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# Returns list of first n elems from list
Take@ := function(list, n)
return List([1..n], i -> list[i]);
end;
# Returns list of all but first n elems from list
Drop@ := function(list, n)
return List([n+1..Length(list)], i -> list[i]);
end;
# Returns a list consisting of n copies of elem
Replicate@ := function(elem, n)
if IsList(elem) then
return List([1..n], i -> ShallowCopy(elem));
else
return List([1..n], i -> elem);
fi;
end;
# Takes a list of blocks (possibly different sizes) and constructs a
# block diagonal matrix with those blocks.
BlockDiagonalMatrix@ := function(blocks)
local combine_blocks, result, block;
# Combines two blocks into a block diagonal matrix
combine_blocks := function(b1, b2)
local len1, len2, new_b1, new_b2;
len1 := Length(b1);
len2 := Length(b2);
# Add len2 zeroes to the end of each row in b1
new_b1 := List(b1, row -> Concatenation(row, Replicate@(0, len2)));
# Add len1 zeroes to the start of each row in b2
new_b2 := List(b2, row -> Concatenation(Replicate@(0, len1), row));
return Concatenation(new_b1, new_b2);
end;
result := [];
for block in blocks do
result := combine_blocks(result, block);
od;
return result;
end;
# Takes 2 lists of reps of G and H and gives a full list of reps of
# GxH you can get from tensor products of them
InstallGlobalFunction( TensorProductRepLists, function(reps1, reps2)
local G, H, GxH, gens, imgs, pi1, pi2, result, rep1, rep2, new_rep;
G := Source(reps1[1]);
H := Source(reps2[1]);
GxH := DirectProduct(G, H);
gens := GeneratorsOfGroup(GxH);
pi1 := Projection(GxH, 1);
pi2 := Projection(GxH, 2);
result := [];
for rep1 in reps1 do
for rep2 in reps2 do
imgs := List(gens, gxh -> KroneckerProduct(Image(rep1, Image(pi1, gxh)),
Image(rep2, Image(pi2, gxh))));
new_rep := GroupHomomorphismByImagesNC(GxH, Group(imgs), gens, imgs);
Add(result, new_rep);
od;
od;
return result;
end );
# Returns a homomorphism g s.t. g(x) = func(hom(x))
InstallGlobalFunction( ComposeHomFunction, function(hom, func)
local G, gens, imgs, H;
G := Source(hom);
gens := GeneratorsOfGroup(G);
imgs := List(gens, gen -> func(Image(hom, gen)));
H := Group(imgs);
return GroupHomomorphismByImagesNC(G, H, gens, imgs);
end );
# Returns homomorphism from G to f(G)
FuncToHom@ := function(G, f)
return ComposeHomFunction(IdentityMapping(G), f);
end;
# Returns a block diagonal representation isomorphic to the direct sum
# of the list of reps
InstallGlobalFunction( DirectSumOfRepresentations, function(reps)
local G, gens, imgs, H;
G := Source(reps[1]);
return FuncToHom@(G, g -> BlockDiagonalMatrix@(List(reps, rep -> Image(rep, g))));
end );
# Takes the inner product of two characters, given as rows of the
# character table
InnerProductOfCharacters@ := function(chi1, chi2, G)
local classes;
# We avoid summing over the whole group
classes := ConjugacyClasses(G);
return (1/Size(G)) * Sum(List([1..Size(classes)],
i -> Size(classes[i]) * chi1[i] * ComplexConjugate(chi2[i])));
end;
# Gives the row of the character table corresponding to irrep
CharacterOfRepresentation@ := function(irrep)
local G;
G := Source(irrep);
return List(ConjugacyClasses(G),
class -> Trace(Image(irrep, Representative(class))));
end;
# Irr(G), but guaranteed to be ordered the same as
# IrreducibleRepresentationsDixon (or the list of irreps given)
IrrWithCorrectOrdering@ := function(G)
local irreps;
irreps := ValueOption("irreps");
if irreps = fail then
irreps := IrreducibleRepresentationsDixon(G);
fi;
return List(irreps, irrep -> CharacterOfRepresentation@(irrep));
end;
# Writes the character of rho as a vector in the basis given by the
# irreducible characters (if chars are not given, use Dixon's method)
IrrVectorOfRepresentation@ := function(rho, irr_chars)
local G, classes, char_rho, char_rho_basis;
G := Source(rho);
# Otherwise, we just compute using characters
classes := ConjugacyClasses(G);
char_rho := List(classes, class -> Trace(Image(rho, Representative(class))));
# Write char_rho in the irr_chars basis for class functions
char_rho_basis := List(irr_chars,
irr_char -> InnerProductOfCharacters@(char_rho, irr_char, G));
return char_rho_basis;
end;
InstallGlobalFunction( DegreeOfRepresentation, function(rep)
if IsPermGroup(Range(rep)) then
return LargestMovedPoint(Range(rep));
else
return Trace(Image(rep, One(Source(rep))));
fi;
end );
# Restricts a matrix to a given space. Resulting rep is given in the
# given basis. This only makes sense if V = span(basis) is
# G-invariant, otherwise results will be nonsense.
RestrictRep@ := function(rho, basis)
local G, restricted_rho;
G := Source(rho);
restricted_rho := function(g)
local imgs;
# where g sends the basis vectors (row vects)
imgs := List(basis, v -> Coefficients(basis, Image(rho, g) * v));
# transpose to get column vectors
return TransposedMat(imgs);
end;
return FuncToHom@(G, restricted_rho);
end;
# Calculates orbital matrix for G given representative
OrbitalMatrix@ := function(G, representative)
local n, orbit, orbital, pair;
# representative is assumed to be a pair [i, j] where i, j <= n
if not IsPermGroup(G) then
Error("G is not a permutation group!");
fi;
n := LargestMovedPoint(G);
# We apply all elements of G to [i,j] and mark all reached
# points with a 1 in the n x n zero matrix.
# The action of G on pairs is just g(i, j) = (gi, gj)
# i.e. OnTuples
orbit := Orbit(G, representative, OnTuples);
orbital := NullMat(n, n);
for pair in orbit do
orbital[pair[1]][pair[2]] := 1;
od;
return orbital;
end;
InstallGlobalFunction( PermToLinearRep, function(rho)
local n;
if not IsPermGroup(Range(rho)) then
return fail;
fi;
n := DegreeOfRepresentation(rho);
return ComposeHomFunction(rho, perm -> PermutationMat(perm, n));
end );
# Check a subspace is really G-invariant (action given by rho)
IsGInvariant@ := function(rho, in_space)
local G, v, g, space;
space := in_space;
G := Source(rho);
if not IsVectorSpace(space) then
space := VectorSpace(Cyclotomics, space);
fi;
for v in Basis(space) do
for g in GeneratorsOfGroup(G) do
if not Image(rho, g) * v in space then
return false;
fi;
od;
od;
return true;
end;
ConjugateTranspose@ := function(mat)
return TransposedMat(List(mat, row -> List(row, ComplexConjugate)));
end;
# This is the matrix inner product used throughout the package
InnerProduct@ := function(A, B)
return Trace(A*ConjugateTranspose@(B));
end;
# v is a list of matrices that are the basis of a space
OrthonormalBasis@ := function(v)
local prod, proj, N, u, e, k;
prod := InnerProduct@;
proj := function(u, v)
return (prod(u, v) / prod(u, u)) * u;
end;
N := Length(v);
# these lists will be replaced so e[k] is the kth orthonormal basis vector
u := [1..N];
e := [1..N];
for k in [1..N] do
u[k] := v[k] - Sum([1..k-1], j -> proj(u[j], v[k]));
e[k] := (1/Sqrt(prod(u[k], u[k]))) * u[k];
od;
return e;
end;
InstallGlobalFunction( IsOrthonormalSet, function(S, prod)
return ForAll(S, v1 -> ForAll(S, function(v2)
if v1 = v2 then
return prod(v1, v2) = 1;
else
return prod(v1, v2) = 0;
fi;
end ));
end );
KroneckerList@ := function(reps)
local prod, G;
prod := function(mats)
local current, mat;
current := [[1]];
for mat in mats do
current := KroneckerProduct(current, mat);
od;
return current;
end;
G := Source(reps[1]);
return FuncToHom@(G, g -> prod(List(reps, rep -> Image(rep, g))));
end;
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