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fromgroup.gi
################################################################################
##
## simpcomp / fromgroup.gi
##
## Compute Simplicial Complexes from given Permutation Groups.
##
## $Id: glprops.gi 68 2010-04-23 14:08:15Z felix $
##
################################################################################
################################################################################
##<#GAPDoc Label="SCsFromGroupExt">
## <ManSection>
## <Func Name="SCsFromGroupExt" Arg="G,n,d,objectType,cache,removeDoubleEntries,
## outfile,maxLinkSize,subset,single"/>
## <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 excluded). A value of <C>objectType = 2</C> behaves like
## <C>objectType = 0</C> but without bistallar flips checking vertex links.
## 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. If <Arg>single</Arg> is
## set to <C>true</C>, only single orbit solutions are returned.
## <Example><![CDATA[
## gap> G:=PrimitiveGroup(8,5);
## gap> Size(G);
## gap> Transitivity(G);
## gap> list:=SCsFromGroupExt(G,8,3,1,0,true,false,0,[]);
## gap> c:=SCFromIsoSig(list[1]);
## gap> SCNeighborliness(c);
## gap> c.F;
## gap> c.IsManifold;
## gap> SCLibDetermineTopologicalType(SCLink(c,1));
## gap> # there are no 3-neighborly 3-manifolds with 8 vertices
## gap> list:=SCsFromGroupExt(PrimitiveGroup(8,5),8,3,0,0,true,false,0,[]);
## ]]></Example>
## </Description>
## </ManSection>
##<#/GAPDoc>
################################################################################
InstallGlobalFunction(SCsFromGroupExt,
function(G,n,d,objectType,cache,removeDoubleEntries,outfile,maxLinkSize,
subset,single)
local
forbiddenHigh,forbiddenLow,stab,repHighForbidden,repHigh,repHighTmp,repLow,
repHighOrbitLen,repHighOrbitLenTmp,repOrbitLen,matrix,matrixRows,matrixCols,
matrixAllowedRows,allCount,repLowCount,repHighCount,cachedHighOrbits,
candidateCount,max,tmp,t,orbit,pos,curRow,complex,complex_collection,
count,maxIdxLow,maxIdxHigh,maxIdxLowReal,maxIdxHighReal,simplex,
maxSimplexPos,simplexPos,add,simplexIdx,rl,ru,rlIdx,rowIdx,singleResults,
element,j,i,o,name,column,s,vectorsum,curComb,stop,stopSize,foundComplex,
colPos,curPos,lastRow,srctr,subsetValid,isoSig,isoSigs,
# functions
examineComplex,
Faces,
assembleComplex,
updateRowVector;
candidateCount:=0;
if not IsString(outfile) then
outfile:=false;
else
tmp:=SmallGeneratingSet(G);
if tmp = [] then
PrintTo(outfile,"G:=Group(());\nc:=[];\n\n");
else
PrintTo(outfile,"G:=Group(\n",SmallGeneratingSet(G),"\n);\nc:=[];\n\n");
fi;
fi;
#upper boundaries for link length
if IsInt(maxLinkSize) and maxLinkSize > 0 then
max:=maxLinkSize;
else
if objectType in [0,2] then #manifolds
if IsInt((d-1)/2) = true then
max:=Binomial((n-1)-Int(d/2)-1,Int(d/2))+
Binomial((n-1)-1-(d-1)/2,(d-1)/2);
else
max:=Binomial((n-1)-d/2,d/2)+Binomial((n-1)-1-
Int((d-1)/2)-1,Int((d-1)/2));
fi;
else #pseudomanifolds
if d = 4 then
max:=Binomial((n-1)-d/2,d/2)+Binomial((n-1)-1-
Int((d-1)/2)-1,Int((d-1)/2));
else
max:=Binomial(n-1,d);
fi;
fi;
fi;
# upper boundary for length of facet list
max:=Int(n*max/(d+1));
complex_collection:=[];
updateRowVector:=function(add,row)
local i;
if(add) then
#add row to chosen combination
for i in matrix[row] do
if vectorsum[i[1]]+i[2] >= 3 then
return false;
fi;
od;
if vectorsum[repLowCount+2]+repHighOrbitLen[row] > max then
return false;
fi;
for i in matrix[row] do
vectorsum[i[1]]:=vectorsum[i[1]]+i[2];
if vectorsum[i[1]] = 1 then
vectorsum[repLowCount+1]:=vectorsum[repLowCount+1]+1;
elif vectorsum[i[1]]=2 and i[2]=1 then
vectorsum[repLowCount+1]:=vectorsum[repLowCount+1]-1;
fi;
od;
vectorsum[repLowCount+2]:=vectorsum[repLowCount+2]+repHighOrbitLen[row];
else
#remove row from chosen combination
for i in matrix[row] do
if vectorsum[i[1]]=2 and i[2]=1 then
vectorsum[repLowCount+1]:=vectorsum[repLowCount+1]+1;
elif vectorsum[i[1]] = 1 then
vectorsum[repLowCount+1]:=vectorsum[repLowCount+1]-1;
fi;
vectorsum[i[1]]:=vectorsum[i[1]]-i[2];
od;
vectorsum[repLowCount+2]:=vectorsum[repLowCount+2]-repHighOrbitLen[row];
fi;
return true;
end;
assembleComplex:=function(combination)
local j,tmp,orbit;
complex:=[];
tmp:=[];
for j in combination do
#generate orbit
if cache=1 then
orbit:=cachedHighOrbits[j];
else
orbit:=Orbit(G,repHigh[j],OnSets);
fi;
Append(complex,orbit);
od;
end;
examineComplex := function(complex,complex_collection,objectType,t)
local k,j,d,isconn,c,lk;
c:=SCFromFacets(complex);
if t = 0 then
lk:=SCLinks(c,0);
else
lk:=[SCLink(c,1)];
fi;
d:=Size(complex[1])-1;
if not SCIsConnected(c) then
Info(InfoSimpcomp,2,"complex not connected.");
return false;
fi;
if not SCIsStronglyConnected(c) then
Info(InfoSimpcomp,2,"complex not strongly connected.");
return false;
fi;
Info(InfoSimpcomp,2,"Testing connectedness of link(s)...");
isconn:= List(lk,x->SCIsConnected(x)) ;
if (objectType in [0,2] and false in isconn) or (objectType=1 and false in isconn
and d > 2) then
Info(InfoSimpcomp,2,"link not connected.");
return false;
fi;
if (objectType = 1 and IsInt(d/2)) or objectType in [0,2] then
Info(InfoSimpcomp,2,"Testing vertex link...");
if Set(List(lk,x->SCEulerCharacteristic(x))) <> [1 + (-1)^(d-1)] then
Info(InfoSimpcomp,2,"link not valid.");
return false;
fi;
fi;
Info(InfoSimpcomp,2,"ok.");
# test lower links only if objectType = 0 is set
if objectType = 0 then
return SCIsManifold(c);
else
return true;
fi;
end;
Info(InfoSimpcomp,2,"Calculating for group ",G);
# check if G operates on [1..n]
if DegreeOperation(G) <> n then
Info(InfoSimpcomp,2,"Group does not operate on ",n," points.");
return [];
fi;
# check group size
if Size(G) > Factorial(d+1)*max and objectType in [0,2] then
Info(InfoSimpcomp,2,"Group too large for combinatorial manifolds.");
return [];
fi;
# check if G operates on [1..n]
if n < d+2 then
Info(InfoSimpcomp,2,"There is no ",d,"-dimensional (pseudo)manifold with ",
n," vertices.");
return [];
fi;
repHigh:=[];
repLow:=[];
repHighOrbitLen:=[];
repOrbitLen:=0;
matrix:=[];
matrixRows:=0;
matrixCols:=0;
matrixAllowedRows:=[];
allCount:=[];
cachedHighOrbits:=[];
t:=Transitivity(G);
if t < 1 then
Info(InfoSimpcomp,1,"Automorphism is not transitive.");
#return fail;
fi;
#complex
complex:=[];
count:=0;
if removeDoubleEntries then
isoSigs:=[];
fi;
##############################################################################
#MAIN
##############################################################################
#number of (possible) simplices calculated
maxIdxLow:=Binomial(n-t,d-t); #maximum number of (d-1)-vertices
maxIdxHigh:=Binomial(n-t,d+1-t); #maximum number of d-vertices
maxIdxLowReal:=Binomial(n,d);
maxIdxHighReal:=Binomial(n,d+1);
if(maxIdxLow<1) then maxIdxLow:=1; fi;
if(maxIdxHigh<1) then maxIdxHigh:=1; fi;
Info(InfoSimpcomp,2,"Simplices count low=",maxIdxLow," high=",maxIdxHigh);
Info(InfoSimpcomp,2,"Generating low simplices count=",maxIdxLow);
forbiddenLow:=[];
#compute low represetative list
maxSimplexPos:=d;
simplex:=[1..d];
simplex[d]:=d-1;
for simplexIdx in [1..maxIdxLow] do
simplexPos:=d;
simplex[simplexPos]:=simplex[simplexPos]+1;
if(simplex[simplexPos]>n) then
simplexPos:=simplexPos-1;
simplex[simplexPos]:=simplex[simplexPos]+1;
while(simplex[simplexPos]>n or simplex[simplexPos]+
maxSimplexPos-simplexPos>n) do
simplexPos:=simplexPos-1;
simplex[simplexPos]:=simplex[simplexPos]+1;
od;
while(simplexPos<maxSimplexPos) do
simplex[simplexPos+1]:=simplex[simplexPos]+1;
simplexPos:=simplexPos+1;
od;
fi;
#generate orbit & look if already represented
add:=1;
orbit:=Orbit(G,simplex,OnSets);
#Info(InfoSimpcomp,2,"orbit ",orbit);
for rl in repLow do
pos:=Position(orbit,rl);
if(pos<>fail) then
#Info(InfoSimpcomp,2,"discarded simplex ",simplex);
add:=0;
break;
fi;
od;
for rl in forbiddenLow do
pos:=Position(orbit,rl);
if(pos<>fail) then
#Info(InfoSimpcomp,2,"discarded simplex ",simplex);
add:=0;
break;
fi;
od;
if(add=1) then
# check, if pointwise stabilizer is valid (<= C2)
stab:=Stabilizer(G,simplex,OnTuples);
if d > 2 then
if Size(stab) <= 2 then
Add(repLow,ShallowCopy(simplex));
else
Info(InfoSimpcomp,3,"simplex ",simplex,
" not valid, stabilizer too large.");
Add(forbiddenLow,ShallowCopy(simplex));
fi;
elif d <= 2 then
Add(repLow,ShallowCopy(simplex));
fi;
repOrbitLen:=repOrbitLen+Length(orbit);
fi;
if((simplexIdx mod 100)=0) then
Info(InfoSimpcomp,2,"Processing (d-1)-simplex ",simplexIdx,"/",maxIdxLow,
", ",Length(repLow)," represenative(s) so far. simplices=",repOrbitLen,
"/",maxIdxLowReal);
GASMAN("collect");
fi;
if(repOrbitLen=maxIdxLowReal) then
Info(InfoSimpcomp,3,"No more low orbits possible. Finished.");
break;
fi;
od;
repLowCount:=Length(repLow);
Info(InfoSimpcomp,2,repLowCount," low represenative(s): ",repLow," orbits=",
repOrbitLen,"/",maxIdxLowReal);
allCount:=[];
allCount[1]:=ListWithIdenticalEntries(repLowCount,0);
allCount[2]:=ListWithIdenticalEntries(repLowCount,0);
repHigh:=[];
forbiddenHigh:=[];
Info(InfoSimpcomp,2,"Generating high simplices count=",maxIdxHigh,
" and matrix...");
#compute high representative list & matrix
repOrbitLen:=0;
maxSimplexPos:=d+1;
simplex:=[1..d+1];
simplex[d+1]:=d;
for simplexIdx in [1..maxIdxHigh] do
simplexPos:=d+1;
simplex[simplexPos]:=simplex[simplexPos]+1;
if(simplex[simplexPos]>n) then
simplexPos:=simplexPos-1;
simplex[simplexPos]:=simplex[simplexPos]+1;
while(simplex[simplexPos]>n or simplex[simplexPos]+maxSimplexPos-
simplexPos>n) do
simplexPos:=simplexPos-1;
simplex[simplexPos]:=simplex[simplexPos]+1;
od;
while(simplexPos<maxSimplexPos) do
simplex[simplexPos+1]:=simplex[simplexPos]+1;
simplexPos:=simplexPos+1;
od;
fi;
#Info(InfoSimpcomp,2,"simplex ",simplex);
#generate orbit & check whether already represented
add:=1;
orbit:=Orbit(G,simplex,OnSets);
for ru in repHigh do
pos:=Position(orbit,ru);
if(pos<>fail) then
#Info(InfoSimpcomp,2,"discarded simplex ",simplex);
add:=0;
break;
fi;
od;
for ru in forbiddenHigh do
pos:=Position(orbit,ru);
if(pos<>fail) then
#Info(InfoSimpcomp,2,"discarded simplex ",simplex);
add:=0;
break;
fi;
od;
if(add=1 and d>2) then
# check, if pointwise stabilizer is trivial
stab:=Stabilizer(G,simplex,OnTuples);
if Size(stab) = 1 then
Add(repHigh,ShallowCopy(simplex));
Add(repHighOrbitLen,Length(orbit));
else
Add(forbiddenHigh,ShallowCopy(simplex));
Info(InfoSimpcomp,3,"simplex ",simplex,
" not valid, stabilizer not trivial.");
fi;
repOrbitLen:=repOrbitLen+Length(orbit);
elif(add=1 and d <= 2) then
Add(repHigh,ShallowCopy(simplex));
Add(repHighOrbitLen,Length(orbit));
fi;
if((simplexIdx mod 100)=0) then
Info(InfoSimpcomp,2,"Processing d-simplex ",simplexIdx,"/",maxIdxHigh,
", ",Length(repHigh)," represenative(s) so far. simplices="
,repOrbitLen,"/",maxIdxHighReal);
GASMAN("collect");
fi;
if(repOrbitLen=maxIdxHighReal) then
Info(InfoSimpcomp,3,"No more high orbits possible. Finished.");
break;
fi;
od;
repHighCount:=Length(repHigh);
Info(InfoSimpcomp,2,repHighCount, " high represenative(s): ",repHigh,
" orbits=",repOrbitLen,"/",maxIdxHighReal);
repHighForbidden:=[];
#check pseudomanifold property & generate matrix
Info(InfoSimpcomp,2,"Calculating matrix...");
cachedHighOrbits:=[];
for simplexIdx in [1..repHighCount] do
simplex:=repHigh[simplexIdx];
if(simplexIdx mod 100 = 1) then Info(InfoSimpcomp,2,
"Calculating incidence matrix row ",simplexIdx,"/",repHighCount); fi;
add:=1;
curRow:=[];
orbit:=Orbit(G,simplex,OnSets);
if(Size(orbit) > max) then
add:=0;
fi;
if(add=1) then
#calculate how often every (d-1)-simplex is contained in this d-simplex
for rlIdx in [1..repLowCount] do
count:=0;
for element in orbit do
if(IsSubset(element,repLow[rlIdx])) then
count:=count+1;
if(count>2) then
break;
fi;
fi;
od;
if(count>2) then
add:=0;
break;
elif(count>0) then
curRow[rlIdx]:=count;
fi;
od;
fi;
if(add=1) then
tmp:=[];
for i in [1..repLowCount] do
if(IsBound(curRow[i])) then
allCount[curRow[i]][i]:=allCount[curRow[i]][i]+1;
Add(tmp,[i,curRow[i]]);
fi;
od;
#save matrix row
Add(matrix,ShallowCopy(tmp));
if(cache=1) then
Add(cachedHighOrbits,ShallowCopy(orbit));
fi;
else
Add(repHighForbidden,simplex);
fi;
od;
repHighTmp:=[];
repHighOrbitLenTmp:=[];
for i in [1..Length(repHigh)] do
if(repHigh[i] in repHighForbidden) then
continue;
fi;
Add(repHighTmp,repHigh[i]);
Add(repHighOrbitLenTmp,repHighOrbitLen[i]);
od;
repHigh:=repHighTmp;
repHighOrbitLen:=repHighOrbitLenTmp;
repHighCount:=Length(repHigh);
Info(InfoSimpcomp,2,repHighCount, " valid high represenative(s).");
Info(InfoSimpcomp,2,"Matrix calculation done.");
#setup matrix
matrixRows:=Length(matrix);
matrixCols:=repLowCount;
matrixAllowedRows:=[1..matrixRows];
if(matrixRows=0 or matrixCols=0) then
Info(InfoSimpcomp,2,"Nothing to compute.");
return [];
fi;
#get single results
singleResults:=[];
for rowIdx in matrixAllowedRows do
if(ForAll(matrix[rowIdx],x->x[2]=2)) then
Add(singleResults,rowIdx);
fi;
od;
Info(InfoSimpcomp,2,"Got ",Length(singleResults)," single results.");
# test subset
subsetValid:=true;
srctr:=0;
if not IsList(subset) or subset = [] then
subsetValid := false;
else
SCIntFunc.DeepSortList(subset);
subset:=Set(subset);
for i in [1..Size(subset)] do
pos:=Position(repHigh,subset[i]);
if pos = fail or not pos in matrixAllowedRows then
subsetValid:=false;
break;
fi;
if pos in singleResults then
srctr:=srctr+1;
fi;
od;
fi;
if srctr > 1 then
return [];
fi;
if subsetValid then
subset:=List(subset,x->Position(repHigh,x));
fi;
if Size(subset) = 1 and srctr = 1 then
#generate complex
complex:=Orbit(G,repHigh[subset[1]],OnSets);
if Union(complex) = [1..n] then
tmp:=[[repHigh[subset[1]],Size(complex)]];
if(examineComplex(complex,complex_collection,objectType,t)) then
complex:=SCFromFacets(complex);
# we cannot perform iso sig computations for 2-pseudomanifolds
if removeDoubleEntries and not (d=2 and objectType=1) then
Info(InfoSimpcomp,2,"Testing if complex is equivalent ",
"to previous one...");
isoSig:=SCExportIsoSig(complex);
if isoSig = fail then
return fail;
fi;
if isoSig in isoSigs then
Info(InfoSimpcomp,2,"yes.");
else
Info(InfoSimpcomp,2,"no.");
Add(isoSigs,isoSig);
candidateCount:=candidateCount+1;
if outfile <> false then
AppendTo(outfile,"Add(c,SCFromGenerators(G,",[repHigh[subset[1]]],"));\n");
fi;
fi;
else
if not HasName(G) then
SetName(G,StructureDescription(G));
fi;
name:=Concatenation("Complex ",String(Size(complex_collection)+1),
" of ",Name(G)," (single orbit)");
SCRename(complex,name);
if outfile <> false then
AppendTo(outfile,"Add(c,SCFromGenerators(G,",
[repHigh[subset[1]]],"));\n");
fi;
Add(complex_collection,complex);
candidateCount:=candidateCount+1;
fi;
fi;
if removeDoubleEntries then
return isoSigs;
else
return complex_collection;
fi;
fi;
elif not subsetValid then
for i in [1..Length(singleResults)] do
Info(InfoSimpcomp,2,"Single result ",i,"/",Length(singleResults),
" : ",matrix[singleResults[i]],"(",singleResults[i],")");
for j in matrix[singleResults[i]] do
allCount[2][j[1]]:=allCount[2][j[1]]-1;
od;
#generate complex
complex:=Orbit(G,repHigh[singleResults[i]],OnSets);
if Union(complex) = [1..n] then
tmp:=[[repHigh[singleResults[i]],Size(complex)]];
if(examineComplex(complex,complex_collection,objectType,1)) then
complex:=SCFromFacets(complex);
# we cannot perform iso sig computations for 2-pseudomanifolds
if removeDoubleEntries and not (d=2 and objectType=1) then
Info(InfoSimpcomp,2,"Testing if complex is equivalent ",
"to previous one...");
isoSig:=SCExportIsoSig(complex);
if isoSig = fail then
return fail;
fi;
if isoSig in isoSigs then
Info(InfoSimpcomp,2,"yes.");
else
Info(InfoSimpcomp,2,"no.");
Add(isoSigs,isoSig);
candidateCount:=candidateCount+1;
if outfile <> false then
AppendTo(outfile,"Add(c,SCFromGenerators(G,",[repHigh[subset[1]]],"));\n");
fi;
fi;
else
if not HasName(G) then
SetName(G,StructureDescription(G));
fi;
name:=Concatenation("Complex ",String(Size(complex_collection)+1),
" of ",Name(G)," (single orbit)");
SCRename(complex,name);
if outfile <> false then
AppendTo(outfile,"Add(c,SCFromGenerators(G,",
[repHigh[subset[1]]],"));\n");
fi;
Add(complex_collection,complex);
candidateCount:=candidateCount+1;
fi;
fi;
fi;
od;
fi;
# note to self: here we could stop and just return single results
SubtractSet(matrixAllowedRows,singleResults);
matrixRows:=Length(matrixAllowedRows);
if(matrixRows<1) or single = true then
#if(matrixRows<1) then
Info(InfoSimpcomp,2,"No more computations needed (no more rows).");
if removeDoubleEntries then
return isoSigs;
else
return complex_collection;
fi;
fi;
Info(InfoSimpcomp,2,"Incidence matrix:\n",matrix);
if(matrixRows<1 or matrixCols<1) then
Info(InfoSimpcomp,2,"No more computations needed (no more rows/cols).");
if removeDoubleEntries then
return isoSigs;
else
return complex_collection;
fi;
fi;
GASMAN("collect");
Info(InfoSimpcomp,2,"Got ",matrixRows," rows, ",matrixCols,
" cols in matrix, computing row combinations...");
#init column vector
column:=[];
for s in [1..repLowCount] do
column[s]:=[];
od;
for i in [1..matrixRows] do
Add(column[matrix[matrixAllowedRows[i]][1][1]],matrixAllowedRows[i]);
od;
if matrixRows <= 1 then
stop:=1;
else
stop:=0;
fi;
# vectorsum{[1..repLowCount]} = current combination of rows
# vectorsum[repLowCount+1] = number of 1-entries in combination
# vectorsum[repLowCount+2] = size of complex (facets)
vectorsum:=ListWithIdenticalEntries(repLowCount+2,0);
##########################
### SETUP BACKTRACKING ###
##########################
if subsetValid then
# case: "subset" is valid
curComb:=[];
for i in subset do
Add(curComb,i);
if (updateRowVector(true,i)=false) then
Info(InfoSimpcomp,2,"argument subset valid but not part ",
"of a candidate complex.");
return [];
fi;
od;
if not matrixAllowedRows[1] in subset then
i:=1;
while (updateRowVector(true,matrixAllowedRows[i])=false) do
i:=i+1;
if i in subset then
break;
fi;
od;
if not i in subset then
Add(curComb,matrixAllowedRows[i]);
colPos:=matrix[matrixAllowedRows[i]][1][1];
i:=Position(column[colPos],matrixAllowedRows[i])+1;
while i <= Size(column[colPos]) and column[colPos][i] in subset do
i:=i+1;
od;
curPos:=i;
else
colPos:=matrix[subset[1]][1][1];
curPos:=Position(column[colPos],subset[1])+1;
fi;
else
colPos:=matrix[subset[1]][1][1];
curPos:=Position(column[colPos],subset[1])+1;
fi;
else
# case: "subset" is invalid or empty
i:=1;
while (updateRowVector(true,matrixAllowedRows[i])=false) do
i:=i+1;
od;
curComb:=[matrixAllowedRows[i]];
colPos:=matrix[matrixAllowedRows[i]][1][1];
curPos:=Position(column[colPos],matrixAllowedRows[i])+1;
fi;
foundComplex:=0;
# column: column[i] contains all row numbers x with row[x][i] <> 0 and
# row[x][j] = 0 for all j < i
# colPos: position of first non zero entry of current row
# curPos: next row where first non zero entry is at the same
# position as in current row
while stop = 0 do
foundComplex:=0;
while foundComplex = 0 and colPos <= repLowCount do
# if there is a row left where the first non zero entry is at the
# same column as colPos
if curPos > Length(column[colPos]) then
if vectorsum[colPos] in [0,2] then
colPos:=colPos+1;
if subsetValid then
i:=1;
while colPos <= Size(column) and i <= Size(column[colPos]) and
column[colPos][i] in subset do
i:=i+1;
od;
curPos:=i;
else
curPos:=1;
fi;
else
if subsetValid and Size(curComb) = Size(subset) then
stop:=1;
break;
fi;
lastRow:=curComb[Length(curComb)];
updateRowVector(false,lastRow);
Unbind(curComb[Length(curComb)]);
if curComb = [] and colPos > repLowCount then
stop:=1;
fi;
colPos:=matrix[lastRow][1][1];
if subsetValid then
i:=Position(column[colPos],lastRow)+1;
while i <= Size(column[colPos]) and column[colPos][i] in subset do
i:=i+1;
od;
curPos:=i;
else
curPos:=Position(column[colPos],lastRow)+1;
fi;
fi;
continue;
fi;
# column colPos is already mapped -> next step
if vectorsum[colPos] = 2 then
colPos:=colPos+1;
if subsetValid then
i:=1;
while colPos <= Size(column) and i <= Size(column[colPos]) and
column[colPos][i] in subset do
i:=i+1;
od;
curPos:=i;
else
curPos:=1;
fi;
# if this is the last row where the first non zero entry is at the
# same column as colPos and orbit cannot be closed anymore -> next step
elif vectorsum[colPos] = 0 and matrix[column[colPos][curPos]][1][2] = 1
and curPos = Length(column[colPos]) then
colPos:=colPos+1;
if subsetValid then
i:=1;
while colPos <= Size(column) and i <= Size(column[colPos])
and column[colPos][i] in subset do
i:=i+1;
od;
curPos:=i;
else
curPos:=1;
fi;
elif subsetValid and column[colPos][curPos] in subset then
curPos:=curPos+1;
else
foundComplex:=1;
stopSize:=0;
if(updateRowVector(true,column[colPos][curPos])=false) then
stopSize:=1;
fi;
if stopSize = 0 then
Add(curComb,column[colPos][curPos]);
if vectorsum[repLowCount+1] = 0 then
assembleComplex(curComb);
if (Union(complex) = [1..n] and
examineComplex(complex,complex_collection,objectType,1)) then
complex:=SCFromFacets(complex);
# we cannot perform iso sig computations for 2-pseudomanifolds
if removeDoubleEntries and not (d=2 and objectType=1) then
Info(InfoSimpcomp,2,"Testing if complex is equivalent ",
"to previous one...");
isoSig:=SCExportIsoSig(complex);
if isoSig = fail then
return fail;
fi;
if isoSig in isoSigs then
Info(InfoSimpcomp,2,"yes.");
else
Info(InfoSimpcomp,2,"no.");
Info(InfoSimpcomp,2,"Found candidate.");
Add(isoSigs,isoSig);
candidateCount:=candidateCount+1;
if outfile <> false then
AppendTo(outfile,"Add(c,SCFromGenerators(G,",repHigh{curComb},"));\n");
fi;
fi;
else
if not HasName(G) then
SetName(G,StructureDescription(G));
fi;
name:=Concatenation("Complex ",
String(Size(complex_collection)+1)," of ",Name(G),
" (multiple orbits)");
SCRename(complex,name);
if outfile <> false then
AppendTo(outfile,"Add(c,SCFromGenerators(G,",
repHigh{curComb},"));\n");
fi;
Info(InfoSimpcomp,2,"Found candidate.");
Add(complex_collection,complex);
candidateCount:=candidateCount+1;
fi;
fi;
else
foundComplex:=0;
fi;
else
foundComplex:=0;
#updateRowVector(false,column[colPos][curPos]);
fi;
if subsetValid then
curPos:=curPos+1;
while curPos <= Length(column[colPos]) and
column[colPos][curPos] in subset do
curPos:=curPos+1;
od;
else
curPos:=curPos+1;
fi;
fi;
od;
if stop = 1 then break; fi;
if curComb = [] and colPos > repLowCount then
stop:=1;
break;
else
if subsetValid and Size(curComb) = Size(subset) then
stop:=1;
break;
fi;
lastRow:=curComb[Length(curComb)];
updateRowVector(false,lastRow);
Unbind(curComb[Length(curComb)]);
colPos:=matrix[lastRow][1][1];
if subsetValid then
i:=Position(column[colPos],lastRow)+1;
while i <= Size(column[colPos]) and column[colPos][i] in subset do
i:=i+1;
od;
curPos:=i;
else
curPos:=Position(column[colPos],lastRow)+1;
fi;
fi;
od;
Info(InfoSimpcomp,2,"All done, ",candidateCount," candidate(s).");
if removeDoubleEntries then
return isoSigs;
else
return complex_collection;
fi;
end);
################################################################################
##<#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);
## ]]></Example>
## </Description>
## </ManSection>
##<#/GAPDoc>
################################################################################
InstallGlobalFunction(SCsFromGroupByTransitivity,
function(n,d,k,maniflag,computeAutGroup,removeDoubleEntries)
local coll, i, G, tmp, dim, verts, sum, Gcollection, g, gg, retList,
retListTmp, c, lk, pm, m, flag, ctr, cc, j, l, warn, isoSigs;
if((not IsPosInt(n) and not (IsList(n) and ForAll(n,x->IsPosInt(x)))) or
(not IsPosInt(d) and not (IsList(d) and ForAll(d,x->IsPosInt(x)))) or
not IsPosInt(k) or not IsBool(maniflag) or not IsBool(computeAutGroup)
or not IsBool(removeDoubleEntries)) then
Info(InfoSimpcomp,1,"SCsFromGroupCheckByTransitivity: 'n' and 'd' must ",
"be positive integers or a list of positive integers, 'k' must be a ",
"positive integer, 'maniflag', computeAutGroup' and ",
"'removeDoubleEntries' must be boolean");
return fail;
fi;
if IsInt(n) then
n:=[n];
fi;
if IsInt(d) then
d:=[d];
fi;
if removeDoubleEntries then
isoSigs:=[];
fi;
# detect group list, do not check groups with k-transitive subgroups
Info(InfoSimpcomp,1,"SCsFromGroupByTransitivity: Building list of groups...");
Gcollection:=[];
if k > 1 then
for verts in n do
Gcollection[verts]:=[];
for i in [1..NrPrimitiveGroups(verts)] do
g:=PrimitiveGroup(verts,i);
if Transitivity(g) <> k then
continue;
fi;
flag := 0;
for gg in Gcollection[verts] do
if IsSubgroup(g,gg) then
flag:=1;
break;
fi;
od;
if flag = 0 then
Add(Gcollection[verts],g);
fi;
od;
od;
elif k = 1 then
for verts in n do
Gcollection[verts]:=[];
for i in [1..NrTransitiveGroups(verts)] do
g:=TransitiveGroup(verts,i);
flag := 0;
for gg in Gcollection[verts] do
if IsSubgroup(g,gg) then
flag:=1;
break;
fi;
od;
if flag = 0 then
Add(Gcollection[verts],g);
fi;
od;
od;
else
Info(InfoSimpcomp,1,"SCsFromGroupByTransitivity: third argument 'k' must ",
"be a positive integer.");
return fail;
fi;
sum:=0;
for i in Gcollection do
sum:=sum+Size(i);
od;
Info(InfoSimpcomp,1,"SCsFromGroupByTransitivity: ...",sum," groups found.");
for i in Gcollection do
if i <> [] then
Info(InfoSimpcomp,1,"degree ",Position(Gcollection,i),": ",i);
fi;
od;
coll:=[];
retList:=[];
if maniflag then
retList[1]:=[];
retList[2]:=[];
retList[3]:=[];
fi;
warn:=false;
for dim in d do
if dim=2 and removeDoubleEntries=true then
warn:=true;
fi;
Info(InfoSimpcomp,1,"SCsFromGroupByTransitivity: Processing dimension ",
dim,".");
coll[dim]:=[];
for verts in n do
Info(InfoSimpcomp,1,"SCsFromGroupByTransitivity: Processing degree ",
verts,".");
sum:=0;
retListTmp:=[];
if maniflag then
retListTmp[1]:=[];
retListTmp[2]:=[];
retListTmp[3]:=[];
fi;
for G in Gcollection[verts] do
tmp:=SCsFromGroupExt(G,verts,dim,1,0,removeDoubleEntries,false,0,[],false);
if removeDoubleEntries then
for i in [1..Size(tmp)] do
if tmp[i] in isoSigs then
Unbind(tmp[i]);
else
Add(isoSigs,tmp[i]);
fi;
od;
tmp:=Compacted(tmp);
fi;
sum:=sum+1;
Info(InfoSimpcomp,1,"SCsFromGroupByTransitivity: ",sum," / ",
Size(Gcollection[verts])," groups calculated, found ",Size(tmp),
" complexes.");
if computeAutGroup then
Info(InfoSimpcomp,1,"SCsFromGroupByTransitivity: Calculating ",
Size(tmp)," automorphism and homology groups...");
cc:=0;
for i in [1..Size(tmp)] do
if removeDoubleEntries then
tmp[i]:=SCFromIsoSig(tmp[i]);
fi;
cc:=cc+1;
SCAutomorphismGroup(tmp[i]);
SCHomology(tmp[i]);
SCGenerators(tmp[i]);
Info(InfoSimpcomp,1,"SCsFromGroupByTransitivity: ",cc," / ",
Size(tmp)," automorphism groups calculated.");
od;
Info(InfoSimpcomp,1,"SCsFromGroupByTransitivity: ...all ",
"automorphism groups calculated for group ",sum," / ",
Size(Gcollection[verts]),".");
fi;
if maniflag then
for c in tmp do
lk:=SCLink(c,1);
pm:=SCIsManifold(lk);
if pm = fail then
Info(InfoSimpcomp,1,"SCsFromGroupByTransitivity: Could not ",
"determine whether candidate is combinatorial pseudomanifold ",
"or not");
Add(retListTmp[3],c);
fi;
if pm then
m:=SCIsManifold(c);
if m then
Add(retListTmp[1],c);
elif not m or m = fail then
Add(retListTmp[2],c);
fi;
elif not pm then
Info(InfoSimpcomp,1,"SCsFromGroupByTransitivity: Combinatorial ",
"pseudomanifold property not fulfilled, discarding complex.");
fi;
od;
else
Append(retListTmp,tmp);
fi;
od;
if maniflag then
Info(InfoSimpcomp,1,"SCsFromGroupByTransitivity: ...done dim = ",dim,
", deg = ",verts,", ",Size(retListTmp[1])," manifolds, ",
Size(retListTmp[2])," pseudomanifolds, ",Size(retListTmp[3]),
" candidates found.");
else
Info(InfoSimpcomp,1,"SCsFromGroupByTransitivity: ...done dim = ",
dim,", deg = ",verts,", ",Size(retListTmp)," candidates found.");
fi;
od;
Info(InfoSimpcomp,1,"SCsFromGroupByTransitivity: ...done dim = ",dim,".");
od;
if warn then
Info(InfoSimpcomp,1,"SCsFromGroupByTransitivity: can not remove double ",
"entries for pp-surfaces.");
fi;
return retList;
end);
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