<?xml version="1.0" encoding="UTF-8"?> <!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> <title>GAP (HAP) - Chapter 19: Coxeter diagrams and graphs of groups</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" /> </head> <body><a href="../www/index.html"><small>HAP home</small></a> <div class="chlinkprevnexttop"> <a href="chap0.html">Top of Book</a> <a href="chap18.html">Previous Chapter</a> <a href="chap20.html">Next Chapter</a> </div> <p><a id="X79D0502085B6734A" name="X79D0502085B6734A"></a></p> <div class="ChapSects"><a href="chap19.html#X79D0502085B6734A">19. <span class="Heading"> Coxeter diagrams and graphs of groups</span></a> </div> <h3>19. <span class="Heading"> Coxeter diagrams and graphs of groups</span></h3> <div class="pcenter"><table cellspacing="10" class="GAPDocTable"> <tr> <td class="tdleft"><code class="code"> CoxeterDiagramComponents(D) </code></p> <p>Inputs a Coxeter diagram D and returns a list [D_1, ..., D_d] of the maximal connected subgraphs D_i.</td> </tr> <tr> <td class="tdleft"><code class="code"> CoxeterDiagramDegree(D,v) </code></p> <p>Inputs a Coxeter diagram D and vertex v. It returns the degree of v (i.e. the number of edges incident with v).</td> </tr> <tr> <td class="tdleft"><code class="code"> CoxeterDiagramDisplay(D) </code> <br /> <code class="code"> CoxeterDiagramDisplay(D,"web browser") </code></p> <p>Inputs a Coxeter diagram D and displays it as a .gif file. It uses the Mozilla web browser as a default to view the diagram. An alternative browser can be set using a second argument.</p> <p>This function requires Graphviz software.</td> </tr> <tr> <td class="tdleft"><code class="code"> CoxeterDiagramFpArtinGroup(D) </code></p> <p>Inputs a Coxeter diagram D and returns the corresponding finitely presented Artin group.</td> </tr> <tr> <td class="tdleft"><code class="code"> CoxeterDiagramFpCoxeterGroup(D) </code></p> <p>Inputs a Coxeter diagram D and returns the corresponding finitely presented Coxeter group.</td> </tr> <tr> <td class="tdleft"><code class="code"> CoxeterDiagramIsSpherical(D) </code></p> <p>Inputs a Coxeter diagram D and returns "true" if the associated Coxeter groups is finite, and returns "false" otherwise.</td> </tr> <tr> <td class="tdleft"><code class="code"> CoxeterDiagramMatrix(D) </code></p> <p>Inputs a Coxeter diagram D and returns a matrix representation of it. The matrix is given as a function DiagramMatrix(D)(i,j) where i,j can range over the vertices.</td> </tr> <tr> <td class="tdleft"><code class="code"> CoxeterSubDiagram(D,V) </code></p> <p>Inputs a Coxeter diagram D and a subset V of its vertices. It returns the full sub-diagram of D with vertex set V.</td> </tr> <tr> <td class="tdleft"><code class="code"> CoxeterDiagramVertices(D) </code></p> <p>Inputs a Coxeter diagram D and returns its set of vertices.</td> </tr> <tr> <td class="tdleft"><code class="code"> EvenSubgroup(G) </code></p> <p>Inputs a group G and returns a subgroup G^+. The subgroup is that generated by all products xy where x and y range over the generating set for G stored by GAP. The subgroup is probably only meaningful when G is an Artin or Coxeter group.</td> </tr> <tr> <td class="tdleft"><code class="code"> GraphOfGroupsDisplay(D) </code> <br /> <code class="code"> GraphOfGroupsDisplay(D,"web browser") </code></p> <p>Inputs a graph of groups D and displays it as a .gif file. It uses the Mozilla web browser as a default to view the diagram. An alternative browser can be set using a second argument.</p> <p>This function requires Graphviz software.</td> </tr> <tr> <td class="tdleft"><code class="code"> GraphOfGroupsTest(D) </code></p> <p>Inputs an object D and itries to test whether it is a Graph of Groups. However, it DOES NOT test the injectivity of any homomorphisms. It returns true if D passes the test, and false otherwise.</p> <p>Note that there is no function IsHapGraphOfGroups() because no special data type has been created for these graphs.</td> </tr> </table><br /><p> </p><br /> </div> <div class="chlinkprevnextbot"> <a href="chap0.html">Top of Book</a> <a href="chap18.html">Previous Chapter</a> <a href="chap20.html">Next Chapter</a> </div> <div class="chlinkbot"><span class="chlink1">Goto Chapter: </span><a href="chap0.html">Top</a> <a href="chap1.html">1</a> <a href="chap2.html">2</a> <a href="chap3.html">3</a> <a href="chap4.html">4</a> <a href="chap5.html">5</a> <a href="chap6.html">6</a> <a href="chap7.html">7</a> <a href="chap8.html">8</a> <a href="chap9.html">9</a> <a href="chap10.html">10</a> <a href="chap11.html">11</a> <a href="chap12.html">12</a> <a href="chap13.html">13</a> <a href="chap14.html">14</a> <a href="chap15.html">15</a> <a href="chap16.html">16</a> <a href="chap17.html">17</a> <a href="chap18.html">18</a> <a href="chap19.html">19</a> <a href="chap20.html">20</a> <a href="chap21.html">21</a> <a href="chap22.html">22</a> <a href="chap23.html">23</a> <a href="chap24.html">24</a> <a href="chap25.html">25</a> <a href="chapInd.html">Ind</a> </div> <hr /> <p class="foot">generated by <a href="http://www.math.rwth-aachen.de/~Frank.Luebeck/GAPDoc">GAPDoc2HTML</a></p> </body> </html>