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#
# subgroupladders: This package provides an algorithm that computes a subgroup ladder from a permutation group up to the parent symmetric group.
#
# Implementations
#
InstallGlobalFunction( YoungGroupFromPartition,
function(part)
local
i,
P,
G,
grps,
olds,
news,
perms,
info,
generators;
grps := [];
olds := [];
news := [];
perms := [];
generators := [];
for i in part do
G := SymmetricGroup(i);
Append(generators, GeneratorsOfGroup(G));
Add(grps, G);
Add(olds, i);
Add(news, i);
Add(perms, ());
od;
P := Group(generators);
info := rec( groups := grps,
olds := olds,
news := news,
perms := perms,
embeddings := [],
projections := [] );
SetDirectProductInfo(P, info);
return P;
end);
InstallGlobalFunction( SubgroupLadder,
function(G)
local
orb,
i,
k,
n,
pair,
ladder,
directfactors,
generators,
mapping,
output,
partition,
FindPos;
FindPos := function(list, x)
local n, i;
n := Length(list);
for i in [1..n] do
if x in list[i] then
return i;
fi;
od;
end;
if (not IsPermGroup(G)) then
ErrorNoReturn("the argument must be a permutation group!\n");
fi;
n := LargestMovedPoint(G);
orb := List(Orbits(G, [1..n]));
SortBy(orb, x->-Length(x));
output := [];
if (YoungGroupFromPartition(orb) <> G) then
output := [G];
fi;
partition := List(orb, Length);
mapping := List([1..n], x -> FindPos(orb, x));
ladder := [[List(partition), List(mapping)]];
while (Length(partition) <> 1 or partition[1] < n) do
if (Length(partition) = 1 and partition[1] < n) then
mapping[Position(mapping, 0)] := 1;
partition[1] := partition[1] + 1;
Add(ladder, [List(partition), List(mapping)]);
else
if (partition[2] = 1) then
Remove(partition, 2);
for i in [1..n] do
if (mapping[i]) > 1 then
mapping[i] := mapping[i] - 1;
fi;
od;
partition[1] := partition[1] + 1;
Add(ladder, [List(partition), List(mapping)]);
else
mapping[Position(mapping, 2)] := Length(partition)+1;
partition[2] := partition[2] - 1;
Add(partition, 1);
Add(ladder, [List(partition), List(mapping)]);
mapping[Position(mapping, Length(partition))] := 1;
Remove(partition);
partition[1] := partition[1] + 1;
Add(ladder, [List(partition), List(mapping)]);
fi;
fi;
od;
for pair in ladder do
k := Length(pair[1]);
mapping := pair[2];
Add(output, YoungGroupFromPartition(List([1..k], i -> Filtered([1..n], x -> i = mapping[x]))));
od;
return output;
end);
# vim: set noet ts=4:
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