  
  [1X18 [33X[0;0YOrbit polytopes and fundamental domains[133X[101X
  
  
  [1X18.1 [33X[0;0Y [133X[101X
  
  [1X18.1-1 CoxeterComplex[101X
  
  [33X[1;0Y[29X[2XCoxeterComplex[102X( [3XD[103X ) [32X function[133X
  [33X[1;0Y[29X[2XCoxeterComplex[102X( [3XD[103X, [3Xn[103X ) [32X function[133X
  
  [33X[0;0YInputs   a  Coxeter  diagram  [22XD[122X  of  finite  type.  It  returns  a  non-free
  ZW-resolution for the associated Coxeter group [22XW[122X. The non-free resolution is
  obtained  from  the  permutahedron  of  type  [22XW[122X. A positive integer [22Xn[122X can be
  entered  as  an  optional  second  variable;  just  the first [22Xn[122X terms of the
  non-free resolution are then returned.[133X
  
  [33X[0;0Y[12XExamples:[112X 1 ([7X../tutorial/chap6.html[107X) [133X
  
  [1X18.1-2 ContractibleGcomplex[101X
  
  [33X[1;0Y[29X[2XContractibleGcomplex[102X( [3Xstr[103X ) [32X function[133X
  
  [33X[0;0YInputs one of the following strings [22Xstr[122X=:[133X
  [33X[0;0Y"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)"[133X
  [33X[0;0Yor one of the following strings[133X
  [33X[0;0Y"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)"[133X
  [33X[0;0YIt returns a non-free ZG-resolution for the group [22XG[122X 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.)[133X
  [33X[0;0YData  for  the  first  list of non-free resolutions was provided provided by
  [12XMathieu Dutour[112X. Data for the second list was provided by [12XAlexander Rahm[112X.[133X
  
  [33X[0;0Y[12XExamples:[112X   1   ([7X../tutorial/chap1.html[107X) ,  2  ([7X../tutorial/chap6.html[107X) ,  3
  ([7X../tutorial/chap8.html[107X) ,       4       ([7X../tutorial/chap10.html[107X) ,       5
  ([7X../www/SideLinks/About/aboutArithmetic.html[107X) ,                            6
  ([7X../www/SideLinks/About/aboutBredon.html[107X) ,                                7
  ([7X../www/SideLinks/About/aboutIntro.html[107X) [133X
  
  [1X18.1-3 QuotientOfContractibleGcomplex[101X
  
  [33X[1;0Y[29X[2XQuotientOfContractibleGcomplex[102X( [3XC[103X, [3XD[103X ) [32X function[133X
  
  [33X[0;0YInputs  a  non-free  [22XZG[122X-resolution [22XC[122X and a finite subgroup [22XD[122X of [22XG[122X which is a
  subgroup  of  each  cell  stabilizer  group  for  [22XC[122X.  Each element of [22XD[122X must
  preserves  the  orientation  of  any  cell  stabilized by it. It returns the
  corresponding  non-free  [22XZ(G/D)[122X-resolution.  (So,  for  instance,  from  the
  [22XSL(2,O)[122X  complex  [22XC=ContractibleGcomplex("SL(2,O-2)");[122X  we  can  construct a
  [22XPSL(2,O)[122X-complex using this function.)[133X
  
  [33X[0;0Y[12XExamples:[112X 1 ([7X../www/SideLinks/About/aboutArithmetic.html[107X) [133X
  
  [1X18.1-4 TruncatedGComplex[101X
  
  [33X[1;0Y[29X[2XTruncatedGComplex[102X( [3XR[103X, [3Xm[103X, [3Xn[103X ) [32X function[133X
  
  [33X[0;0YInputs  a  non-free  [22XZG[122X-resolution  [22XR[122X  and two positive integers [22Xm[122X and [22Xn[122X. It
  returns  the  non-free  [22XZG[122X-resolution  consisting  of  those modules in [22XR[122X of
  degree at least [22Xm[122X and at most [22Xn[122X.[133X
  
  [33X[0;0Y[12XExamples:[112X[133X
  
  [1X18.1-5 FundamentalDomainStandardSpaceGroup[101X
  
  [33X[1;0Y[29X[2XFundamentalDomainStandardSpaceGroup[102X( [3Xv[103X, [3XG[103X ) [32X function[133X
  
  [33X[0;0YInputs 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  [22XR^n[122X  on  which  [22XG[122X  acts.  It returns the Dirichlet-Voronoi
  fundamental cell for the action of [22XG[122X on euclidean space corresponding to the
  vector  [22Xv[122X.  The fundamental cell is a fundamental domain if [22XG[122X is Bieberbach.
  The  fundamental cell/domain is returned as a [21XPolymake object[121X. Currently the
  function only applies to certain crystallographic groups. See the manuals to
  HAPcryst and HAPpolymake for full details.[133X
  
  [33X[0;0YThis  function is part of the HAPcryst package written by [12XMarc Roeder[112X and is
  thus only available if HAPcryst is loaded.[133X
  
  [33X[0;0YThe function requires the use of Polymake software.[133X
  
  [33X[0;0Y[12XExamples:[112X 1 ([7X../www/SideLinks/About/aboutIntro.html[107X) [133X
  
  [1X18.1-6 OrbitPolytope[101X
  
  [33X[1;0Y[29X[2XOrbitPolytope[102X( [3XG[103X, [3Xv[103X, [3XL[103X ) [32X function[133X
  
  [33X[0;0YInputs  a  permutation  group  or  matrix group [22XG[122X of degree [22Xn[122X and a rational
  vector [22Xv[122X of length [22Xn[122X. In both cases there is a natural action of [22XG[122X on [22Xv[122X. Let
  [22XP(G,v)[122X  be  the  convex polytope arising as the convex hull of the Euclidean
  points  in  the orbit of [22Xv[122X under the action of [22XG[122X. The function also inputs a
  sublist [22XL[122X of the following list of strings:[133X
  
  [33X[0;0Y["dimension","vertex_degree", "visual_graph", "schlegel","visual"][133X
  
  [33X[0;0YDepending on the sublist, the function:[133X
  
  [30X    [33X[0;6Yprints the dimension of the orbit polytope [22XP(G,v)[122X;[133X
  
  [30X    [33X[0;6Yprints the degree of a vertex in the graph of [22XP(G,v)[122X;[133X
  
  [30X    [33X[0;6Yvisualizes the graph of [22XP(G,v)[122X;[133X
  
  [30X    [33X[0;6Yvisualizes the Schlegel diagram of [22XP(G,v)[122X;[133X
  
  [30X    [33X[0;6Yvisualizes [22XP(G,v)[122X if the polytope is of dimension 2 or 3.[133X
  
  [33X[0;0YThe function uses Polymake software.[133X
  
  [33X[0;0Y[12XExamples:[112X 1 ([7X../www/SideLinks/About/aboutPolytopes.html[107X) [133X
  
  [1X18.1-7 PolytopalComplex[101X
  
  [33X[1;0Y[29X[2XPolytopalComplex[102X( [3XG[103X, [3Xv[103X ) [32X function[133X
  [33X[1;0Y[29X[2XPolytopalComplex[102X( [3XG[103X, [3Xv[103X, [3Xn[103X ) [32X function[133X
  
  [33X[0;0YInputs  a  permutation  group  or  matrix group [22XG[122X of degree [22Xn[122X and a rational
  vector [22Xv[122X of length [22Xn[122X. In both cases there is a natural action of [22XG[122X on [22Xv[122X. Let
  [22XP(G,v)[122X  be  the  convex polytope arising as the convex hull of the Euclidean
  points  in  the orbit of [22Xv[122X under the action of [22XG[122X. The cellular chain complex
  [22XC_*=C_*(P(G,v))[122X  is  an exact sequence of (not necessarily free) [22XZG[122X-modules.
  The function returns a component object [22XR[122X with components:[133X
  
  [30X    [33X[0;6Y[22XR!.dimension(k)[122X  is a function which returns the number of [22XG[122X-orbits of
        the  [22Xk[122X-dimensional  faces  in  [22XP(G,v)[122X.  If  each  [22Xk[122X-face  has  trivial
        stabilizer  subgroup  in  [22XG[122X  then  [22XC_k[122X  is  a  free  [22XZG[122X-module of rank
        [22XR.dimension(k)[122X.[133X
  
  [30X    [33X[0;6Y[22XR!.stabilizer(k,n)[122X is a function which returns the stabilizer subgroup
        for a face in the [22Xn[122X-th orbit of [22Xk[122X-faces.[133X
  
  [30X    [33X[0;6YIf  all faces of dimension <[22Xk+1[122X have trivial stabilizer group then the
        first  [22Xk[122X  terms  of  [22XC_*[122X  constitute part of a free [22XZG[122X-resolution. The
        boundary  map  is  described  by the function [22Xboundary(k,n)[122X . (If some
        faces  have  non-trivial  stabilizer group then [22XC_*[122X is not free and no
        attempt is made to determine signs for the boundary map.)[133X
  
  [30X    [33X[0;6Y[22XR!.elements[122X, [22XR!.group[122X, [22XR!.properties[122X are as in a [22XZG[122X-resolution.[133X
  
  [33X[0;0YIf  an  optional third input variable [22Xn[122X is used, then only the first [22Xn[122X terms
  of the resolution [22XC_*[122X will be computed.[133X
  
  [33X[0;0YThe function uses Polymake software.[133X
  
  [33X[0;0Y[12XExamples:[112X 1 ([7X../www/SideLinks/About/aboutPolytopes.html[107X) [133X
  
  [1X18.1-8 PolytopalGenerators[101X
  
  [33X[1;0Y[29X[2XPolytopalGenerators[102X( [3XG[103X, [3Xv[103X ) [32X function[133X
  
  [33X[0;0YInputs  a  permutation  group  or  matrix group [22XG[122X of degree [22Xn[122X and a rational
  vector [22Xv[122X of length [22Xn[122X. In both cases there is a natural action of [22XG[122X on [22Xv[122X, and
  the vector [22Xv[122X must be chosen so that it has trivial stabilizer subgroup in [22XG[122X.
  Let  [22XP(G,v)[122X  be  the  convex  polytope  arising  as  the  convex hull of the
  Euclidean  points  in  the  orbit  of  [22Xv[122X under the action of [22XG[122X. The function
  returns a record [22XP[122X with components:[133X
  
  [30X    [33X[0;6Y[22XP.generators[122X is a list of all those elements [22Xg[122X in [22XG[122X such that [22Xg⋅ v[122X has
        an edge in common with [22Xv[122X. The list is a generating set for [22XG[122X.[133X
  
  [30X    [33X[0;6Y[22XP.vector[122X is the vector [22Xv[122X.[133X
  
  [30X    [33X[0;6Y[22XP.hasseDiagram[122X is the Hasse diagram of the cone at [22Xv[122X.[133X
  
  [33X[0;0YThe  function uses Polymake software. The function is joint work with Seamus
  Kelly.[133X
  
  [33X[0;0Y[12XExamples:[112X[133X
  
  [1X18.1-9 VectorStabilizer[101X
  
  [33X[1;0Y[29X[2XVectorStabilizer[102X( [3XG[103X, [3Xv[103X ) [32X function[133X
  
  [33X[0;0YInputs  a  permutation  group  or  matrix group [22XG[122X of degree [22Xn[122X and a rational
  vector  of  degree  [22Xn[122X. In both cases there is a natural action of [22XG[122X on [22Xv[122X and
  the function returns the group of elements in [22XG[122X that fix [22Xv[122X.[133X
  
  [33X[0;0Y[12XExamples:[112X[133X
  
