def get_unscaled_proj_35_8888_sd_asd(dtype=numpy.int32):
"""Computes the [35, 8, 8, 8, 8]-projector to the (anti)self-dual 4-forms."""
# In principle, we could use the so8.gamma_vvvvss and so8.gamma_vvvvcc here,
# but we opt for instead constructing some explicit orthonormal bases for the
# 35-dimensional irreps as this makes it easier to stay exact despite using
# floating point numbers.
#
# We first need some basis for the 35 self-dual 4-forms.
# Our convention is that we lexicographically list those 8-choose-4
# combinations that contain the index 0.
ret_sd = numpy.zeros([35, 8, 8, 8, 8], dtype=dtype)
ret_asd = numpy.zeros([35, 8, 8, 8, 8], dtype=dtype)
#
def get_complementary(ijkl):
mnpq = tuple(n for n in range(8) if n not in ijkl)
return (algebra.permutation_sign(ijkl + mnpq), ijkl, mnpq)
#
complements = [get_complementary(ijkl)
for ijkl in itertools.combinations(range(8), 4)
if 0 in ijkl]
for num_sd, (sign, ijkl, mnpq) in enumerate(complements):
for abcd in itertools.permutations(range(4)):
sign_abcd = algebra.permutation_sign(abcd)
for r, part2_sign in ((ret_sd, 1), (ret_asd, -1)):
r[num_sd,
ijkl[abcd[0]], ijkl[abcd[1]],
ijkl[abcd[2]], ijkl[abcd[3]]] = sign_abcd
r[num_sd,
mnpq[abcd[0]], mnpq[abcd[1]], mnpq[abcd[2]],
mnpq[abcd[3]]] = sign_abcd * sign * part2_sign
return ret_sd, ret_asd
def get_complementary(ijkl): mnpq = tuple(n for n in range(8) if n not in ijkl) return (algebra.permutation_sign(ijkl + mnpq), ijkl, mnpq)