Symmetry of Toric Initial Ideal on H_4

### ### This program finds all symmetry classes of the 1002 ### initial ideals for the toric ideal on H_4. For each ### symmetry class, a representative is output, along ### with the number of ideals in its entire G-orbit. ### monomial_order = 'deglex' R.<x1,x2,x3,x4,y1,y2,y3,y4,z1,z2,z3,z4> = PolynomialRing(QQ, 12, ['x1','x2','x3','x4','y1','y2','y3','y4','z1','z2','z3','z4'], order=monomial_order) x = [x1, x2, x3, x4] y = [y1, y2, y3, y4] z = [z1, z2, z3, z4] R_gens = x+y+z ### ### Set up the permutation group. ### #x1=1; x2=2; x3=3; x4=4; y1=5; y2=6; y3=7; y4=8; z1=9; z2=10; z3=11; z4=12 #permute the copies of P^2 perm_coords = ['(1,2)(5,6)(9,10)', '(1,3)(5,7)(9,11)', '(1,4)(5,8)(9,12)',\ '(2,3)(6,7)(10,11)', '(2,4)(6,8)(10,12)', '(3,4)(7,8)(11,12)'] perm_p1 = ['(1,5)', '(1,9)', '(5,9)'] #permute the first copy of P^2 perm_p2 = ['(2,6)', '(2,10)', '(6,10)'] #permute the second copy of P^2 perm_p3 = ['(3,7)', '(3,11)', '(7,11)'] #permute the third copy of P^2 perm_p4 = ['(4,8)', '(4,12)', '(8,12)'] #permute the fourth copy of P^2 G = PermutationGroup(perm_coords + perm_p1 + perm_p2 + perm_p3 + perm_p4) #INPUT : g = an element of the permutation group G # m = a monomial in R #OUTPUT : the monomial returned when g acts on m def act(g,m): permuted = [R_gens[g(i+1)-1] for i in range(len(R_gens))] n = 1 for var in m.variables(): n = n*(permuted[R_gens.index(var)]^(m.degree(var))) return n ### ### Read in the 1002 initial ideals ### #Each element of mon_ideals is a frozenset which contains the #generators for the different monomial ideals. We use the frozenset #data type because it speeds up the computation. mon_ideals = set() filename = DATA + 'toric_4_initials.txt' infile = open(filename, 'r') str_data = infile.readline() while str_data != '' and str_data != '\n': data = str_data.split(',') #remove the beginning and ending brackets data[0] = data[0][1:] data[-1] = data[-1][:len(data[-1])-2] I = set() for m in data: I.add(R(m)) I = frozenset(I) mon_ideals.add(I) str_data = infile.readline() print 'Finished populating mon_ideals.' print 'There are', len(mon_ideals), 'distinct initial ideals in the input file.\n' 

Finished populating mon_ideals.
There are 1002 distinct initial ideals in the input file.

Finished populating mon_ideals.
There are 1002 distinct initial ideals in the input file.

### ### Calculate the orbit representatives ### #reps = one representative from each G-orbit reps = [] while len(mon_ideals) != 0: I = mon_ideals.pop() reps.append(I) print 'Calculating the orbit of ideal', list(I) I_list = set() #Cycle through the orbit of I under G for g in G: permuted = frozenset([act(g,m) for m in I]) I_list.add(permuted) #Remove all the elements in mon_ideals #which are in the orbit of I if permuted in mon_ideals: mon_ideals.remove(permuted) print 'There are', len(I_list), 'elements in this orbit.\n' print 'There are', len(reps), 'monomial ideals up to symmetry.' print 'Representatives are:' for r in reps: print list(r), '\n' 

Calculating the orbit of ideal
Traceback (click to the left of this block for traceback)
...
AttributeError: 'frozenset' object has no attribute 'gens'

Calculating the orbit of ideal
Traceback (most recent call last):    
  File "", line 1, in <module>
    
  File "/tmp/tmpp0msby/___code___.py", line 12, in <module>
    print 'Calculating the orbit of ideal', I.gens()
AttributeError: 'frozenset' object has no attribute 'gens'