules series</span></h4> <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">gens :=</span> [ [ [ 1746/1405, 524/7025, 418/1405, -77/2810 ], [ 815/843, 899/843, -1675/843, 415/281 ], [ -3358/4215, -3512/21075, 4631/4215, -629/1405 ], [ 258/1405, 792/7025, 1404/1405, 832/1405 ] ], [ [ -2389/2810, 3664/21075, 8942/4215, -35851/16860 ], [ 395/281, 2498/2529, -5105/5058, 3260/2529 ], [ 3539/2810, -13832/63225, -12001/12645, 87053/50580 ], [ 5359/1405, -3128/21075, -13984/4215, 40561/8430 ] ] ]; <span class="GAPprompt">gap></span> <span class="GAPinput">H := Group( gens );</span> <matrix group with 2 generators> <span class="GAPprompt">gap></span> <span class="GAPinput">RadicalSeriesSolvableMatGroup( H [ [ [ 1, 0, 0, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 0, 0, 1 ] ], [ [ 1, 0, 0, 79/138 ], [ 0, 1, 0, -275/828 ], [ 0, 0, 1, -197/414 ] ], [ [ 1, 0, -3, 2 ], [ 0, 1, 55/4, -55/8 ] ], [ [ 1, 4/15, 2/3, 1/6 ] ], [ ] ] </pre></div> <p><a id="X7BA34CD28059D6CD" name="X7BA34CD28059D6CD"></a></p> <h4>3.3 <span class="Heading">Triangularizable subgroups</span></h4> <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">G := PolExamples(3);</span> <matrix group with 2 generators> <span class="GAPprompt">gap></span> <span class="GAPinput">GeneratorsOfGroup( G );</span> [ [ [ 73/10, -35/2, 42/5, 63/2 ], [ 27/20, -11/4, 9/5, 27/4 ], [ -3/5, 1, -4/5, -9 ], [ -11/20, 7/4, -2/5, 1/4 ] ], [ [ -42/5, 423/10, 27/5, 479/10 ], [ -23/10, 227/20, 13/10, 231/20 ], [ 14/5, -63/5, -4/5, -79/5 ], [ -1/10, 9/20, 1/10, 37/20 ] ] ] <span class="GAPprompt">gap></span> <span class="GAPinput">subgroups := SubgroupsUnipotentBy rec( T := <matrix group with 2 generators>, U := <matrix group with 4 generators> ) <span class="GAPprompt">gap></span> <span class="GAPinput">GeneratorsOfGroup( subgroups.T );< [ [ [ 73/10, -35/2, 42/5, 63/2 ], [ 27/20, -11/4, 9/5, 27/4 ], [ -3/5, 1, -4/5, -9 ], [ -11/20, 7/4, -2/5, 1/4 ] ], [ [ -42/5, 423/10, 27/5, 479/10 ], [ -23/10, 227/20, 13/10, 231/20 ], [ 14/5, -63/5, -4/5, -79/5 ], [ -1/10, 9/20, 1/10, 37/20 ] ] ] # so G is triangularizable! </pre></div> <div class="chlinkprevnextbot"> <a href="chap0_mj.html">[Top of Book]</a> <a href="chap0_mj.ht <div class="chlinkbot"><span class="chlink1">Goto Chapter: </span><a href="chap0_mj.html">Top< <hr /> <p class="foot">generated by <a href="https://www.math.rwth-aachen.de/~Frank.Luebeck/GA </body> </html> ¤ Dauer der Verarbeitung: 0.8 Sekunden ¤*© Formatika GbR, Deutschland |
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