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<Section><Heading>Face Lattice and Resolution</Heading>
For Bieberbach groups (torsion free crystallographic groups), the following
functions calculate free resolutions. This calculation is done by finding a
fundamental domain for the group.
For a description of the <K>HapResolution</K> datatype, see the
<Package>Hap</Package> data types documentation or the experimental datatypes
documentation <Ref Label="Resolutions in Hap" BookName="HAPprog"/>
<ManSection>
<Meth Name="ResolutionBieberbachGroup" Arg="G [v]"/>
<Returns>a <K>HAPresolution</K></Returns>
<Description>
Let <Arg>G</Arg> be a Bieberbach group given as an
<K>AffineCrystGroupOnRight</K> and <A>v</A> a vector. Then a Dirichlet domain
with respect to <A>v</A> is calculated using
<Ref Meth="FundamentalDomainBieberbachGroup"/>. From this domain, a
resolution is calculated using
<Ref Meth="FaceLatticeAndBoundaryBieberbachGroup"/> and
<Ref Meth="ResolutionFromFLandBoundary"/>. If <A>v</A> is not given, the
origin is used.
</Description>
</ManSection>
<Example><![CDATA[
gap> R:=ResolutionBieberbachGroup(SpaceGroup(3,9));
Resolution of length 3 in characteristic
0 for SpaceGroupOnRightBBNWZ( 3, 2, 2, 2, 2 ) .
No contracting homotopy available.
gap> List([0..3],Dimension(R));
[ 1, 3, 3, 1 ]
gap> R:=ResolutionBieberbachGroup(SpaceGroup(3,9),[1/2,0,0]);
Resolution of length 3 in characteristic
0 for SpaceGroupOnRightBBNWZ( 3, 2, 2, 2, 2 ) .
No contracting homotopy available.
gap> List([0..3],Dimension(R));
[ 6, 12, 7, 1 ]
]]>
</Example>
<ManSection>
<Meth Name="FaceLatticeAndBoundaryBieberbachGroup" Arg="poly, group"/>
<Returns> Record with entries <K>.hasse</K> and <K>.elts</K>
representing a part of the hasse diagram and a lookup table of group
elements</Returns>
<Description>
Let <A>group</A> be a torsion free <K>AffineCrystGroupOnRight</K> (that
is, a Bieberbach group).
Given a <K>PolymakeObject</K> <A>poly</A> representing a fundamental
domain for <A>group</A>, this method uses <Package>polymaking</Package>
to calculate the face lattice of <A>poly</A>.
From the set of faces, a system of representatives for <A>group</A>-
orbits is chosen. For each representative, the boundary is then
calculated. The list <K>.elts</K> contains elements of <A>group</A> (in
fact, it is even a set). The structure of the returned list
<K>.hasse</K> is as follows:
<List>
<Item>The <M>i</M>-th entry contains a system of representatives
for the <M>i-1</M> dimensional faces of <A>poly</A>.
</Item>
<Item>Each face is represented by a pair of lists
<C>[vertices,boundary]</C>. The list of integers
<C>vertices</C> represents the vertices of
<A>poly</A> which are contained in this face. The enumeration
is chosen such that an <C>i</C> in the list represents the
<M>i</M>-th entry of the list
<C>Polymake(poly,"VERTICES");</C>
</Item>
<Item> The list <C>boundary</C> represents the boundary of
the respective face. It is a list of pairs of integers
<C>[j,g]</C>. The first entry lies between <M>-n</M> and
<M>n</M>, where <M>n</M> is the number of faces of dimension
<M>i-1</M>. This entry represents a face of dimension
<M>i-1</M> (or its additive inverse as a module generator).
The second entry <C>g</C> is the position of the matrix in
<K>.elts</K>.
</Item>
</List>
This representation is compatible with the representation of free
<M>\mathbb Z G</M> modules in <Package>Hap</Package> and this method essentially
calculates a free resolution of <A>group</A>. If the value of
<Ref InfoClass="InfoHAPcryst" /> 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.
</Description>
</ManSection>
<Example><![CDATA[
gap> SetInfoLevel(InfoHAPcryst,2);
gap> G:=SpaceGroup(3,165);
SpaceGroupOnRightBBNWZ( 3, 6, 1, 1, 4 )
gap> fd:=FundamentalDomainBieberbachGroup(G);
<polymake object>
gap> fl:=FaceLatticeAndBoundaryBieberbachGroup(fd,G);;
#I 1(4/8): 0:00:00.004
#I 2(5/18): 0:00:00.000
#I 3(2/12): 0:00:00.000
#I Face lattice done ( 0:00:00.004). Calculating boundary
#I done ( 0:00:00.004) Reformating...
gap> RecNames(fl);
[ "hasse", "elts", "groupring" ]
gap> fl.groupring;
<free left module over Integers, and ring-with-one, with 10 generators>
]]></Example>
<ManSection>
<Meth Name="ResolutionFromFLandBoundary" Arg="fl group"/>
<Returns>Free resolution</Returns>
<Description>
If <A>fl</A> is the record output by <Ref
Meth="FaceLatticeAndBoundaryBieberbachGroup"/> and <A>group</A> is
the corresponding group, this function returns a <K>HapResolution</K>.
Of course, <A>fl</A> has to be generated from a fundamental domain
for <A>group</A>
</Description>
</ManSection>
<Example><![CDATA[
gap> G:=SpaceGroup(3,165);
SpaceGroupOnRightBBNWZ( 3, 6, 1, 1, 4 )
gap> fd:=FundamentalDomainBieberbachGroup(G);
<polymake object>
gap> fl:=FaceLatticeAndBoundaryBieberbachGroup(fd,G);;
gap> ResolutionFromFLandBoundary(fl,G);
Resolution of length 3 in characteristic
0 for SpaceGroupOnRightBBNWZ( 3, 6, 1, 1, 4 ) .
No contracting homotopy available.
gap> ResolutionFromFLandBoundary(fl,G);
Resolution of length 3 in characteristic
0 for SpaceGroupOnRightBBNWZ( 3, 6, 1, 1, 4 ) .
No contracting homotopy available.
gap> List([0..4],Dimension(last));
[ 2, 5, 4, 1, 0 ]
]]></Example>
</Section>
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