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##
## simpcomp / fromgroup.gi
##
## Compute Simplicial Complexes from given Permutation Groups.
##
## $Id: glprops.gi 68 2010-04-23 14:08:15Z felix $
##
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##<#GAPDoc Label="SCsFromGroupExt">
## <ManSection>
## <Func Name="SCsFromGroupExt" Arg="G,n,d,objectType,cache,removeDoubleEntries,
## outfile,maxLinkSize,subset"/>
## <Returns>a list of simplicial complexes of type <C>SCSimplicialComplex</C>
## upon success, <K>fail</K> otherwise.</Returns>
## <Description>
## Computes all combinatorial <Arg>d</Arg>-pseudomanifolds, <M>d=2</M> / all
## strongly connected combinatorial <Arg>d</Arg>-pseudomanifolds,
## <M>d \geq 3</M>, as a union of orbits of the group action of <Arg>G</Arg>
## on <C>(d+1)</C>-tuples on the set of <Arg>n</Arg> vertices, see
## <Cite Key="Lutz03TrigMnfFewVertVertTrans" />. The integer argument
## <Arg>objectType</Arg> specifies, whether complexes exceeding the maximal
## size of each vertex link for combinatorial manifolds are sorted out
## (<C>objectType = 0</C>) or not (<C>objectType = 1</C>, in this case some
## combinatorial pseudomanifolds won't be found, but no combinatorial manifold
## will be sorted out). The integer argument <Arg>cache</Arg> specifies if the
## orbits are held in memory during the computation, a value of <C>0</C> means
## that the orbits are discarded, trading speed for memory, any other value
## means that they are kept, trading memory for speed. The boolean argument
## <Arg>removeDoubleEntries</Arg> specifies whether the results are checked for
## combinatorial isomorphism, preventing isomorphic entries. The argument
## <Arg>outfile</Arg> specifies an output file containing all complexes found
## by the algorithm, if <Arg>outfile</Arg> is anything else than a string, not
## output file is generated. The argument <Arg>maxLinkSize</Arg> determines a
## maximal link size of any output complex. If <Arg>maxLinkSize</Arg><M>=0</M>
## or if <Arg>maxLinkSize</Arg> is anything else than an integer the argument
## is ignored. The argument <Arg>subset</Arg> specifies a set of orbits (given
## by a list of indices of <C>repHigh</C>) which have to be contained in any
## output complex. If <Arg>subset</Arg> is anything else than a subset of
## <C>matrixAllowedRows</C> the argument is ignored.
## <Example><![CDATA[
## gap> G:=PrimitiveGroup(8,5);
## PGL(2, 7)
## gap> Size(G);
## 336
## gap> Transitivity(G);
## 3
## gap> list:=SCsFromGroupExt(G,8,3,1,0,true,false,0,[]);
## [ "defgh.g.h.fah.e.gaf.h.eag.e.faf.a.haa.g.fah.a.gjhzh" ]
## gap> c:=SCFromIsoSig(list[1]);
## <SimplicialComplex: unnamed complex 6 | dim = 3 | n = 8>
## gap> SCNeighborliness(c);
## 3
## gap> c.F;
## [ 8, 28, 56, 28 ]
## gap> c.IsManifold;
## false
## gap> SCLibDetermineTopologicalType(SCLink(c,1));
## <SimplicialComplex: lk([ 1 ]) in unnamed complex 6 | dim = 2 | n = 7>
## gap> # there are no 3-neighborly 3-manifolds with 8 vertices
## gap> list:=SCsFromGroupExt(PrimitiveGroup(8,5),8,3,0,0,true,false,0,[]);
## gap> [ ]
## ]]></Example>
## </Description>
## </ManSection>
##<#/GAPDoc>
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### SETUP BACKTRACKING ###
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##<#GAPDoc Label="SCsFromGroupByTransitivity">
## <ManSection>
## <Func Name="SCsFromGroupByTransitivity" Arg="n,d,k,maniflag,computeAutGroup,
## removeDoubleEntries"/>
## <Returns>a list of simplicial complexes of type <C>SCSimplicialComplex</C>
## upon success, <K>fail</K> otherwise.</Returns>
## <Description>
## Computes all combinatorial <Arg>d</Arg>-pseudomanifolds, <M>d = 2</M> / all
## strongly connected combinatorial <Arg>d</Arg>-pseudomanifolds,
## <M>d \geq 3</M>, as union of orbits of group actions for all
## <Arg>k</Arg>-transitive groups on <C>(d+1)</C>-tuples on the set of
## <Arg>n</Arg> vertices, see <Cite Key="Lutz03TrigMnfFewVertVertTrans" />.
## The boolean argument <Arg>maniflag</Arg> specifies, whether the resulting
## complexes should be listed separately by combinatorial manifolds,
## combinatorial pseudomanifolds and complexes where the verification that the
## object is at least a combinatorial pseudomanifold failed. The boolean
## argument <Arg>computeAutGroup</Arg> specifies whether or not the real
## automorphism group should be computed (note that a priori the generating
## group is just a subgroup of the automorphism group). The boolean argument
## <Arg>removeDoubleEntries</Arg> specifies whether the results are checked
## for combinatorial isomorphism, preventing isomorphic entries. Internally
## calls <Ref Func="SCsFromGroupExt" /> for every group.
## <Example><![CDATA[
## gap> list:=SCsFromGroupByTransitivity(8,3,2,true,true,true);
## #I SCsFromGroupByTransitivity: Building list of groups...
## #I SCsFromGroupByTransitivity: ...2 groups found.
## #I degree 8: [ AGL(1, 8), PSL(2, 7) ]
## #I SCsFromGroupByTransitivity: Processing dimension 3.
## #I SCsFromGroupByTransitivity: Processing degree 8.
## #I SCsFromGroupByTransitivity: 1 / 2 groups calculated, found 0 complexes.
## #I SCsFromGroupByTransitivity: Calculating 0 automorphism and homology groups...
## #I SCsFromGroupByTransitivity: ...all automorphism groups calculated for group 1 / 2.
## #I SCsFromGroupByTransitivity: 2 / 2 groups calculated, found 1 complexes.
## #I SCsFromGroupByTransitivity: Calculating 1 automorphism and homology groups...
## #I group not listed
## #I SCsFromGroupByTransitivity: 1 / 1 automorphism groups calculated.
## #I SCsFromGroupByTransitivity: ...all automorphism groups calculated for group 2 / 2.
## #I SCsFromGroupByTransitivity: ...done dim = 3, deg = 8, 0 manifolds, 1 pseudomanifolds, 0 candidates found.
## #I SCsFromGroupByTransitivity: ...done dim = 3.
## [ [ ], [ ], [ ] ]
## ]]></Example>
## </Description>
## </ManSection>
##<#/GAPDoc>
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