from sympy.combinatorics import Permutation
from sympy.combinatorics.util import _distribute_gens_by_base
from sympy.core.compatibility import range
rmul = Permutation.rmul
[docs]def _cmp_perm_lists(first, second):
"""
Compare two lists of permutations as sets.
This is used for testing purposes. Since the array form of a
permutation is currently a list, Permutation is not hashable
and cannot be put into a set.
Examples
========
>>> from sympy.combinatorics.permutations import Permutation
>>> from sympy.combinatorics.testutil import _cmp_perm_lists
>>> a = Permutation([0, 2, 3, 4, 1])
>>> b = Permutation([1, 2, 0, 4, 3])
>>> c = Permutation([3, 4, 0, 1, 2])
>>> ls1 = [a, b, c]
>>> ls2 = [b, c, a]
>>> _cmp_perm_lists(ls1, ls2)
True
"""
return {tuple(a) for a in first} == \
{tuple(a) for a in second}
[docs]def _naive_list_centralizer(self, other, af=False):
from sympy.combinatorics.perm_groups import PermutationGroup
"""
Return a list of elements for the centralizer of a subgroup/set/element.
This is a brute force implementation that goes over all elements of the
group and checks for membership in the centralizer. It is used to
test ``.centralizer()`` from ``sympy.combinatorics.perm_groups``.
Examples
========
>>> from sympy.combinatorics.testutil import _naive_list_centralizer
>>> from sympy.combinatorics.named_groups import DihedralGroup
>>> D = DihedralGroup(4)
>>> _naive_list_centralizer(D, D)
[Permutation([0, 1, 2, 3]), Permutation([2, 3, 0, 1])]
See Also
========
sympy.combinatorics.perm_groups.centralizer
"""
from sympy.combinatorics.permutations import _af_commutes_with
if hasattr(other, 'generators'):
elements = list(self.generate_dimino(af=True))
gens = [x._array_form for x in other.generators]
commutes_with_gens = lambda x: all(_af_commutes_with(x, gen) for gen in gens)
centralizer_list = []
if not af:
for element in elements:
if commutes_with_gens(element):
centralizer_list.append(Permutation._af_new(element))
else:
for element in elements:
if commutes_with_gens(element):
centralizer_list.append(element)
return centralizer_list
elif hasattr(other, 'getitem'):
return _naive_list_centralizer(self, PermutationGroup(other), af)
elif hasattr(other, 'array_form'):
return _naive_list_centralizer(self, PermutationGroup([other]), af)
[docs]def _verify_bsgs(group, base, gens):
"""
Verify the correctness of a base and strong generating set.
This is a naive implementation using the definition of a base and a strong
generating set relative to it. There are other procedures for
verifying a base and strong generating set, but this one will
serve for more robust testing.
Examples
========
>>> from sympy.combinatorics.named_groups import AlternatingGroup
>>> from sympy.combinatorics.testutil import _verify_bsgs
>>> A = AlternatingGroup(4)
>>> A.schreier_sims()
>>> _verify_bsgs(A, A.base, A.strong_gens)
True
See Also
========
sympy.combinatorics.perm_groups.PermutationGroup.schreier_sims
"""
from sympy.combinatorics.perm_groups import PermutationGroup
strong_gens_distr = _distribute_gens_by_base(base, gens)
current_stabilizer = group
for i in range(len(base)):
candidate = PermutationGroup(strong_gens_distr[i])
if current_stabilizer.order() != candidate.order():
return False
current_stabilizer = current_stabilizer.stabilizer(base[i])
if current_stabilizer.order() != 1:
return False
return True
[docs]def _verify_centralizer(group, arg, centr=None):
"""
Verify the centralizer of a group/set/element inside another group.
This is used for testing ``.centralizer()`` from
``sympy.combinatorics.perm_groups``
Examples
========
>>> from sympy.combinatorics.named_groups import (SymmetricGroup,
... AlternatingGroup)
>>> from sympy.combinatorics.perm_groups import PermutationGroup
>>> from sympy.combinatorics.permutations import Permutation
>>> from sympy.combinatorics.testutil import _verify_centralizer
>>> S = SymmetricGroup(5)
>>> A = AlternatingGroup(5)
>>> centr = PermutationGroup([Permutation([0, 1, 2, 3, 4])])
>>> _verify_centralizer(S, A, centr)
True
See Also
========
_naive_list_centralizer,
sympy.combinatorics.perm_groups.PermutationGroup.centralizer,
_cmp_perm_lists
"""
if centr is None:
centr = group.centralizer(arg)
centr_list = list(centr.generate_dimino(af=True))
centr_list_naive = _naive_list_centralizer(group, arg, af=True)
return _cmp_perm_lists(centr_list, centr_list_naive)
[docs]def _verify_normal_closure(group, arg, closure=None):
from sympy.combinatorics.perm_groups import PermutationGroup
"""
Verify the normal closure of a subgroup/subset/element in a group.
This is used to test
sympy.combinatorics.perm_groups.PermutationGroup.normal_closure
Examples
========
>>> from sympy.combinatorics.named_groups import (SymmetricGroup,
... AlternatingGroup)
>>> from sympy.combinatorics.testutil import _verify_normal_closure
>>> S = SymmetricGroup(3)
>>> A = AlternatingGroup(3)
>>> _verify_normal_closure(S, A, closure=A)
True
See Also
========
sympy.combinatorics.perm_groups.PermutationGroup.normal_closure
"""
if closure is None:
closure = group.normal_closure(arg)
conjugates = set()
if hasattr(arg, 'generators'):
subgr_gens = arg.generators
elif hasattr(arg, '__getitem__'):
subgr_gens = arg
elif hasattr(arg, 'array_form'):
subgr_gens = [arg]
for el in group.generate_dimino():
for gen in subgr_gens:
conjugates.add(gen ^ el)
naive_closure = PermutationGroup(list(conjugates))
return closure.is_subgroup(naive_closure)
def canonicalize_naive(g, dummies, sym, *v):
"""
Canonicalize tensor formed by tensors of the different types
g permutation representing the tensor
dummies list of dummy indices
msym symmetry of the metric
v is a list of (base_i, gens_i, n_i, sym_i) for tensors of type `i`
base_i, gens_i BSGS for tensors of this type
n_i number ot tensors of type `i`
sym_i symmetry under exchange of two component tensors of type `i`
None no symmetry
0 commuting
1 anticommuting
Return 0 if the tensor is zero, else return the array form of
the permutation representing the canonical form of the tensor.
Examples
========
>>> from sympy.combinatorics.testutil import canonicalize_naive
>>> from sympy.combinatorics.tensor_can import get_symmetric_group_sgs
>>> from sympy.combinatorics import Permutation, PermutationGroup
>>> g = Permutation([1, 3, 2, 0, 4, 5])
>>> base2, gens2 = get_symmetric_group_sgs(2)
>>> canonicalize_naive(g, [2, 3], 0, (base2, gens2, 2, 0))
[0, 2, 1, 3, 4, 5]
"""
from sympy.combinatorics.perm_groups import PermutationGroup
from sympy.combinatorics.tensor_can import gens_products, dummy_sgs
from sympy.combinatorics.permutations import Permutation, _af_rmul
v1 = []
for i in range(len(v)):
base_i, gens_i, n_i, sym_i = v[i]
v1.append((base_i, gens_i, [[]]*n_i, sym_i))
size, sbase, sgens = gens_products(*v1)
dgens = dummy_sgs(dummies, sym, size-2)
if isinstance(sym, int):
num_types = 1
dummies = [dummies]
sym = [sym]
else:
num_types = len(sym)
dgens = []
for i in range(num_types):
dgens.extend(dummy_sgs(dummies[i], sym[i], size - 2))
S = PermutationGroup(sgens)
D = PermutationGroup([Permutation(x) for x in dgens])
dlist = list(D.generate(af=True))
g = g.array_form
st = set()
for s in S.generate(af=True):
h = _af_rmul(g, s)
for d in dlist:
q = tuple(_af_rmul(d, h))
st.add(q)
a = list(st)
a.sort()
prev = (0,)*size
for h in a:
if h[:-2] == prev[:-2]:
if h[-1] != prev[-1]:
return 0
prev = h
return list(a[0])
def graph_certificate(gr):
"""
Return a certificate for the graph
gr adjacency list
The graph is assumed to be unoriented and without
external lines.
Associate to each vertex of the graph a symmetric tensor with
number of indices equal to the degree of the vertex; indices
are contracted when they correspond to the same line of the graph.
The canonical form of the tensor gives a certificate for the graph.
This is not an efficient algorithm to get the certificate of a graph.
Examples
========
>>> from sympy.combinatorics.testutil import graph_certificate
>>> gr1 = {0:[1, 2, 3, 5], 1:[0, 2, 4], 2:[0, 1, 3, 4], 3:[0, 2, 4], 4:[1, 2, 3, 5], 5:[0, 4]}
>>> gr2 = {0:[1, 5], 1:[0, 2, 3, 4], 2:[1, 3, 5], 3:[1, 2, 4, 5], 4:[1, 3, 5], 5:[0, 2, 3, 4]}
>>> c1 = graph_certificate(gr1)
>>> c2 = graph_certificate(gr2)
>>> c1
[0, 2, 4, 6, 1, 8, 10, 12, 3, 14, 16, 18, 5, 9, 15, 7, 11, 17, 13, 19, 20, 21]
>>> c1 == c2
True
"""
from sympy.combinatorics.permutations import _af_invert
from sympy.combinatorics.tensor_can import get_symmetric_group_sgs, canonicalize
items = list(gr.items())
items.sort(key=lambda x: len(x[1]), reverse=True)
pvert = [x[0] for x in items]
pvert = _af_invert(pvert)
# the indices of the tensor are twice the number of lines of the graph
num_indices = 0
for v, neigh in items:
num_indices += len(neigh)
# associate to each vertex its indices; for each line
# between two vertices assign the
# even index to the vertex which comes first in items,
# the odd index to the other vertex
vertices = [[] for i in items]
i = 0
for v, neigh in items:
for v2 in neigh:
if pvert[v] < pvert[v2]:
vertices[pvert[v]].append(i)
vertices[pvert[v2]].append(i+1)
i += 2
g = []
for v in vertices:
g.extend(v)
assert len(g) == num_indices
g += [num_indices, num_indices + 1]
size = num_indices + 2
assert sorted(g) == list(range(size))
g = Permutation(g)
vlen = [0]*(len(vertices[0])+1)
for neigh in vertices:
vlen[len(neigh)] += 1
v = []
for i in range(len(vlen)):
n = vlen[i]
if n:
base, gens = get_symmetric_group_sgs(i)
v.append((base, gens, n, 0))
v.reverse()
dummies = list(range(num_indices))
can = canonicalize(g, dummies, 0, *v)
return can
