def test_inverse(n):
for p in perm.group(n):
ip = perm.inverse(p)
assert perm.compose(p, ip) == perm.identity(n)
assert perm.compose(ip, p) == perm.identity(n)
def reduce_tensor(formula, eps=1e-9, has_parity=None, **kw_Rs):
"""
Usage
Rs, Q = rs.reduce_tensor('ijkl=jikl=ikjl=ijlk', i=[(1, 1)])
Rs = 0,2,4
Q = tensor of shape [15, 81]
"""
dtype = torch.get_default_dtype()
with torch_default_dtype(torch.float64):
# reformat `formulas` and make checks
formulas = [(-1 if f.startswith('-') else 1, f.replace('-', ''))
for f in formula.split('=')]
s0, f0 = formulas[0]
assert s0 == 1
for _s, f in formulas:
if len(set(f)) != len(f) or set(f) != set(f0):
raise RuntimeError(f'{f} is not a permutation of {f0}')
if len(f0) != len(f):
raise RuntimeError(
f'{f0} and {f} don\'t have the same number of indices')
# `formulas` is a list of (sign, permutation of indices)
# each formula can be viewed as a permutation of the original formula
formulas = {(s, tuple(f.index(i) for i in f0))
for s, f in formulas} # set of generators (permutations)
# they can be composed, for instance if you have ijk=jik=ikj
# you also have ijk=jki
# applying all possible compositions creates an entire group
while True:
n = len(formulas)
formulas = formulas.union([(s, perm.inverse(p))
for s, p in formulas])
formulas = formulas.union([(s1 * s2, perm.compose(p1, p2))
for s1, p1 in formulas
for s2, p2 in formulas])
if len(formulas) == n:
break # we break when the set is stable => it is now a group \o/
# lets clean the `kw_Rs` before checking that they are compatible with the formulas
for i in kw_Rs:
if not callable(kw_Rs[i]):
Rs = convention(kw_Rs[i])
if has_parity is None:
has_parity = any(p != 0 for _, _, p in Rs)
if not has_parity and not all(p == 0 for _, _, p in Rs):
raise RuntimeError(
f'{format_Rs(Rs)} parity has to be specified everywhere or nowhere'
)
if has_parity and any(p == 0 for _, _, p in Rs):
raise RuntimeError(
f'{format_Rs(Rs)} parity has to be specified everywhere or nowhere'
)
kw_Rs[i] = Rs
if has_parity is None:
raise RuntimeError(f'please specify the argument `has_parity`')
# here we check that each index has one and only one representation
for _s, p in formulas:
f = "".join(f0[i] for i in p)
for i, j in zip(f0, f):
if i in kw_Rs and j in kw_Rs and kw_Rs[i] != kw_Rs[j]:
raise RuntimeError(
f'Rs of {i} (Rs={format_Rs(kw_Rs[i])}) and {j} (Rs={format_Rs(kw_Rs[j])}) should be the same'
)
if i in kw_Rs:
kw_Rs[j] = kw_Rs[i]
if j in kw_Rs:
kw_Rs[i] = kw_Rs[j]
for i in f0:
if i not in kw_Rs:
raise RuntimeError(f'index {i} has not Rs associated to it')
e = (0, 0, 0, 0) if has_parity else (0, 0, 0)
dims = {
i: len(kw_Rs[i](*e)) if callable(kw_Rs[i]) else dim(kw_Rs[i])
for i in f0
} # dimension of each index
full_base = list(itertools.product(
*(range(dims[i])
for i in f0))) # (0, 0, 0), (0, 0, 1), (0, 0, 2), ... (3, 3, 3)
# len(full_base) degrees of freedom in an unconstrained tensor
# but there is constraints given by the group `formulas`
# For instance if `ij=-ji`, then 00=-00, 01=-01 and so on
base = set()
for x in full_base:
# T[x] is a coefficient of the tensor T and is related to other coefficient T[y]
# if x and y are related by a formula
xs = {(s, tuple(x[i] for i in p)) for s, p in formulas}
# s * T[x] are all equal for all (s, x) in xs
# if T[x] = -T[x] it is then equal to 0 and we lose this degree of freedom
if not (-1, x) in xs:
# the sign is arbitrary, put both possibilities
base.add(
frozenset(
{frozenset(xs),
frozenset({(-s, x)
for s, x in xs})}))
# len(base) is the number of degrees of freedom in the tensor.
# Now we want to decompose these degrees of freedom into irreps
base = sorted([
sorted([sorted(xs) for xs in x]) for x in base
]) # requested for python 3.7 but not for 3.8 (probably a bug in 3.7)
# First we compute the change of basis (projection) between full_base and base
d_sym = len(base)
d = len(full_base)
Q = torch.zeros(d_sym, d)
for i, x in enumerate(base):
x = max(x, key=lambda xs: sum(s for s, x in xs))
for s, e in x:
j = full_base.index(e)
Q[i, j] = s / len(x)**0.5
assert torch.allclose(Q @ Q.T, torch.eye(d_sym))
if d_sym == 0:
return [], torch.zeros(d_sym, d).to(dtype=dtype)
# We project the representation on the basis `base`
def representation(alpha, beta, gamma, parity=None):
def re(r):
if callable(r):
if has_parity:
return r(alpha, beta, gamma, parity)
return r(alpha, beta, gamma)
return rep(r, alpha, beta, gamma, parity)
m = o3.kron(*(re(kw_Rs[i]) for i in f0))
return Q @ m @ Q.T
# And check that after this projection it is still a representation
assert _is_representation(representation, eps, has_parity)
# The rest of the code simply extract the irreps present in this representation
Rs_out = []
A = Q.clone()
for l in range(int((d_sym - 1) // 2) + 1):
for p in [-1, 1] if has_parity else [0]:
if 2 * l + 1 > d_sym - dim(Rs_out):
break
mul, B, representation = o3.reduce(representation,
partial(rep, [(1, l, p)]),
eps, has_parity)
A = o3.direct_sum(torch.eye(d_sym - B.shape[0]), B) @ A
A = _round_sqrt(A, eps)
Rs_out += [(mul, l, p)]
if dim(Rs_out) == d_sym:
break
if dim(Rs_out) != d_sym:
raise RuntimeError(
f'unable to decompose into irreducible representations')
return simplify(Rs_out), A.to(dtype=dtype)
def reduce_tensor(formula, eps=1e-9, has_parity=None, **kw_Rs):
"""
Usage
Rs, Q = rs.reduce_tensor('ijkl=jikl=ikjl=ijlk', i=[(1, 1)])
Rs = 0,2,4
Q = tensor of shape [15, 81]
"""
with torch_default_dtype(torch.float64):
formulas = [(-1 if f.startswith('-') else 1, f.replace('-', ''))
for f in formula.split('=')]
s0, f0 = formulas[0]
assert s0 == 1
for _s, f in formulas:
if len(set(f)) != len(f) or set(f) != set(f0):
raise RuntimeError(f'{f} is not a permutation of {f0}')
if len(f0) != len(f):
raise RuntimeError(
f'{f0} and {f} don\'t have the same number of indices')
formulas = {(s, tuple(f.index(i) for i in f0))
for s, f in formulas} # set of generators (permutations)
# create the entire group
while True:
n = len(formulas)
formulas = formulas.union([(s, perm.inverse(p))
for s, p in formulas])
formulas = formulas.union([(s1 * s2, perm.compose(p1, p2))
for s1, p1 in formulas
for s2, p2 in formulas])
if len(formulas) == n:
break
for i in kw_Rs:
if not callable(kw_Rs[i]):
Rs = convention(kw_Rs[i])
if has_parity is None:
has_parity = any(p != 0 for _, _, p in Rs)
if not has_parity and not all(p == 0 for _, _, p in Rs):
raise RuntimeError(
f'{format_Rs(Rs)} parity has to be specified everywhere or nowhere'
)
if has_parity and any(p == 0 for _, _, p in Rs):
raise RuntimeError(
f'{format_Rs(Rs)} parity has to be specified everywhere or nowhere'
)
kw_Rs[i] = Rs
if has_parity is None:
raise RuntimeError(f'please specify the argument `has_parity`')
for _s, p in formulas:
f = "".join(f0[i] for i in p)
for i, j in zip(f0, f):
if i in kw_Rs and j in kw_Rs and kw_Rs[i] != kw_Rs[j]:
raise RuntimeError(
f'Rs of {i} (Rs={format_Rs(kw_Rs[i])}) and {j} (Rs={format_Rs(kw_Rs[j])}) should be the same'
)
if i in kw_Rs:
kw_Rs[j] = kw_Rs[i]
if j in kw_Rs:
kw_Rs[i] = kw_Rs[j]
for i in f0:
if i not in kw_Rs:
raise RuntimeError(f'index {i} has not Rs associated to it')
e = (0, 0, 0, 0) if has_parity else (0, 0, 0)
full_base = list(
itertools.product(*(range(
len(kw_Rs[i](*e)) if callable(kw_Rs[i]) else dim(kw_Rs[i]))
for i in f0)))
base = set()
for x in full_base:
xs = {(s, tuple(x[i] for i in p)) for s, p in formulas}
# s * T[x] all equal for (s, x) in xs
if not (-1, x) in xs:
# the sign is arbitrary, put both possibilities
base.add(
frozenset(
{frozenset(xs),
frozenset({(-s, x)
for s, x in xs})}))
base = sorted([
sorted([sorted(xs) for xs in x]) for x in base
]) # requested for python 3.7 but not for 3.8 (probably a bug in 3.7)
d_sym = len(base)
d = len(full_base)
Q = torch.zeros(d_sym, d)
for i, x in enumerate(base):
x = max(x, key=lambda xs: sum(s for s, x in xs))
for s, e in x:
j = full_base.index(e)
Q[i, j] = s / len(x)**0.5
assert torch.allclose(Q @ Q.T, torch.eye(d_sym))
if d_sym == 0:
return [], torch.zeros(d_sym, d)
def representation(alpha, beta, gamma, parity=None):
def re(r):
if callable(r):
if has_parity:
return r(alpha, beta, gamma, parity)
return r(alpha, beta, gamma)
return rep(r, alpha, beta, gamma, parity)
m = o3.kron(*(re(kw_Rs[i]) for i in f0))
return Q @ m @ Q.T
assert _is_representation(representation, eps, has_parity)
Rs_out = []
A = Q.clone()
for l in range(int((d_sym - 1) // 2) + 1):
for p in [-1, 1] if has_parity else [0]:
if 2 * l + 1 > d_sym - dim(Rs_out):
break
mul, B, representation = o3.reduce(representation,
partial(rep, [(1, l, p)]),
eps, has_parity)
A = o3.direct_sum(torch.eye(d_sym - B.shape[0]), B) @ A
A = _round_sqrt(A, eps)
Rs_out += [(mul, l, p)]
if dim(Rs_out) == d_sym:
break
if dim(Rs_out) != d_sym:
raise RuntimeError(
f'unable to decompose into irreducible representations')
return simplify(Rs_out), A