<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ 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">‣ 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>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ 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>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ 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>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ 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>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ 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>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ 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>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ 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">‣ 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>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ 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 <spanclass="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>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ 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>
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