|
InstallGlobalFunction( TensorProductOfMatricesObj, function(Fam, A, B)
# check dimensions match the family dimensions
if DimensionsMat(A) <> Fam!.dimA or DimensionsMat(B) <> Fam!.dimB then
Error("Dimensions of tensor product don't match family!");
fi;
return Objectify(NewType(Fam, IsTensorProductOfMatricesObj and IsTensorProductPairRep),
[A, B]);
end );
InstallGlobalFunction( TensorProductOfMatrices, function(A, B)
local F;
F := TensorProductOfMatricesFamily(DimensionsMat(A), DimensionsMat(B));
return TensorProductOfMatricesObj(F, A, B);
end );
IsDimensions@ := function(x)
return IsList(x) and Length(x) = 2 and ForAll(x, IsPosInt);
end;
InstallGlobalFunction( TensorProductOfMatricesFamily, function(dimA, dimB)
local F, G;
if not IsDimensions@(dimA) or not IsDimensions@(dimB) then
Error("<dimA> and <dimB> must be pairs of positive integers");
fi;
F := NewFamily( Concatenation("TensorProductOfMatrices", String(dimA), "x", String(dimB)),
IsTensorProductOfMatricesObj );
F!.dimA := dimA;
F!.dimB := dimB;
return F;
end );
InstallMethod( \*,
"for tensor products of matrices",
[IsTensorProductOfMatricesObj and IsTensorProductPairRep,
IsTensorProductOfMatricesObj and IsTensorProductPairRep],
function(x, y)
return TensorProductOfMatricesObj(FamilyObj(x),
x![1]*y![1],
x![2]*y![2]);
end );
InstallMethod( \*,
"for applying tensor products to matrices",
[IsTensorProductOfMatricesObj and IsTensorProductPairRep,
IsMatrix],
function(prod, A)
# A pure tensor product x tensor y acts on e_i
# tensor e_j, giving x(e_i) tensor y(e_j). The
# action extends to all matrices by linearity.
#
# Linearity also lets us avoid materialising the
# Kronecker product, but we have to spend O(n^4)
# time looping over indices
local i, j, result, e, n, m, ith_col, jth_col;
n := Length(A);
m := Length(A[1]);
# check the sizes are all correct
if n <> Length(prod![1][1]) or m <> Length(prod![2][1]) then
Error("Tensor product is wrong size to act on <A>!");
fi;
result := NullMat(n, m);
for i in [1..n] do
for j in [1..m] do
ith_col := TransposedMat([TransposedMat(prod![1])[i]]);
jth_col := [TransposedMat(prod![2])[j]];
result := result + A[i][j] * KroneckerProduct(ith_col, jth_col);
od;
od;
return result;
end );
InstallMethod( \=,
"for tensor products of matrices",
[IsTensorProductOfMatricesObj and IsTensorProductPairRep,
IsTensorProductOfMatricesObj and IsTensorProductPairRep],
function(x, y)
return x![1] = y![1] and x![2] = y![2];
end );
InstallMethod( OneOp,
"for tensor products of matrices",
[IsTensorProductOfMatricesObj and IsTensorProductPairRep],
function(x)
local F;
F := FamilyObj(x);
return TensorProductOfMatricesObj(F,
IdentityMat(F!.dimA[1]),
IdentityMat(F!.dimB[1]));
end );
InstallMethod( InverseOp,
"for tensor products of matrices",
[IsTensorProductOfMatricesObj and IsTensorProductPairRep],
function(x)
return TensorProductOfMatricesObj(FamilyObj(x),
x![1]^-1,
x![2]^-1);
end );
# We don't want to have to print the kronecker product matrix, would
# use too much memory.
InstallMethod( PrintObj,
"for tensor products of matrices",
[IsTensorProductOfMatricesObj and IsTensorProductPairRep],
function(x)
Print(x![1], " tensor ", x![2]);
end );
InstallMethod( IsGeneratorsOfMagmaWithInverses,
"for tensor products of matrices",
[CategoryCollections(IsTensorProductOfMatricesObj)],
function(coll)
# just check if all matrices are invertible
return ForAll(coll, function(tensor)
if IsTensorProductPairRep(tensor) then
return IsGeneratorsOfMagmaWithInverses([tensor![1]])
and IsGeneratorsOfMagmaWithInverses([tensor![2]]);
elif IsTensorProductKroneckerRep(tensor) then
return IsGeneratorsOfMagmaWithInverses([tensor![1]]);
else
return fail;
fi;
end );
end );
# takes tensor product of two matrix representations
InstallGlobalFunction( TensorProductOfRepresentations, function(rho, tau)
local F, G, deg_rho, deg_tau;
deg_rho := DegreeOfRepresentation(rho);
deg_tau := DegreeOfRepresentation(tau);
F := TensorProductOfMatricesFamily([deg_rho, deg_rho], [deg_tau, deg_tau]);
if Source(rho) <> Source(tau) then
Error("Not representations of the same group!");
fi;
G := Source(rho);
return FuncToHom@(G, g -> TensorProductOfMatricesObj(F, Image(rho, g), Image(tau, g)));
end );
# this is the old way, sometimes useful if you need to sum
InstallGlobalFunction( KroneckerProductOfRepresentations, function(rho, tau)
return FuncToHom@(Source(rho), g -> KroneckerProduct(Image(rho, g), Image(tau, g)));
end );
# don't have to materialise matrices if you just want the character
InstallGlobalFunction( CharacterOfTensorProductOfRepresentations, function(rho)
return function(g)
local im;
im := Image(rho, g);
return Trace(im![1]) * Trace(im![2]);
end;
end );
[ 0.33Quellennavigators
Projekt
]
|
