  
  [1X4 [33X[0;0YResolutions of Crystallographic Groups[133X[101X
  
  
  [1X4.1 [33X[0;0YFundamental Domains[133X[101X
  
  [33X[0;0YLet  [22XS[122X  be a crystallographic group. A Fundamental domain is a closed convex
  set  containing  a  system  of  representatives  for  the Orbits of [22XS[122X in its
  natural action on euclidian space.[133X
  [33X[0;0YThere  are  two  algorithms for calculating fundamental domains in [5XHAPcryst[105X.
  One  uses the geometry and relies on having the standard rule for evaluating
  the  scalar product (i.e. the gramian matrix is the identity). The other one
  is  independent  of  the  gramian  matrix  but does only work for Bieberbach
  groups,   while  the  first  ("geometric")  algorithm  works  for  arbitrary
  crystallographic groups given a point with trivial stabilizer.[133X
  
  [1X4.1-1 FundamentalDomainStandardSpaceGroup[101X
  
  [33X[1;0Y[29X[2XFundamentalDomainStandardSpaceGroup[102X( [[3Xv[103X, ][3XG[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya [9XPolymakeObject[109X[133X
  
  [33X[0;0YLet  [3XG[103X  be  an  [9XAffineCrystGroupOnRight[109X and [3Xv[103X a vector. A fundamental domain
  containing [3Xv[103X is calculated and returned as a [9XPolymakeObject[109X. The vector [3Xv[103X is
  used  as the starting point for a Dirichlet-Voronoi construction. If no [3Xv[103X is
  given,  the  origin  is used as starting point if it has trivial stabiliser.
  Otherwise an error is cast.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xfd:=FundamentalDomainStandardSpaceGroup([1/2,0,1/5],SpaceGroup(3,9));[127X[104X
    [4X[28X<polymake object>[128X[104X
    [4X[25Xgap>[125X [27XPolymake(fd,"N_VERTICES");[127X[104X
    [4X[28X24[128X[104X
    [4X[25Xgap>[125X [27Xfd:=FundamentalDomainStandardSpaceGroup(SpaceGroup(3,9));[127X[104X
    [4X[28X<polymake object>[128X[104X
    [4X[25Xgap>[125X [27XPolymake(fd,"N_VERTICES");[127X[104X
    [4X[28X8[128X[104X
  [4X[32X[104X
  
  [1X4.1-2 FundamentalDomainBieberbachGroup[101X
  
  [33X[1;0Y[29X[2XFundamentalDomainBieberbachGroup[102X( [[3Xv[103X, ][3XG[103X[, [3Xgram[103X] ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya [9XPolymakeObject[109X[133X
  
  [33X[0;0YGiven  a  starting  vector [3Xv[103X and a Bieberbach group [3XG[103X in standard form, this
  method  calculates  the  Dirichlet  domain with respect to [3Xv[103X. If [3Xgram[103X is not
  supplied,     the     average     gramian     matrix     is     used    (see
  [2XGramianOfAverageScalarProductFromFiniteMatrixGroup[102X   ([14X2.3-1[114X)).   It  is  not
  tested if [3Xgram[103X is symmetric and positive definite. It is also not tested, if
  the product defined by [3Xgram[103X is invariant under the point group of [3XG[103X.[133X
  
  [33X[0;0YThe  behaviour  of  this function is influenced by the option [9XineqThreshold[109X.
  The   algorithm   calculates  approximations  to  a  fundamental  domain  by
  iteratively  adding  inequalities.  For  an  approximating polyhedron, every
  vertex  is  tested  to  find  new  inequalities. When all vertices have been
  considered or the number of new inequalities already found exceeds the value
  of  [9XineqThreshold[109X, a new approximating polyhedron in calculated. The default
  for [9XineqThreshold[109X is 200. Roughly speaking, a large threshold means shifting
  work  from  [9Xpolymake[109X  to [5XGAP[105X, a small one means more calls of (and work for)
  [9Xpolymake[109X.[133X
  
  [33X[0;0YIf  the  value  of [2XInfoHAPcryst[102X ([14X1.3-1[114X) is 2 or more, for each approximation
  the  number  of  vertices  of the approximation, the number of vertices that
  have  to be considered during the calculation, the number of facets, and new
  inequalities is shown.[133X
  
  [33X[0;0YNote  that  the  algorithm  chooses vertices in random order and also writes
  inequalities for [9Xpolymake[109X in random order.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xa0:=[[ 1, 0, 0, 0, 0, 0, 0 ], [ 0, -1, 0, 0, 0, 0, 0 ], [127X[104X
    [4X[25X>[125X [27X    [ 0, 0, 1, 0, 0, 0, 0 ], [ 0, 0, 0, 1, 0, 0, 0 ], [127X[104X
    [4X[25X>[125X [27X    [ 0, 0, 0, 0, 0, 1, 0 ], [ 0, 0, 0, 0, -1, -1, 0 ],[127X[104X
    [4X[25X>[125X [27X    [ -1/2, 0, 0, 1/6, 0, 0, 1 ] [127X[104X
    [4X[25X>[125X [27X    ];;[127X[104X
    [4X[25Xgap>[125X [27Xa1:=[[ 0, -1, 0, 0, 0, 0, 0 ],[ 0, 0, -1, 0, 0, 0, 0 ],[127X[104X
    [4X[25X>[125X [27X        [ 1, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 1, 0, 0, 0 ], [127X[104X
    [4X[25X>[125X [27X        [ 0, 0, 0, 0, 1, 0, 0 ], [ 0, 0, 0, 0, 0, 1, 0 ],[127X[104X
    [4X[25X>[125X [27X        [ 0, 0, 0, 0, 1/3, -1/3, 1 ] [127X[104X
    [4X[25X>[125X [27X       ];;[127X[104X
    [4X[25Xgap>[125X [27Xtrans:=List(Group((1,2,3,4,5,6)),g->[127X[104X
    [4X[25X>[125X [27X          TranslationOnRightFromVector(Permuted([1,0,0,0,0,0],g)));;[127X[104X
    [4X[25Xgap>[125X [27XS:=AffineCrystGroupOnRight(Concatenation(trans,[a0,a1]));[127X[104X
    [4X[28X<matrix group with 8 generators>[128X[104X
    [4X[25Xgap>[125X [27XSetInfoLevel(InfoHAPcryst,2);[127X[104X
    [4X[25Xgap>[125X [27XFundamentalDomainBieberbachGroup(S:ineqThreshold:=10);[127X[104X
    [4X[28X#I  v: 104/104 f:15[128X[104X
    [4X[28X#I  new: 201[128X[104X
    [4X[28X#I  v: 961/961 f:58[128X[104X
    [4X[28X#I  new: 20[128X[104X
    [4X[28X#I  v: 1143/805 f:69[128X[104X
    [4X[28X#I  new: 12[128X[104X
    [4X[28X#I  v: 1059/555 f:64[128X[104X
    [4X[28X#I  new: 15[128X[104X
    [4X[28X#I  v: 328/109 f:33[128X[104X
    [4X[28X#I  new: 12[128X[104X
    [4X[28X#I  v: 336/58 f:32[128X[104X
    [4X[28X#I  new: 0[128X[104X
    [4X[28X<polymake object>[128X[104X
    [4X[25Xgap>[125X [27XFundamentalDomainBieberbachGroup(S:ineqThreshold:=1000);[127X[104X
    [4X[28X#I  v: 104/104 f:15[128X[104X
    [4X[28X#I  new: 149[128X[104X
    [4X[28X#I  v: 635/635 f:41[128X[104X
    [4X[28X#I  new: 115[128X[104X
    [4X[28X#I  v: 336/183 f:32[128X[104X
    [4X[28X#I  new: 0[128X[104X
    [4X[28X#I  out of inequalities[128X[104X
    [4X[28X<polymake object>[128X[104X
  [4X[32X[104X
  
  [1X4.1-3 FundamentalDomainFromGeneralPointAndOrbitPartGeometric[101X
  
  [33X[1;0Y[29X[2XFundamentalDomainFromGeneralPointAndOrbitPartGeometric[102X( [3Xv[103X, [3Xorbit[103X ) [32X method[133X
  [6XReturns:[106X  [33X[0;10Ya [9XPolymakeObject[109X[133X
  
  [33X[0;0YThis  uses an alternative algorithm based on geometric considerations. It is
  not  used  in  any  of the high-level methods. Let [3Xv[103X be a vector and [3Xorbit[103X a
  sufficiently  large  part  of  the orbit of [3Xv[103X under a crystallographic group
  with  standard-  orthogonal  point  group (satisfying [22XA^t=A^-1[122X). A geometric
  algorithm  is then used to calculate the Dirichlet domain with respect to [3Xv[103X.
  This  also  works  for crystallographic groups which are not Bieberbach. The
  point [3Xv[103X has to have trivial stabilizer.[133X
  [33X[0;0YThe  intersection  of  the  full  orbit  with  the  unit  cube  around  [3Xv[103X is
  sufficiently large.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XG:=SpaceGroup(3,9);;[127X[104X
    [4X[25Xgap>[125X [27Xv:=[0,0,0];[127X[104X
    [4X[28X[ 0, 0, 0 ][128X[104X
    [4X[25Xgap>[125X [27Xorbit:=OrbitStabilizerInUnitCubeOnRight(G,v).orbit;[127X[104X
    [4X[28X[ [ 0, 0, 0 ], [ 0, 0, 1/2 ] ][128X[104X
    [4X[25Xgap>[125X [27Xfd:=FundamentalDomainFromGeneralPointAndOrbitPartGeometric(v,orbit);[127X[104X
    [4X[28X<polymake object>[128X[104X
    [4X[25Xgap>[125X [27XPolymake(fd,"N_VERTICES");[127X[104X
    [4X[28X8[128X[104X
  [4X[32X[104X
  
  [1X4.1-4 IsFundamentalDomainStandardSpaceGroup[101X
  
  [33X[1;0Y[29X[2XIsFundamentalDomainStandardSpaceGroup[102X( [3Xpoly[103X, [3XG[103X ) [32X method[133X
  [6XReturns:[106X  [33X[0;10Ytrue or false[133X
  
  [33X[0;0YThis  tests  if a [9XPolymakeObject[109X [3Xpoly[103X is a fundamental domain for the affine
  crystallographic group [3XG[103X in standard form.[133X
  [33X[0;0YThe  function  tests  the  following: First, does the orbit of any vertex of
  [3Xpoly[103X  have  a  point  inside  [3Xpoly[103X (if this is the case, [10Xfalse[110X is returned).
  Second:  Is every facet of [3Xpoly[103X the image of a different facet under a group
  element which does not fix [3Xpoly[103X. If this is satisfied, [10Xtrue[110X is returned.[133X
  
  [1X4.1-5 IsFundamentalDomainBieberbachGroup[101X
  
  [33X[1;0Y[29X[2XIsFundamentalDomainBieberbachGroup[102X( [3Xpoly[103X, [3XG[103X ) [32X method[133X
  [6XReturns:[106X  [33X[0;10Ytrue, false or fail[133X
  
  [33X[0;0YThis  tests  if a [9XPolymakeObject[109X [3Xpoly[103X is a fundamental domain for the affine
  crystallographic  group [3XG[103X in standard form and if this group is torsion free
  (ie a Bieberbach group)[133X
  [33X[0;0YIt returns [9Xtrue[109X if [3XG[103X is torsion free and [3Xpoly[103X is a fundamental domain for [3XG[103X.
  If  [3Xpoly[103X  is  not  a fundamental domain, [9Xfalse[109X is returned regardless of the
  structure  of [3XG[103X. And if [3XG[103X is not torsion free, the method returns [9Xfail[109X. If [3XG[103X
  is  polycyclic,  torsion  freeness  is  tested using a representation as pcp
  group. Otherwise the stabilisers of the faces of the fundamental domain [3Xpoly[103X
  are  calculated  ([3XG[103X  is torsion free if and only if it all these stabilisers
  are trivial).[133X
  
  
  [1X4.2 [33X[0;0YFace Lattice and Resolution[133X[101X
  
  [33X[0;0YFor  Bieberbach groups (torsion free crystallographic groups), the following
  functions  calcualte free resolutions. This calculation is done by finding a
  fundamental  domain  for  the  group. For a description of the [9XHapResolution[109X
  datatype, see the [5XHap[105X data types documentation or the experimental datatypes
  documentation [14XA[114X[133X
  
  [1X4.2-1 ResolutionBieberbachGroup[101X
  
  [33X[1;0Y[29X[2XResolutionBieberbachGroup[102X( [3XG[103X[, [3Xv[103X] ) [32X method[133X
  [6XReturns:[106X  [33X[0;10Ya [9XHAPresolution[109X[133X
  
  [33X[0;0YLet  [3XG[103X  be  a  Bieberbach  group given as an [9XAffineCrystGroupOnRight[109X and [3Xv[103X a
  vector.  Then  a  Dirichlet  domain  with  respect  to [3Xv[103X is calculated using
  [2XFundamentalDomainBieberbachGroup[102X  ([14X4.1-2[114X). From this domain, a resolution is
  calculated    using    [2XFaceLatticeAndBoundaryBieberbachGroup[102X   ([14X4.2-2[114X)   and
  [2XResolutionFromFLandBoundary[102X ([14X4.2-3[114X). If [3Xv[103X is not given, the origin is used.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XR:=ResolutionBieberbachGroup(SpaceGroup(3,9));[127X[104X
    [4X[28XResolution of length 3 in characteristic[128X[104X
    [4X[28X0 for SpaceGroupOnRightBBNWZ( 3, 2, 2, 2, 2 ) .[128X[104X
    [4X[28XNo contracting homotopy available.[128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XList([0..3],Dimension(R));[127X[104X
    [4X[28X[ 1, 3, 3, 1 ][128X[104X
    [4X[25Xgap>[125X [27XR:=ResolutionBieberbachGroup(SpaceGroup(3,9),[1/2,0,0]);[127X[104X
    [4X[28XResolution of length 3 in characteristic[128X[104X
    [4X[28X0 for SpaceGroupOnRightBBNWZ( 3, 2, 2, 2, 2 ) .[128X[104X
    [4X[28XNo contracting homotopy available.[128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XList([0..3],Dimension(R));[127X[104X
    [4X[28X[ 6, 12, 7, 1 ][128X[104X
  [4X[32X[104X
  
  [1X4.2-2 FaceLatticeAndBoundaryBieberbachGroup[101X
  
  [33X[1;0Y[29X[2XFaceLatticeAndBoundaryBieberbachGroup[102X( [3Xpoly[103X, [3Xgroup[103X ) [32X method[133X
  [6XReturns:[106X  [33X[0;10YRecord  with  entries  [9X.hasse[109X and [9X.elts[109X representing a part of the
            hasse diagram and a lookup table of group elements[133X
  
  [33X[0;0YLet  [3Xgroup[103X  be a torsion free [9XAffineCrystGroupOnRight[109X (that is, a Bieberbach
  group).  Given  a  [9XPolymakeObject[109X [3Xpoly[103X representing a fundamental domain for
  [3Xgroup[103X,  this  method  uses [5Xpolymaking[105X to calculate the face lattice of [3Xpoly[103X.
  From  the  set  of  faces,  a system of representatives for [3Xgroup[103X- orbits is
  chosen.  For  each representative, the boundary is then calculated. The list
  [9X.elts[109X  contains elements of [3Xgroup[103X (in fact, it is even a set). The structure
  of the returned list [9X.hasse[109X is as follows:[133X
  
  [30X    [33X[0;6YThe  [22Xi[122X-th  entry  contains  a  system  of  representatives for the [22Xi-1[122X
        dimensional faces of [3Xpoly[103X.[133X
  
  [30X    [33X[0;6YEach  face  is represented by a pair of lists [10X[vertices,boundary][110X. The
        list  of  integers  [10Xvertices[110X represents the vertices of [3Xpoly[103X which are
        contained  in  this  face. The enumeration is chosen such that an [10Xi[110X in
        the    list    represents    the    [22Xi[122X-th    entry    of    the    list
        [10XPolymake(poly,"VERTICES");[110X[133X
  
  [30X    [33X[0;6YThe  list  [10Xboundary[110X represents the boundary of the respective face. It
        is  a list of pairs of integers [10X[j,g][110X. The first entry lies between [22X-n[122X
        and  [22Xn[122X,  where  [22Xn[122X  is the number of faces of dimension [22Xi-1[122X. This entry
        represents  a  face  of  dimension  [22Xi-1[122X  (or its additive inverse as a
        module generator). The second entry [10Xg[110X is the position of the matrix in
        [9X.elts[109X.[133X
  
  [33X[0;0YThis  representation  is  compatible  with  the  representation  of free [22XZ G[122X
  modules  in  [5XHap[105X and this method essentially calculates a free resolution of
  [3Xgroup[103X.  If  the  value  of  [2XInfoHAPcryst[102X  ([14X1.3-1[114X)  is  2 or more, additional
  information  about  the  number of faces in every codimension, the number of
  orbits  of  the  group  on the free module generated by those faces, and the
  time it took to calculate the orbit decomposition is output.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XSetInfoLevel(InfoHAPcryst,2);[127X[104X
    [4X[25Xgap>[125X [27XG:=SpaceGroup(3,165);[127X[104X
    [4X[28XSpaceGroupOnRightBBNWZ( 3, 6, 1, 1, 4 )[128X[104X
    [4X[25Xgap>[125X [27Xfd:=FundamentalDomainBieberbachGroup(G);[127X[104X
    [4X[28X<polymake object>[128X[104X
    [4X[25Xgap>[125X [27Xfl:=FaceLatticeAndBoundaryBieberbachGroup(fd,G);;[127X[104X
    [4X[28X#I  1(4/8): 0:00:00.004[128X[104X
    [4X[28X#I  2(5/18): 0:00:00.000[128X[104X
    [4X[28X#I  3(2/12): 0:00:00.000[128X[104X
    [4X[28X#I  Face lattice done ( 0:00:00.004). Calculating boundary[128X[104X
    [4X[28X#I  done ( 0:00:00.004) Reformating...[128X[104X
    [4X[25Xgap>[125X [27XRecNames(fl);[127X[104X
    [4X[28X[ "hasse", "elts", "groupring" ][128X[104X
    [4X[25Xgap>[125X [27Xfl.groupring;[127X[104X
    [4X[28X<free left module over Integers, and ring-with-one, with 10 generators>[128X[104X
  [4X[32X[104X
  
  [1X4.2-3 ResolutionFromFLandBoundary[101X
  
  [33X[1;0Y[29X[2XResolutionFromFLandBoundary[102X( [3Xfl[103X, [3Xgroup[103X ) [32X method[133X
  [6XReturns:[106X  [33X[0;10YFree resolution[133X
  
  [33X[0;0YIf  [3Xfl[103X is the record output by [2XFaceLatticeAndBoundaryBieberbachGroup[102X ([14X4.2-2[114X)
  and [3Xgroup[103X is the corresponding group, this function returns a [9XHapResolution[109X.
  Of course, [3Xfl[103X has to be generated from a fundamental domain for [3Xgroup[103X[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XG:=SpaceGroup(3,165);[127X[104X
    [4X[28XSpaceGroupOnRightBBNWZ( 3, 6, 1, 1, 4 )[128X[104X
    [4X[25Xgap>[125X [27Xfd:=FundamentalDomainBieberbachGroup(G);[127X[104X
    [4X[28X<polymake object>[128X[104X
    [4X[25Xgap>[125X [27Xfl:=FaceLatticeAndBoundaryBieberbachGroup(fd,G);;[127X[104X
    [4X[25Xgap>[125X [27XResolutionFromFLandBoundary(fl,G);[127X[104X
    [4X[28XResolution of length 3 in characteristic[128X[104X
    [4X[28X0 for SpaceGroupOnRightBBNWZ( 3, 6, 1, 1, 4 ) .[128X[104X
    [4X[28XNo contracting homotopy available.[128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XResolutionFromFLandBoundary(fl,G);[127X[104X
    [4X[28XResolution of length 3 in characteristic[128X[104X
    [4X[28X0 for SpaceGroupOnRightBBNWZ( 3, 6, 1, 1, 4 ) .[128X[104X
    [4X[28XNo contracting homotopy available.[128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XList([0..4],Dimension(last));[127X[104X
    [4X[28X[ 2, 5, 4, 1, 0 ][128X[104X
  [4X[32X[104X
  
