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##
## The main setup function for the algorithm <MajoranaRepresentation>
##
InstallGlobalFunction( MAJORANA_SetUp,
function( input, index, options )
local rep, s, t, i, dim, ev, orbs;
if not IsBound(options.axioms) then options.axioms := "AllAxioms"; fi;
if not IsBound(options.form) then options.form := true; fi;
rep := rec( group := input.group,
involutions := input.involutions,
eigenvalues := [0, 1/4, 1/32], # The eigenvalues not equal to one
generators := StructuralCopy(input.generators),
shape := input.shapes[index],
axioms := options.axioms );
t := Size(rep.involutions);
rep.setup := rec( coords := [1..t],
coordmap := HashMap( t*t ),
pairorbit := StructuralCopy(input.pairorbit),
pairconj := StructuralCopy(input.pairconj),
pairconjelts := StructuralCopy(input.pairconjelts),
pairreps := ShallowCopy(input.pairreps) );
for i in [1..t] do
rep.setup.coordmap[i] := i;
rep.setup.coordmap[rep.involutions[i]] := i;
od;
## Orbits on axes for eigenvectors
orbs := MAJORANA_Orbits(input.generators, rep.setup);
rep.setup.conjelts := orbs.conjelts;
rep.setup.orbitreps := orbs.orbitreps;
## Set up products and eigenvectors
s := Size(rep.setup.pairreps);
rep.algebraproducts := List([1..s], x -> false);
rep.evecs := [];
if options.form = true then
rep.innerproducts := List([1..s], x -> false);
fi;
for i in rep.setup.orbitreps do
rep.evecs[i] := rec( );
for ev in rep.eigenvalues do
rep.evecs[i].(String(ev)) := SparseMatrix(0, t, [], [], Rationals);
od;
od;
## Embed dihedral algebras
for i in Positions(rep.shape, "4B") do
MAJORANA_EmbedDihedralAlgebra( i, rep, MAJORANA_DihedralAlgebras("4B") );
od;
for i in Positions(rep.shape, "6A") do
MAJORANA_EmbedDihedralAlgebra( i, rep, MAJORANA_DihedralAlgebras("6A") );
od;
for i in PositionsProperty(rep.shape, x -> not x in [ "1A", "4B", "6A" ]) do
MAJORANA_EmbedDihedralAlgebra( i, rep, MAJORANA_DihedralAlgebras(rep.shape[i]) );
od;
## Finish off setup
dim := Size(rep.setup.coords);
rep.setup.nullspace := rec( heads := [1 .. dim]*0,
vectors := SparseMatrix( 0, dim, [], [], Rationals) );
MAJORANA_AddConjugateEvecs(rep);
for i in rep.generators do MAJORANA_ExtendPerm( i, rep); od;
MAJORANA_Orbitals( rep.generators, t, rep.setup);
for i in [1..Size(rep.setup.pairreps)] do
if not IsBound(rep.algebraproducts[i]) then
rep.algebraproducts[i] := false;
if IsBound(rep.innerproducts) then
rep.innerproducts[i] := false;
fi;
elif rep.algebraproducts[i] <> false then
rep.algebraproducts[i]!.ncols := dim;
fi;
od;
return rep;
end );
##
## Given the dihedral algebra generated by the axes in <rep.setup.pairreps[i]>,
## adds any new 2A, 3A, 4A or 5A axes to <rep.setup.coords> and <rep.setup.coordmap>
## and adds any new products and eigenvectors coming from the dihedral algebra.
##
InstallGlobalFunction( MAJORANA_EmbedDihedralAlgebra,
function( i, rep, subrep )
local dim, t, x, elts, emb, j, im, orbit, y, k, sign;
dim := Size(rep.setup.coords);
t := Size(rep.involutions);
x := rep.setup.pairreps[i];
## Add new basis vector(s) and their orbit(s) and extend pairconj and pairorbit matrices
MAJORANA_AddNewVectors( rep, subrep, x);
## Find the embedding of the subrep into the main algebra
emb := MAJORANA_FindEmbedding( rep, subrep, x);
## Add any new orbits
for j in [1 .. Size(subrep.setup.pairreps)] do
im := emb{subrep.setup.pairreps[j]};
if im[1] < 0 then im[1] := -im[1]; fi;
if im[2] < 0 then im[2] := -im[2]; fi;
orbit := rep.setup.pairorbit[im[1], im[2]];
## If need be, add a new orbit
if orbit = 0 then
MAJORANA_NewOrbital(im, rep.generators, rep.setup);
fi;
od;
## Embed products and evecs
MAJORANA_Embed( rep, subrep, emb );
end );
##
## Finds the embedding of a dihedral algebra <subrep> into <rep>
##
InstallGlobalFunction( MAJORANA_FindEmbedding,
function( rep, subrep, inv)
local imgs, emb, pos, x, gens;
## Find the images of the embedding of the subrep into the main rep
gens := GeneratorsOfGroup(subrep.group);
if IsBound(subrep.setup.orbits) then
imgs := List(subrep.setup.coords, w -> MAJORANA_TauMappedWord(rep, subrep, w, gens, inv) );
else
imgs := List(subrep.setup.coords, w -> MAJORANA_MappedWord(rep, subrep, w, gens, inv) );
fi;
emb := [];
for x in imgs do
pos := rep.setup.coordmap[x];
if pos = fail then
pos := rep.setup.coordmap[ Product( rep.involutions{x} )];
fi;
Add( emb, pos);
od;
return emb;
end );
##
## If new vectors have been added to <setup.coords> then extend the action
## of an existing perm to these new vectors.
##
InstallGlobalFunction( MAJORANA_ExtendPerm,
function(perm, rep)
local dim, new_dim, i, im, sign, pos;
new_dim := Size(rep.setup.coords);
dim := Size(perm);
for i in [dim + 1 .. new_dim] do
im := perm{rep.setup.coords[i]};
## Keep track of sign changes
sign := 1;
if im[1] < 0 then im[1] := -im[1]; sign := -sign; fi;
if im[2] < 0 then im[2] := -im[2]; sign := -sign; fi;
if im[1] > im[2] then
im := im{[2,1]};
fi;
## Find the new vector in <setup.coords>
pos := rep.setup.coordmap[im];
if pos = fail then
pos := rep.setup.coordmap[ Product( rep.involutions{im} ) ];
fi;
Add(perm, sign*pos);
od;
end);
##
## Adds any additional 2A, 3A, 4A or 5A vectors coming from the dihedral
## algebra <subrep>.
##
InstallGlobalFunction( MAJORANA_AddNewVectors,
function(rep, subrep, inv)
local i, list, list_5A, new, new_5A, x, vec, g, im, k, dim, gens;
dim := Size(rep.setup.coords);
gens := GeneratorsOfGroup(subrep.group);
for i in [Size(subrep.involutions) + 1 .. Size(subrep.setup.coords)] do
## Find the new vectors to be added to <setup.coordmap>
## TODO - do we want to change this to hashmaps?
list := Positions(subrep.setup.poslist, i);
list_5A := Positions(subrep.setup.poslist, -i);
new := []; new_5A := [];
for x in subrep.setup.longcoords{list} do
if IsBound(subrep.setup.orbits) then
Add( new, MAJORANA_TauMappedWord(rep, subrep, x, gens, inv));
else
Add( new, MAJORANA_MappedWord(rep, subrep, x, gens, inv));
fi;
od;
for x in subrep.setup.longcoords{list_5A} do
if IsBound(subrep.setup.orbits) then
Add( new_5A, MAJORANA_TauMappedWord(rep, subrep, x, gens, inv));
else
Add( new_5A, MAJORANA_MappedWord(rep, subrep, x, gens, inv));
fi;
od;
MAJORANA_AddConjugateVectors( rep, new, new_5A );
od;
for x in rep.setup.pairorbit do
Append(x, [dim + 1 .. Size(rep.setup.coords)]*0 );
od;
for x in rep.setup.pairconj do
Append(x, [dim + 1 .. Size(rep.setup.coords)]*0 );
od;
Append(rep.setup.pairorbit, NullMat( Size(rep.setup.coords) - dim , Size(rep.setup.coords) ));
Append(rep.setup.pairconj, NullMat( Size(rep.setup.coords) - dim , Size(rep.setup.coords) ));
for g in rep.setup.pairconjelts do MAJORANA_ExtendPerm( g, rep); od;
for g in rep.generators do MAJORANA_ExtendPerm( g, rep); od;
for g in rep.setup.conjelts do MAJORANA_ExtendPerm( g, rep); od;
end );
##
## When adding a new set of vectors to <rep.setup.coordmap>, also adds
## all of their images under the group action.
##
InstallGlobalFunction( MAJORANA_AddConjugateVectors,
function( rep, new, new_5A )
local vec, g, im, im_5A, k, elts, x;
## Find which (if any) new vectors are not yet in <setup.coordmap>
vec := First(new, x -> x in rep.setup.coordmap);
if vec <> fail then
new := Filtered(new, x -> not x in rep.setup.coordmap);
new_5A := Filtered(new_5A, x -> not x in rep.setup.coordmap);
fi;
if vec = fail and rep.axioms <> "NoAxioms" then
vec := First(new, x -> Product( rep.involutions{x} ) in rep.setup.coordmap );
if vec <> fail then
new := Filtered(new, x -> not Product( rep.involutions{x} ) in rep.setup.coordmap);
new_5A := Filtered(new_5A, x -> not Product( rep.involutions{x} ) in rep.setup.coordmap);
fi;
fi;
if new = [] then return; fi;
## If vec = fail then all vectors are new and a new rep will be added
## to <setup.coords>. Otherwise, there are some new vectors but these
## are equal to an existing element of <setup.coords>
for g in rep.setup.pairconjelts do
im := List(new, x -> SortedList( g{ x } ));
im := Filtered( im, x -> not x in rep.setup.coordmap );
if rep.axioms = "AllAxioms" then
im := Filtered( im, x -> not Product( rep.involutions{x} ) in rep.setup.coordmap);
fi;
if im <> [] then
im_5A := List(new_5A, x -> SortedList( g{ x } ));
## If need be, add a new vector to coords, otherwise find index of existing vector
if vec = fail then
Add( rep.setup.coords, im[1] );
k := Size(rep.setup.coords);
else
k := rep.setup.coordmap[ SortedList( g{ vec } )];
if k = fail then
k := rep.setup.coordmap[ Product(rep.involutions{ g{ vec } }) ];
fi;
fi;
## Add new vectors to <setup.coordmap>
for x in im do rep.setup.coordmap[ x ] := k; od;
for x in im_5A do rep.setup.coordmap[ x ] := -k; od;
if rep.axioms = "AllAxioms" then
for x in im do
rep.setup.coordmap[ Product( rep.involutions{ x } )] := k;
od;
for x in im_5A do
rep.setup.coordmap[ Product( rep.involutions{ x } )] := -k;
od;
fi;
fi;
od;
end );
##
## Use the group action to find more eigenvectors
##
InstallGlobalFunction( MAJORANA_AddConjugateEvecs,
function(rep)
local i, new, ev, v, g, im;
for i in rep.setup.orbitreps do
# Setup a record to record the new evecs
new := rec();
# Loop over eigenvalues
for ev in RecNames(rep.evecs[i]) do
new.(ev) := SparseMatrix(0, Size(rep.setup.coords), [], [], Rationals);
for v in Iterator(rep.evecs[i].(ev)) do
for g in Filtered(rep.setup.pairconjelts, h -> h[i] = i) do
# Find the image of each eigenvector under g
im := MAJORANA_ConjugateVec(v, g);
# Add the image to the new eigenspaces
if not _IsRowOfSparseMatrix(new.(ev), im) then
new.(ev) := MAJORANA_UnionOfRows(new.(ev), im);
fi;
od;
od;
# Find a basis of the new eigenspaces
rep.evecs[i].(ev) := ReversedEchelonMatDestructive(new.(ev)).vectors;
od;
od;
end );
##
## <MappedWord> for indices or pairs of indices referring to elements of coords
##
InstallGlobalFunction(MAJORANA_MappedWord,
function(rep, subrep, w, gens, inv)
local im, imgs;
imgs := rep.involutions{inv};
if IsRowVector(w) then
im := List(w, i -> MappedWord(subrep.setup.coords[i], gens, imgs));
return SortedList(List(im, x -> Position(rep.involutions, x )));
else
return Position(rep.involutions, MappedWord(w, gens, imgs) );
fi;
end );
##
## This is used only the the main algorithm to record algebra products once found
##
InstallGlobalFunction(SP_Inverse,
function(perm)
local l, inv, i;
if perm = [] then return []; fi;
l := Length(perm);
inv := [1..l];
for i in [1..l] do
if perm[i] > 0 then
inv[perm[i]] := i;
else
inv[-perm[i]] := -i;
fi;
od;
return inv;
end);
##
## Not currently in use, here for reference
##
InstallGlobalFunction(SP_Product,
function( perm1, perm2) # Perms must be of same length!
local prod, i;
prod := [];
for i in perm1 do
if i > 0 then
Add(prod, perm2[i]);
else
Add(prod, -perm2[-i]);
fi;
od;
return prod;
end );
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