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ENHANCE: normalform option for FpGroupCocycle
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Using the RWS and normal form. This often costs little extra, and is just nicer overall.

Also adds required forward declaration of utility function.
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@hulpke @fingolfin
hulpke authored and fingolfin committed May 3, 2021
1 parent b1c4988 commit 69f0389
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2 changes: 1 addition & 1 deletion lib/gpfpiso.gi
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Original file line number Diff line number Diff line change
Expand Up @@ -1029,7 +1029,7 @@ local hom;
return hom;
end);

BindGlobal("MakeFpGroupToMonoidHomType1",function(fp,m)
InstallGlobalFunction(MakeFpGroupToMonoidHomType1,function(fp,m)
local fam,mfam,fpfam,mfpfam,hom;
fam:=FamilyObj(UnderlyingElement(One(fp)));
mfam:=FamilyObj(UnderlyingElement(One(m)));
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3 changes: 3 additions & 0 deletions lib/grpfp.gd
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Expand Up @@ -1352,5 +1352,8 @@ DeclareGlobalFunction("StringFactorizationWord");
# used to test whether abeliniazation can be mapped in GQuotients
DeclareGlobalFunction("CanMapFiniteAbelianInvariants");

# map fpgrp->fpmon creator
DeclareGlobalFunction("MakeFpGroupToMonoidHomType1");

# used in homomorphisms
DeclareGlobalName("TRIVIAL_FP_GROUP");
8 changes: 7 additions & 1 deletion lib/twocohom.gd
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Expand Up @@ -210,8 +210,9 @@ DeclareOperation( "TwoCohomology", [ IsPcGroup, IsObject ] );
## routine, this generating system corresponds to the components
## <C>group</C> and <C>module</C> of the record returned.
## The extension corresponding to a cocyle <C>c</C> can be constructed as
## <C>Extension(r,c)</C> where <C>r</C> is the cohomology record. This is
## <C>FpGroupCocycle(r,c)</C> where <C>r</C> is the cohomology record. This is
## currently done as a finitely presented group.
##
## <Example><![CDATA[
## gap> g:=Group((1,2,3,4,5),(1,2,3));;
## gap> mats:=[[[2,0,0,1],[1,2,1,0],[2,1,1,1],[2,1,1,0]],
Expand Down Expand Up @@ -274,6 +275,8 @@ DeclareOperation( "TwoCohomologyGeneric", [ IsGroup, IsObject ] );
## If the optional parameter <A>doperm</A> is given as <A>true</A>, a
## faithful permutation representation is computed and stored in the
## attribute <Ref Attr="IsomorphismPermGroup"/> of the computed group.
## If the option <C>normalform</C> is given as <C>true</C>, arithmetic in
## the resulting finitely presented group will bring words into normal form.
## <Example><![CDATA[
## gap> g:=Group((2,15,8,16)(3,17,14,21)(4,23,20,6)(5,9,22,11)(7,13,19,25),
## > (2,12,7,17)(3,18,13,23)(4,24,19,9)(5,10,25,15)(6,11,16,21));;
Expand All @@ -299,6 +302,9 @@ DeclareOperation( "TwoCohomologyGeneric", [ IsGroup, IsObject ] );
## [ [ 480, 3 ], [ 3888, 1 ], [ 6480, 1 ], [ 7776, 1 ], [ 19440, 1 ] ]
## gap> Collected(List(MaximalSubgroupClassReps(g2),Size));
## [ [ 3888, 1 ], [ 6480, 1 ], [ 7776, 1 ], [ 19440, 1 ] ]
## gap> p:=FpGroupCocycle(coh,coh.cohomology[1],true:normalform);;
## gap> p.7*p.1; # i.e. m1*F1, but in normal form
## F1*m2^-1
## ]]></Example>
## </Description>
## </ManSection>
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124 changes: 118 additions & 6 deletions lib/twocohom.gi
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Expand Up @@ -1501,7 +1501,8 @@ end);
InstallGlobalFunction(FpGroupCocycle,function(arg)
local r,z,ogens,n,gens,str,dim,i,j,f,rels,new,quot,g,p,collect,m,e,fp,old,sim,
it,hom,trysy,prime,mindeg,fps,ei,mgens,mwrd,nn,newfree,mfpi,mmats,sub,
tab,tab0,evalprod,gensmrep,invsmrep,zerob,step,simi,simiq,wasbold;
tab,tab0,evalprod,gensmrep,invsmrep,zerob,step,simi,simiq,wasbold,
mon,ord,mn,melmvec;

# function to evaluate product (as integer list) in gens (and their
# inverses invs) with corresponding action mats
Expand All @@ -1518,17 +1519,124 @@ local r,z,ogens,n,gens,str,dim,i,j,f,rels,new,quot,g,p,collect,m,e,fp,old,sim,
fi;
od;
return new;

end;


melmvec:=function(v)
local i,a,w;
w:=One(f);
for i in [1..Length(v)] do
a:=Int(v[i]);
if prime=2 or a*2<prime then
w:=w*gens[2*i-1+mn]^a;
else
a:=prime-a;
w:=w*gens[2*i+mn]^a;
fi;
od;
return w;
end;

r:=arg[1];
z:=arg[2];
ogens:=GeneratorsOfGroup(r.presentation.group);
dim:=r.module.dimension;
prime:=Size(r.module.field);

if ValueOption("normalform")<>true then
mon:=fail;
else
# Construct the monoid and rewriting system, as this is cheap to do but
# will allow for reduced multiplication. Do so before the group, so there is no
# issue about overwriting variables that might be needed later
mon:=Image(r.monhom);
str:=List(GeneratorsOfMonoid(mon),String);
mn:=Length(str);
for i in [1..dim] do
Add(str,Concatenation("m",String(i)));
Add(str,Concatenation("m",String(i),"^-1"));
od;
f:=FreeMonoid(str);
gens:=GeneratorsOfMonoid(f);

rels:=[];
# inverse rules
for i in [1,3..Length(gens)-1] do
Add(rels,[gens[i]*gens[i+1],One(f)]);
Add(rels,[gens[i+1]*gens[i],One(f)]);
od;
# rules with tails
for i in [1..Length(r.presentation.monrulpos)] do
new:=RelationsOfFpMonoid(mon)[r.presentation.monrulpos[i]];
new:=List(new,x->MappedWord(x,FreeGeneratorsOfFpMonoid(mon),gens{[1..mn]}));
new[2]:=new[2]*melmvec(z{[(i-1)*dim+1..i*dim]});
Add(rels,new);
od;

if r.presentation.killrelators<>[] then Error("TODO killrelators");fi;

# elementary abelian
for i in [mn+1,mn+3..Length(str)-1] do
if prime=2 then
Add(rels,[gens[i]^2,One(f)]);
Add(rels,[gens[i+1],gens[i]]);
else
j:=QuoInt(prime+1,2); # power that changes the exponent sign
Add(rels,[gens[i]^j,gens[i+1]^(j-1)]);
Add(rels,[gens[i+1]^j,gens[i]^(j-1)]);
fi;
for j in [i+2..Length(str)] do
Add(rels,[gens[j]*gens[i],gens[i]*gens[j]]);
Add(rels,[gens[j]*gens[i+1],gens[i+1]*gens[j]]);
od;
od;
# module rules
for i in [1..mn] do
for j in [mn+1..Length(str)] do
new:=r.module.generators[QuoInt(i+1,2)];
if IsEvenInt(i) then new:=new^-1; fi;
new:=new[QuoInt(j-mn+1,2)];
if IsEvenInt(j) then new:=-new;fi;
Add(rels,[gens[j]*gens[i],gens[i]*melmvec(new)]);
od;
od;
if HasReducedConfluentRewritingSystem(mon) then
ord:=OrderingOfRewritingSystem(ReducedConfluentRewritingSystem(mon));
if HasLevelsOfGenerators(ord) then
ord:=WreathProductOrdering(f,
Concatenation(LevelsOfGenerators(ord)+1,
ListWithIdenticalEntries(2*dim,1)));
else
ord:=WreathProductOrdering(f,
Concatenation(ListWithIdenticalEntries(mn,2),
ListWithIdenticalEntries(2*dim,1)));
fi;
else
ord:=WreathProductOrdering(f,
Concatenation(ListWithIdenticalEntries(mn,2),
ListWithIdenticalEntries(2*dim,1)));
fi;

# if there is any [x^2,1] relation, this means the inverse needs to be
# mapped to the generator. (This will have been skipped in the
# monrulpos.)
for i in rels do
if Length(i[2])=0 and Length(i[1])=2
and Length(Set(LetterRepAssocWord(i[1])))=1 then
new:=LetterRepAssocWord(i[1])[1];
if IsOddInt(new) then
Add(rels,[gens[new+1],gens[new]]);
fi;
fi;
od;

mon:=FactorFreeMonoidByRelations(f,rels);
new:=KnuthBendixRewritingSystem(FamilyObj(One(mon)),ord);
MakeConfluent(new); # will add rules to kill inverses, if needed
SetReducedConfluentRewritingSystem(mon,new);
fi;

# now construct the group
str:=List(ogens,String);
dim:=r.module.dimension;
zerob:=ImmutableVector(r.module.field,
ListWithIdenticalEntries(dim,Zero(r.module.field)));
n:=Length(ogens);
Expand Down Expand Up @@ -1560,7 +1668,7 @@ local r,z,ogens,n,gens,str,dim,i,j,f,rels,new,quot,g,p,collect,m,e,fp,old,sim,
od;

for i in [n+1..Length(gens)] do
Add(rels,gens[i]^r.prime);
Add(rels,gens[i]^prime);
for j in [i+1..Length(gens)] do
Add(rels,Comm(gens[i],gens[j]));
od;
Expand All @@ -1572,11 +1680,15 @@ local r,z,ogens,n,gens,str,dim,i,j,f,rels,new,quot,g,p,collect,m,e,fp,old,sim,
od;
od;
fp:=f/rels;
prime:=Size(r.module.field);
SetSize(fp,Size(r.group)*prime^r.module.dimension);
simi:=fail;
wasbold:=false;

if mon<>fail then
rels:=MakeFpGroupToMonoidHomType1(fp,mon);
SetReducedMultiplication(fp);
fi;

if Length(arg)>2 and arg[3]=true then
if IsZero(z) and MTX.IsIrreducible(r.module) then
# make SDP directly
Expand Down

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