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<div class="ChapSects"><a href="chap18.html#X7CD67FEA7A1B6345">18 <span class="Heading"> Orbit polytopes and fundamental domains</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap18.html#X7CFDEEC07F15CF82">18.1 <span class="Heading"> </span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap18.html#X802CD86983E8384B">18.1-1 CoxeterComplex</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap18.html#X7B60301179A6E7D2">18.1-2 ContractibleGcomplex</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap18.html#X82B7AF6A876AA021">18.1-3 QuotientOfContractibleGcomplex</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap18.html#X8092E70D8425CEBB">18.1-4 TruncatedGComplex</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap18.html#X79F4F7938116201E">18.1-5 FundamentalDomainStandardSpaceGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap18.html#X80EC50C27EFF2E12">18.1-6 OrbitPolytope</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap18.html#X7A4234867B232E34">18.1-7 PolytopalComplex</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap18.html#X7E03550580383C01">18.1-8 PolytopalGenerators</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap18.html#X797A90E17C18FC89">18.1-9 VectorStabilizer</a></span>
</div></div>
</div>
<h3>18 <span class="Heading"> Orbit polytopes and fundamental domains</span></h3>
<p><a id="X7CFDEEC07F15CF82" name="X7CFDEEC07F15CF82"></a></p>
<h4>18.1 <span class="Heading"> </span></h4>
<p><a id="X802CD86983E8384B" name="X802CD86983E8384B"></a></p>
<h5>18.1-1 CoxeterComplex</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; CoxeterComplex</code>( <var class="Arg">D</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; CoxeterComplex</code>( <var class="Arg">D</var>, <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a Coxeter diagram <span class="SimpleMath">D</span> of finite type. It returns a non-free ZW-resolution for the associated Coxeter group <span class="SimpleMath">W</span>. The non-free resolution is obtained from the permutahedron of type <span class="SimpleMath">W</span>. A positive integer <span class="SimpleMath">n</span> can be entered as an optional second variable; just the first <span class="SimpleMath">n</span> terms of the non-free resolution are then returned.</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap7.html">1</a></span> </p>
<p><a id="X7B60301179A6E7D2" name="X7B60301179A6E7D2"></a></p>
<h5>18.1-2 ContractibleGcomplex</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; ContractibleGcomplex</code>( <var class="Arg">str</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs one of the following strings <span class="SimpleMath">str</span>=: <br /> <br /> "SL(2,Z)" , "SL(3,Z)" , "PGL(3,Z[i])" , "PGL(3,Eisenstein_Integers)" , "PSL(4,Z)" , "PSL(4,Z)_b" , "PSL(4,Z)_c" , "PSL(4,Z)_d" , "Sp(4,Z)" <br /> <br /> or one of the following strings <br /> <br /> "SL(2,O-2)" , "SL(2,O-7)" , "SL(2,O-11)" , "SL(2,O-19)" , "SL(2,O-43)" , "SL(2,O-67)" , "SL(2,O-163)" <br /> <br /> It returns a non-free ZG-resolution for the group <span class="SimpleMath">G</span> described by the string. The stabilizer groups of cells are finite. (Subscripts _b , _c , _d denote alternative non-free ZG-resolutions for a given group G.)<br /> <br /> Data for the first list of non-free resolutions was provided provided by <strong class="button">Mathieu Dutour</strong>. Data for the second list was provided by <strong class="button">Alexander Rahm</strong>.</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap1.html">1</a></span> , <span class="URL"><a href="../tutorial/chap6.html">2</a></span> , <span class="URL"><a href="../tutorial/chap7.html">3</a></span> , <span class="URL"><a href="../tutorial/chap9.html">4</a></span> , <span class="URL"><a href="../tutorial/chap11.html">5</a></span> , <span class="URL"><a href="../tutorial/chap13.html">6</a></span> , <span class="URL"><a href="../tutorial/chap14.html">7</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutArithmetic.html">8</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutBredon.html">9</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutIntro.html">10</a></span> </p>
<p><a id="X82B7AF6A876AA021" name="X82B7AF6A876AA021"></a></p>
<h5>18.1-3 QuotientOfContractibleGcomplex</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; QuotientOfContractibleGcomplex</code>( <var class="Arg">C</var>, <var class="Arg">D</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a non-free <span class="SimpleMath">ZG</span>-resolution <span class="SimpleMath">C</span> and a finite subgroup <span class="SimpleMath">D</span> of <span class="SimpleMath">G</span> which is a subgroup of each cell stabilizer group for <span class="SimpleMath">C</span>. Each element of <span class="SimpleMath">D</span> must preserves the orientation of any cell stabilized by it. It returns the corresponding non-free <span class="SimpleMath">Z(G/D)</span>-resolution. (So, for instance, from the <span class="SimpleMath">SL(2,O)</span> complex <span class="SimpleMath">C=ContractibleGcomplex("SL(2,O-2)");</span> we can construct a <span class="SimpleMath">PSL(2,O)</span>-complex using this function.)</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap13.html">1</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutArithmetic.html">2</a></span> </p>
<p><a id="X8092E70D8425CEBB" name="X8092E70D8425CEBB"></a></p>
<h5>18.1-4 TruncatedGComplex</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; TruncatedGComplex</code>( <var class="Arg">R</var>, <var class="Arg">m</var>, <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a non-free <span class="SimpleMath">ZG</span>-resolution <span class="SimpleMath">R</span> and two positive integers <span class="SimpleMath">m</span> and <span class="SimpleMath">n</span>. It returns the non-free <span class="SimpleMath">ZG</span>-resolution consisting of those modules in <span class="SimpleMath">R</span> of degree at least <span class="SimpleMath">m</span> and at most <span class="SimpleMath">n</span>.</p>
<p><strong class="button">Examples:</strong></p>
<p><a id="X79F4F7938116201E" name="X79F4F7938116201E"></a></p>
<h5>18.1-5 FundamentalDomainStandardSpaceGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; FundamentalDomainStandardSpaceGroup</code>( <var class="Arg">v</var>, <var class="Arg">G</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a crystallographic group G (represented using AffineCrystGroupOnRight as in the GAP package Cryst). It also inputs a choice of vector v in the euclidean space <span class="SimpleMath">R^n</span> on which <span class="SimpleMath">G</span> acts. It returns the Dirichlet-Voronoi fundamental cell for the action of <span class="SimpleMath">G</span> on euclidean space corresponding to the vector <span class="SimpleMath">v</span>. The fundamental cell is a fundamental domain if <span class="SimpleMath">G</span> is Bieberbach. The fundamental cell/domain is returned as a <q>Polymake object</q>. Currently the function only applies to certain crystallographic groups. See the manuals to HAPcryst and HAPpolymake for full details.</p>
<p>This function is part of the HAPcryst package written by <strong class="button">Marc Roeder</strong> and is thus only available if HAPcryst is loaded.</p>
<p>The function requires the use of Polymake software.</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap1.html">1</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutIntro.html">2</a></span> </p>
<p><a id="X80EC50C27EFF2E12" name="X80EC50C27EFF2E12"></a></p>
<h5>18.1-6 OrbitPolytope</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; OrbitPolytope</code>( <var class="Arg">G</var>, <var class="Arg">v</var>, <var class="Arg">L</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a permutation group or matrix group <span class="SimpleMath">G</span> of degree <span class="SimpleMath">n</span> and a rational vector <span class="SimpleMath">v</span> of length <span class="SimpleMath">n</span>. In both cases there is a natural action of <span class="SimpleMath">G</span> on <span class="SimpleMath">v</span>. Let <span class="SimpleMath">P(G,v)</span> be the convex polytope arising as the convex hull of the Euclidean points in the orbit of <span class="SimpleMath">v</span> under the action of <span class="SimpleMath">G</span>. The function also inputs a sublist <span class="SimpleMath">L</span> of the following list of strings:</p>
<p>["dimension","vertex_degree", "visual_graph", "schlegel","visual"]</p>
<p>Depending on the sublist, the function:</p>
<ul>
<li><p>prints the dimension of the orbit polytope <span class="SimpleMath">P(G,v)</span>;</p>
</li>
<li><p>prints the degree of a vertex in the graph of <span class="SimpleMath">P(G,v)</span>;</p>
</li>
<li><p>visualizes the graph of <span class="SimpleMath">P(G,v)</span>;</p>
</li>
<li><p>visualizes the Schlegel diagram of <span class="SimpleMath">P(G,v)</span>;</p>
</li>
<li><p>visualizes <span class="SimpleMath">P(G,v)</span> if the polytope is of dimension 2 or 3.</p>
</li>
</ul>
<p>The function uses Polymake software.</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap11.html">1</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutPolytopes.html">2</a></span> </p>
<p><a id="X7A4234867B232E34" name="X7A4234867B232E34"></a></p>
<h5>18.1-7 PolytopalComplex</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; PolytopalComplex</code>( <var class="Arg">G</var>, <var class="Arg">v</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; PolytopalComplex</code>( <var class="Arg">G</var>, <var class="Arg">v</var>, <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a permutation group or matrix group <span class="SimpleMath">G</span> of degree <span class="SimpleMath">n</span> and a rational vector <span class="SimpleMath">v</span> of length <span class="SimpleMath">n</span>. In both cases there is a natural action of <span class="SimpleMath">G</span> on <span class="SimpleMath">v</span>. Let <span class="SimpleMath">P(G,v)</span> be the convex polytope arising as the convex hull of the Euclidean points in the orbit of <span class="SimpleMath">v</span> under the action of <span class="SimpleMath">G</span>. The cellular chain complex <span class="SimpleMath">C_*=C_*(P(G,v))</span> is an exact sequence of (not necessarily free) <span class="SimpleMath">ZG</span>-modules. The function returns a component object <span class="SimpleMath">R</span> with components:</p>
<ul>
<li><p><span class="SimpleMath">R!.dimension(k)</span> is a function which returns the number of <span class="SimpleMath">G</span>-orbits of the <span class="SimpleMath">k</span>-dimensional faces in <span class="SimpleMath">P(G,v)</span>. If each <span class="SimpleMath">k</span>-face has trivial stabilizer subgroup in <span class="SimpleMath">G</span> then <span class="SimpleMath">C_k</span> is a free <span class="SimpleMath">ZG</span>-module of rank <span class="SimpleMath">R.dimension(k)</span>.</p>
</li>
<li><p><span class="SimpleMath">R!.stabilizer(k,n)</span> is a function which returns the stabilizer subgroup for a face in the <span class="SimpleMath">n</span>-th orbit of <span class="SimpleMath">k</span>-faces.</p>
</li>
<li><p>If all faces of dimension <<span class="SimpleMath">k+1</span> have trivial stabilizer group then the first <span class="SimpleMath">k</span> terms of <span class="SimpleMath">C_*</span> constitute part of a free <span class="SimpleMath">ZG</span>-resolution. The boundary map is described by the function <span class="SimpleMath">boundary(k,n)</span> . (If some faces have non-trivial stabilizer group then <span class="SimpleMath">C_*</span> is not free and no attempt is made to determine signs for the boundary map.)</p>
</li>
<li><p><span class="SimpleMath">R!.elements</span>, <span class="SimpleMath">R!.group</span>, <span class="SimpleMath">R!.properties</span> are as in a <span class="SimpleMath">ZG</span>-resolution.</p>
</li>
</ul>
<p>If an optional third input variable <span class="SimpleMath">n</span> is used, then only the first <span class="SimpleMath">n</span> terms of the resolution <span class="SimpleMath">C_*</span> will be computed.</p>
<p>The function uses Polymake software.</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap11.html">1</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutPolytopes.html">2</a></span> </p>
<p><a id="X7E03550580383C01" name="X7E03550580383C01"></a></p>
<h5>18.1-8 PolytopalGenerators</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; PolytopalGenerators</code>( <var class="Arg">G</var>, <var class="Arg">v</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a permutation group or matrix group <span class="SimpleMath">G</span> of degree <span class="SimpleMath">n</span> and a rational vector <span class="SimpleMath">v</span> of length <span class="SimpleMath">n</span>. In both cases there is a natural action of <span class="SimpleMath">G</span> on <span class="SimpleMath">v</span>, and the vector <span class="SimpleMath">v</span> must be chosen so that it has trivial stabilizer subgroup in <span class="SimpleMath">G</span>. Let <span class="SimpleMath">P(G,v)</span> be the convex polytope arising as the convex hull of the Euclidean points in the orbit of <span class="SimpleMath">v</span> under the action of <span class="SimpleMath">G</span>. The function returns a record <span class="SimpleMath">P</span> with components:</p>
<ul>
<li><p><span class="SimpleMath">P.generators</span> is a list of all those elements <span class="SimpleMath">g</span> in <span class="SimpleMath">G</span> such that <span class="SimpleMath">g⋅ v</span> has an edge in common with <span class="SimpleMath">v</span>. The list is a generating set for <span class="SimpleMath">G</span>.</p>
</li>
<li><p><span class="SimpleMath">P.vector</span> is the vector <span class="SimpleMath">v</span>.</p>
</li>
<li><p><span class="SimpleMath">P.hasseDiagram</span> is the Hasse diagram of the cone at <span class="SimpleMath">v</span>.</p>
</li>
</ul>
<p>The function uses Polymake software. The function is joint work with Seamus Kelly.</p>
<p><strong class="button">Examples:</strong></p>
<p><a id="X797A90E17C18FC89" name="X797A90E17C18FC89"></a></p>
<h5>18.1-9 VectorStabilizer</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; VectorStabilizer</code>( <var class="Arg">G</var>, <var class="Arg">v</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a permutation group or matrix group <span class="SimpleMath">G</span> of degree <span class="SimpleMath">n</span> and a rational vector of degree <span class="SimpleMath">n</span>. In both cases there is a natural action of <span class="SimpleMath">G</span> on <span class="SimpleMath">v</span> and the function returns the group of elements in <span class="SimpleMath">G</span> that fix <span class="SimpleMath">v</span>.</p>
<p><strong class="button">Examples:</strong></p>
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