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#############################################################################
##
#M Order( <e> ) . . . . . . . . . . . . . . . . . for wreath product element
##
InstallMethod( Order, "wreath cycle wreath elements", true, [IsWreathCycle], 1,
function(x)
local info;
info := FamilyObj(x)!.info;
return Order(Yade(x)) * Order(WPE_TopComponent(x));
end);
InstallMethod( Order, "generic wreath elements", true, [IsWreathProductElement], 0,
function(x)
local info, decomposition;
info := FamilyObj(x)!.info;
decomposition := WreathCycleDecomposition(x);
if IsEmpty(decomposition) then
return 1;
fi;
return Lcm(List(decomposition, Order));
end);
#############################################################################
##
#M ConjugacyClasses( <G> ) . . . . . . . . . . . . . . . for wreath product
##
InstallMethod( ConjugacyClasses, "for wreath product", true,
[HasWreathProductInfo], OVERRIDENICE + 4200,
function(G)
local iso;
# TODO: remove this hack
# Ugly Hack: Multiplications in Generic Wreath Product
# are faster than in Matrix Wreath Product
if IsMatrixGroup(G) then
iso := IsomorphismWreathProduct(G);
return List(WPE_ConjugacyClasses(Image(iso)), C -> ConjugacyClass(G, PreImage(iso, Representative(C))));
else
return WPE_ConjugacyClasses(G);
fi;
end);
#############################################################################
##
#M RepresentativeAction( <G> [,<Omega>], <d>, <e> [,<gens>,<acts>] [,<act>] ) . . . . . . . . . . . . for wreath product
##
InstallOtherMethod( RepresentativeActionOp, "for wreath product", true,
[HasWreathProductInfo, IsObject, IsObject, IsFunction], OVERRIDENICE + 4200,
function(G, g, h, act)
local x, y;
if act <> OnPoints then
TryNextMethod();
fi;
# Check if g and h are elements in the same representation as the group G
if IsPermGroup(G) then
if not (IsPerm(g) and IsPerm(h)) then
TryNextMethod();
fi;
elif IsMatrixGroup(G) then
if not (IsMatrix(g) and IsMatrix(h)) then
TryNextMethod();
fi;
else
if not (IsWreathProductElement(g) and IsWreathProductElement(h) and
FamilyObj(g) = FamilyObj(h) and FamilyObj(One(G)) = FamilyObj(g)) then
TryNextMethod();
fi;
fi;
# Check if elements are living inside the wreath product
x := ListWreathProductElement(G, g);
if x = fail then
TryNextMethod();
fi;
y := ListWreathProductElement(G, h);
if y = fail then
TryNextMethod();
fi;
return WPE_RepresentativeAction(G, x, y);
end);
InstallOtherMethod( RepresentativeActionOp, "for wreath product", true,
[HasWreathProductInfo, IsObject, IsObject], OVERRIDENICE + 4200,
function(G, g, h)
return RepresentativeActionOp(G, g, h, OnPoints);
end);
#############################################################################
##
#M Centralizer( <G>, <e> ) . . . . . . . . . . . . . . . for wreath products
##
InstallMethod( CentralizerOp, "for wreath product", IsCollsElms,
[ HasWreathProductInfo, IsObject ], OVERRIDENICE + 4200,
function( G, g )
local x;
# Check if g is an element in the same representation as the group G
if IsPermGroup(G) then
if not (IsPerm(g)) then
TryNextMethod();
fi;
elif IsMatrixGroup(G) then
if not (IsMatrix(g)) then
TryNextMethod();
fi;
else
if not (IsWreathProductElement(g) and FamilyObj(One(G)) = FamilyObj(g)) then
TryNextMethod();
fi;
fi;
x := ListWreathProductElement(G, g);
if x = fail then
TryNextMethod();
fi;
return WPE_Centraliser(G, x);
end );
#############################################################################
##
#M CycleIndex( <G>, <dom>, <act> ) . . . . . . . . . . . . . . . for wreath products
##
InstallMethod(CycleIndexOp, "wreath product", true,
[HasWreathProductInfo, IsListOrCollection, IsFunction], 4200,
function( G, dom, act )
local info, comp, K, H;
if dom <> MovedPoints(G) then
TryNextMethod();
fi;
if act <> OnPoints then
TryNextMethod();
fi;
info := WreathProductInfo(G);
comp := ComponentsOfWreathProduct(G);
K := comp[1];
H := comp[2];
if IsBound(info.permimpr) and info.permimpr then
return CycleIndexWreathProductImprimitiveAction(K, H);
elif IsBound(info.productType) and info.productType then
return CycleIndexWreathProductProductAction(K, H);
else
TryNextMethod();
fi;
end);
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