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#
# Ugaly: Universal Groups Acting LocallY
#
# Implementations
#
##################################################################################################################
InstallGlobalFunction( AreCompatibleBallElements,
function(d,k,aut1,aut2,dir)
local lf, im_dir, im_lf;
if not (IsInt(d) and d>=3) then
Error("input argument d=",d," must be an integer greater than or equal to 3");
elif not (IsInt(k) and k>=1) then
Error("input argument k=",k," must be an integer greater than or equal to 1");
elif not IsPerm(aut1) then
Error("input argument aut1=",aut1," must be an element of AutBall(d,k)");
elif not IsPerm(aut2) then
Error("input argument aut1=",aut2," must be an element of AutBall(d,k)");
elif not (IsInt(dir) and dir in [1..d]) then
Error("input argument dir=",dir," must be in the range [1..",d,"]");
else
if k=1 then
if dir^aut1=dir^aut2 then
return true;
else
return false;
fi;
else
# k>=2
for lf in [(dir-1)*(d-1)^(k-1)+1..dir*(d-1)^(k-1)] do
im_lf:=AddressOfLeaf(d,k,lf^aut1);
if not im_lf{[2..k]}=ImageAddress(d,k,aut2,AddressOfLeaf(d,k,lf){[2..k]}) then
return false;
fi;
im_lf:=AddressOfLeaf(d,k,lf^aut2);
if not im_lf{[2..k]}=ImageAddress(d,k,aut1,AddressOfLeaf(d,k,lf){[2..k]}) then
return false;
fi;
od;
return true;
fi;
fi;
end );
##################################################################################################################
InstallGlobalFunction( CompatibleBallElement,
function(F,aut,dir)
local d, k, pr, K, aut_dir, r, reps, b, compatible, lf, im_lf;
if not IsLocalAction(F) then
Error("input argument F=",F," must be a local action");
elif not IsPerm(aut) then
Error("input argument aut=",aut," must be an element of AutBall(d,k)");
elif not (IsInt(dir) and dir in [1..LocalActionDegree(F)]) then
Error("input argument dir=",dir," must be an integer in the range [1..d=",LocalActionDegree(F),"], where d is the degree of input argument F=",F);
else
d:=LocalActionDegree(F);
k:=LocalActionRadius(F);
if k=0 or k=1 then
return aut;
else
# k>=2
# search in the correct coset of the kernel (inner local action condition)
pr:=RestrictedMapping(Projection(AutBall(d,k)),F);
K:=Kernel(pr);
aut_dir:=LocalAction(k-1,d,k,aut,[dir]);
if not aut_dir in Range(pr) then
return fail;
else
r:=PreImagesRepresentative(pr,aut_dir);
if r=fail then return fail; fi;
fi;
# generate representatives for the action on the relevant leaves
reps:=ShallowCopy(AsList(RightTransversal(K,Stabilizer(K,[(dir-1)*(d-1)^(k-1)+1..dir*(d-1)^(k-1)],OnTuples))));
Apply(reps,x->x*r);
# test outer local action condition
for b in reps do
compatible:=true;
for lf in [(dir-1)*(d-1)^(k-1)+1..dir*(d-1)^(k-1)] do
im_lf:=AddressOfLeaf(d,k,lf^b);
if not im_lf{[2..k]}=ImageAddress(d,k,aut,AddressOfLeaf(d,k,lf){[2..k]}) then
compatible:=false;
break;
fi;
od;
if compatible then return b; fi;
od;
return fail;
fi;
fi;
end );
##################################################################################################################
InstallMethod( CompatibilitySet, "for a local action F, an element aut of F and a direction dir", [IsLocalAction, IsPerm, IsInt],
function(F,aut,dir)
local d, k, r, K;
if not dir in [1..LocalActionDegree(F)] then
Error("input argument dir=",dir," must be in the range [1..",LocalActionDegree(F),"]");
else
d:=LocalActionDegree(F);
k:=LocalActionRadius(F);
if k=0 then
return F;
elif k=1 then
return RightCoset(Stabilizer(F,dir),aut);
else
# k>=2
r:=CompatibleBallElement(F,aut,dir);
if r=fail then return []; fi;
K:=Kernel(RestrictedMapping(Projection(AutBall(d,k)),F));
return RightCoset(Stabilizer(K,[(dir-1)*(d-1)^(k-1)+1..dir*(d-1)^(k-1)],OnTuples),r);
fi;
fi;
end );
InstallMethod( CompatibilitySet, "for a local action F, an element aut of F and a list of directions dirs", [IsLocalAction, IsPerm, IsList],
function(F,aut,dirs)
local d, k, comp_sets, dir, r, p, K;
if not IsSubset([1..LocalActionDegree(F)],dirs) then
Error("input argument dirs=",dirs," must be a subset of [1..",LocalActionDegree(F),"]");
else
d:=LocalActionDegree(F);
k:=LocalActionRadius(F);
if k=0 then
return F;
elif k=1 then
return RightCoset(Stabilizer(F,dirs,OnTuples),aut);
else
# k>=2
comp_sets:=[];
for dir in dirs do
Add(comp_sets,CompatibilitySet(F,aut,dir));
od;
return Intersection(comp_sets);
fi;
fi;
end );
##################################################################################################################
InstallGlobalFunction( AssembleAutomorphism,
function(d,k,auts)
local aut, i, l, addr, im_addr;
if not d>=3 then
Error("input argument d=",d," must be an integer greater than or equal to 3");
elif not k>=1 then
Error("input argument k=",k," must be an integer greater than or equal to 1");
elif not IsList(auts) then
Error("input argument auts=",auts," must be a list of automorphisms of B_{d,k}");
else
aut:=[];
for i in [1..d] do
for l in [(i-1)*(d-1)^k+1..i*(d-1)^k] do
addr:=AddressOfLeaf(d,k+1,l);
im_addr:=ImageAddress(d,k,auts[i],[addr[1]]);
Append(im_addr,ImageAddress(d,k,auts[i],addr{[2..k+1]}));
Add(aut,LeafOfAddress(d,k+1,im_addr));
od;
od;
return PermList(aut);
fi;
end );
##################################################################################################################
InstallMethod( MaximalCompatibleSubgroup, "for a local action F", [IsLocalAction],
function(F)
local d, k, grps, poss, pos, i, G;
d:=LocalActionDegree(F);
k:=LocalActionRadius(F);
if k=0 then
return F;
elif SatisfiesC(F) then
return F;
else
# go through subgroup lattice, starting at the top
grps:=[F];
while not IsEmpty(grps) do
Apply(grps,H->ShallowCopy(MaximalSubgroups(H)));
grps:=DuplicateFreeList(Flat(grps));
poss:=Positions(grps,Group(()));
for pos in poss do Remove(grps,pos); od;
for G in grps do
if SatisfiesC(LocalActionNC(d,k,G)) then return LocalActionNC(d,k,G); fi;
od;
od;
# no non-trivial compatible subgroup
return LocalActionNC(d,k,Group(()));
fi;
end );
##################################################################################################################
InstallMethod( SatisfiesC, "for a local action F", [IsLocalAction],
function(F)
local d, k, gens, a, dir, r, b;
d:=LocalActionDegree(F);
k:=LocalActionRadius(F);
if k=0 or k=1 then
return true;
else
gens:=SmallGeneratingSet(F);
for a in gens do
for dir in [1..d] do
if CompatibleBallElement(F,a,dir)=fail then
return false;
fi;
od;
od;
return true;
fi;
end );
##################################################################################################################
InstallGlobalFunction( CompatibleSubgroups,
function(F)
local d, k, grps, H;
d:=LocalActionDegree(F);
k:=LocalActionRadius(F);
if k=0 or k=1 then
return AllSubgroups(F);
else
grps:=[];
for H in AllSubgroups(F) do
if SatisfiesC(LocalActionNC(d,k,H)) then Add(grps,LocalActionNC(d,k,H)); fi;
od;
return grps;
fi;
end );
##################################################################################################################
InstallMethod( ConjugacyClassRepsCompatibleSubgroups, "for a local action F", [IsLocalAction],
function(F)
local d, k, reps, H, class;
d:=LocalActionDegree(F);
k:=LocalActionRadius(F);
if k=0 or k=1 then
return ConjugacyClassesSubgroups(F);
else
reps:=[];
for class in ConjugacyClassesSubgroups(F) do
for H in class do
if SatisfiesC(LocalActionNC(d,k,H)) then
Add(reps,LocalActionNC(d,k,H));
break;
fi;
od;
od;
return reps;
fi;
end );
##################################################################################################################
InstallGlobalFunction( ConjugacyClassRepsCompatibleGroupsWithProjection,
function(l,F)
local d, k, reps, G_k, G_l, C, class, H, new, i;
if not IsLocalAction(F) then
Error("input argument F=",F," must be a local action");
elif not (IsInt(l) and l>=LocalActionRadius(F)) then return
Error("input argument l=",l," must be an integer greater than or equal to the radius k=",LocalActionRadius(F)," of input argument F=",F);
else
d:=LocalActionDegree(F);
k:=LocalActionRadius(F);
if l=k then
return F;
else
reps:=[];
G_k:=AutBall(d,k);
G_l:=AutBall(d,l);
# initialize Phi^{l}(F)
C:=LocalActionPhi(l,F);
# search
for class in ConjugacyClassesSubgroups(C) do
if not IsConjugate(G_k,ImageOfProjection(LocalActionNC(d,l,class[1]),k),F) then continue; fi;
for H in class do
if not ImageOfProjection(LocalActionNC(d,l,class[1]),k)=F then continue; fi;
if SatisfiesC(LocalActionNC(d,l,H)) then
new:=true;
for i in [Length(reps),Length(reps)-1..1] do
if IsConjugate(G_l,H,reps[i]) then new:=false; break; fi;
od;
if new then Add(reps,LocalActionNC(d,l,H)); fi;
break;
fi;
od;
od;
return reps;
fi;
fi;
end );
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