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##
## Calculates the orbits of the group generated by gens on the set
## setup.coords. Stores the representatives of the orbits in setup.orbitreps
## and the elements that map the representatives to the elements of setup.coords
## in setup.conjelts.
##
## <gens> must be (signed) permutations in list format.
##
InstallGlobalFunction(MAJORANA_Orbits,
function(gens, setup)
local conjelts,
orbitreps,
i, j,
orb,
gen,
elts,
count,
dim,
p,
q,
h,
g;
dim := Size(setup.coords);
# The list <orbitreps> form the representative elements of the orbits
# and the list <conjelts> contains signed permutations (as lists) that map
# the representatives to the elements of setup.coords
conjelts := [1..dim]*0;
orbitreps := [];
# For all elements of rep.setup.coords
for i in [1..dim] do
# If the element is not yet in an orbit
if conjelts[i] = 0 then
# Add a new orbit
Add(orbitreps, i);
conjelts[i] := [1..dim];
# Store the orbit and the group elements that map the
# representative of this orbit to its elements. The integer
# <count> keeps track of where we are in the elts list.
orb := [i];
elts := [[1..dim]];
count := 0;
# For anything already in the orbit
for p in orb do
count := count + 1;
h := elts[count];
# Apply the generators of the group to q
for gen in gens do
q := gen[p];
if q < 0 then q := -q; fi;
# If this element has not already been added to the orbit then
# do so now, and also store the corresponding group elt
if conjelts[q] = 0 then
# Multiply h and gen
g := [];
for j in h do
if j > 0 then
Add(g, gen[j]);
else
Add(g, -gen[-j]);
fi;
od;
Add(orb, q);
Add(elts, g);
conjelts[q] := g;
fi;
od;
od;
fi;
od;
return rec( conjelts := conjelts,
orbitreps := orbitreps );
end );
##
## Calculates the orbitals of the group generated by gens on the set
## setup.coords. Stores the representatives of the orbitals in setup.pairreps
## and the elements that map the representatives to the pairs of elts of setup.coords
## in setup.pairconjelts.
##
## <gens> must be (signed) permutations in list format.
##
## The argument <t> is an integer. If it is greater than 0 then it indicates that
## the orbitals have already been found on the first <t> elements of setup.coords.
##
InstallGlobalFunction(MAJORANA_Orbitals,
function(gens,t,setup)
local dim, i, j;
dim := Size(setup.coords);
# Loop over pairs of elements of setup.coords
for i in [1..dim] do
for j in [Maximum(i,t + 1)..dim] do
# If this pair is not already in an orbit
if setup.pairorbit[i, j] = 0 then
# Find the orbital containing this pair
MAJORANA_NewOrbital([i,j], gens, setup);
fi;
od;
od;
end );
##
## Computes the orbital under the action of the group generated by <gens> that
## contains the pair <pnt>.
##
## <gens> must be (signed) permutations in list format.
##
InstallGlobalFunction( MAJORANA_NewOrbital,
function( pnt, gens, setup)
local orb, elts, count, y, p, h, q, z, g, i, pos, sign, gen;
# Added <pnt> as the orbital representative
Add(setup.pairreps, pnt);
# Store the orbital and the group elements that map the
# representative of this orbital to its elements. The integer
# <count> keeps track of where we are in the elts list.
orb := [ pnt ];
elts := [ [1 .. Size(setup.coords)] ];
count := 0;
# Store the pairorbit and pairconj data for this representative element
y := Size(setup.pairreps);
setup.pairorbit[pnt[1], pnt[2]] := y;
setup.pairorbit[pnt[2], pnt[1]] := y;
setup.pairconj[pnt[1], pnt[2]] := 1;
setup.pairconj[pnt[2], pnt[1]] := 1;
# For anything already in the orbital
for p in orb do
count := count + 1;
h := elts[count];
# Apply the generators of the group to q
for gen in gens do
q := gen{p};
if q[1] < 0 then q[1] := -q[1]; fi;
if q[2] < 0 then q[2] := -q[2]; fi;
z := setup.pairorbit[q[1], q[2]];
# If this element has not already been added to the orbital then
# do so now, and also store the corresponding data
if z = 0 then
# Multiply h and gen
g := [];
for i in h do
if i > 0 then
Add(g, gen[i]);
else
Add(g, -gen[-i]);
fi;
od;
# Add the pair and the group element to the orbital and elt list
Add( orb, q );
Add( elts, g);
# Store the pair orbit data (adjusting for sign)
if Product(g{orb[1]}) < 0 then
sign := -1;
else
sign := 1;
fi;
setup.pairorbit[q[1], q[2]] := sign*y;
setup.pairorbit[q[2], q[1]] := sign*y;
# Store the pair conj data (adding g to pairconjelts if necessary)
pos := Position(setup.pairconjelts, g);
if pos = fail then
Add(setup.pairconjelts, g);
pos := Size(setup.pairconjelts);
fi;
setup.pairconj[q[1], q[2]] := pos;
setup.pairconj[q[2], q[1]] := pos;
fi;
od;
od;
return;
end );
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