# This file generates expressions for the positive_roots in the parameters d1,...,d6 starting from the positive roots of E6 constructed in Sage.

# IMPORTANT: We load a saved version of RootSystem('E6') since the ordering of the roots changes with the version of sage, and we use a fixed order in our calculations. 

# A list of variables d_1, ..., d_6 for the coefficients
ds = vector([var("d%s"%i) for i in range(1,7)])

RootSystemE6 = load("L6.sobj")

L6 = RootSystemE6.root_lattice()

alpha = {i:L6.alpha()[i] for i in range(1,7)}

positive_roots = load("posRootsE6.sobj")

# # We save the ordered list as a plain text file (one root per line):
# f = file("posRootsE6.txt", 'w')
# f.write('positive_roots = [' + '\n')
# for i in range(0, len(positive_roots)-1):
#     f.write(str(positive_roots[i]) + ',\n')
# f.write(str(positive_roots[-1]) + ']')
# f.close()

# # We record the ordering of the positive roots:
# positive_roots = [alpha[1],
#                   alpha[2],
#                   alpha[3],
#                   alpha[4],
#                   alpha[5],
#                   alpha[6],
#                   alpha[3] + alpha[4],
#                   alpha[2] + alpha[4],
#                   alpha[5] + alpha[6],
#                   alpha[1] + alpha[3],
#                   alpha[4] + alpha[5],
#                   alpha[3] + alpha[4] + alpha[5],
#                   alpha[1] + alpha[3] + alpha[4],
#                   alpha[2] + alpha[4] + alpha[5],
#                   alpha[4] + alpha[5] + alpha[6],
#                   alpha[2] + alpha[3] + alpha[4],
#                   alpha[1] + alpha[2] + alpha[3] + alpha[4],
#                   alpha[2] + alpha[4] + alpha[5] + alpha[6],
#                   alpha[3] + alpha[4] + alpha[5] + alpha[6],
#                   alpha[1] + alpha[3] + alpha[4] + alpha[5],
#                   alpha[2] + alpha[3] + alpha[4] + alpha[5],
#                   alpha[2] + alpha[3] + 2*alpha[4] + alpha[5],
#                   alpha[1] + alpha[2] + alpha[3] + alpha[4] + alpha[5],
#                   alpha[2] + alpha[3] + alpha[4] + alpha[5] + alpha[6],
#                   alpha[1] + alpha[3] + alpha[4] + alpha[5] + alpha[6],
#                   alpha[1] + alpha[2] + alpha[3] + alpha[4] + alpha[5] + alpha[6],
#                   alpha[1] + alpha[2] + alpha[3] + 2*alpha[4] + alpha[5],
#                   alpha[2] + alpha[3] + 2*alpha[4] + alpha[5] + alpha[6],
#                   alpha[2] + alpha[3] + 2*alpha[4] + 2*alpha[5] + alpha[6],
#                   alpha[1] + alpha[2] + 2*alpha[3] + 2*alpha[4] + alpha[5],
#                   alpha[1] + alpha[2] + alpha[3] + 2*alpha[4] + alpha[5] + alpha[6],
#                   alpha[1] + alpha[2] + alpha[3] + 2*alpha[4] + 2*alpha[5] + alpha[6],
#                   alpha[1] + alpha[2] + 2*alpha[3] + 2*alpha[4] + alpha[5] + alpha[6],
#                   alpha[1] + alpha[2] + 2*alpha[3] + 2*alpha[4] + 2*alpha[5] + alpha[6],
#                   alpha[1] + alpha[2] + 2*alpha[3] + 3*alpha[4] + 2*alpha[5] + alpha[6],
#                   alpha[1] + 2*alpha[2] + 2*alpha[3] + 3*alpha[4] + 2*alpha[5] + alpha[6]]



# The following matrix allows to convert these roots to expressions in the ds, following [RSS]

AtoD = matrix([[-1, 1, 0, 0, 0, 0],
               [ 1, 1,-1, 0, 0, 0],
               [ 0, 1, 1,-1, 0, 0],
               [ 0, 0, 0, 1,-1, 0],
               [ 0, 0, 0, 0, 1,-1],
               [ 0, 0, 0, 0, 0, 1]])



roots = [ds.dot_product(AtoD*v.to_vector()) for v in positive_roots]


# For the record, we print the result, and save it as an sobj file:

# [-d1 + d2,
#  d1 + d2 + d3,
#  -d2 + d3,
#  -d3 + d4,
#  -d4 + d5,
#  -d5 + d6,
#  -d2 + d4,
#  d1 + d2 + d4,
#  -d4 + d6,
#  -d1 + d3,
#  -d3 + d5,
#  -d2 + d5,
#  -d1 + d4,
#  d1 + d2 + d5,
#  -d3 + d6,
#  d1 + d3 + d4,
#  d2 + d3 + d4,
#  d1 + d2 + d6,
#  -d2 + d6,
#  -d1 + d5,
#  d1 + d3 + d5,
#  d1 + d4 + d5,
#  d2 + d3 + d5,
#  d1 + d3 + d6,
#  -d1 + d6,
#  d2 + d3 + d6,
#  d2 + d4 + d5,
#  d1 + d4 + d6,
#  d1 + d5 + d6,
#  d3 + d4 + d5,
#  d2 + d4 + d6,
#  d2 + d5 + d6,
#  d3 + d4 + d6,
#  d3 + d5 + d6,
#  d4 + d5 + d6,
#  d1 + d2 + d3 + d4 + d5 + d6]

# # We save the ordered list as a plain text file (one root per line):
# f = file("../Yoshida/Scripts/Input/rootsE6.txt", 'w')
# f.write('positive_roots_in_ds = [' + '\n')
# for i in range(0, len(roots)-1):
#     f.write(str(roots[i]) + ',\n')
# f.write(str(roots[-1]) + ']')
# f.close()


# save(roots, "../Yoshida/Scripts/Input/rootsE6.sobj")