def verify_representation(self):
r"""
Verify the representation: tests that the images of the simple
transpositions are involutions and tests that the braid relations
hold.
EXAMPLES::
sage: spc = SymmetricGroupRepresentation([1,1,1])
sage: spc.verify_representation()
True
sage: spc = SymmetricGroupRepresentation([4,2,1])
sage: spc.verify_representation()
True
"""
n = self._n
transpositions = [from_cycles(n, ((i, i + 1), )) for i in range(1, n)]
repn_matrices = [self.representation_matrix(t) for t in transpositions]
for i, si in enumerate(repn_matrices):
for j, sj in enumerate(repn_matrices):
if i == j:
if si * sj != si.parent().identity_matrix():
return False, "si si != 1 for i = %s" % (i, )
elif abs(i - j) > 1:
if si * sj != sj * si:
return False, "si sj != sj si for (i,j) =(%s,%s)" % (i,
j)
else:
if si * sj * si != sj * si * sj:
return False, "si sj si != sj si sj for (i,j) = (%s,%s)" % (
i, j)
return True
def __iter__(self):
"""
Iterate over ``self``.
EXAMPLES::
sage: G = Permutations(4)
sage: C = G.conjugacy_class(Partition([3,1]))
sage: for x in C: x
[1, 3, 4, 2]
[1, 4, 2, 3]
[3, 2, 4, 1]
[4, 2, 1, 3]
[2, 4, 3, 1]
[4, 1, 3, 2]
[2, 3, 1, 4]
[3, 1, 2, 4]
"""
if self._set:
for x in self._set:
yield x
return
for x in conjugacy_class_iterator(self._part, self._domain):
yield from_cycles(self._parent.n, x, self._parent)
def set(self):
r"""
The set of all elements in the conjugacy class ``self``.
EXAMPLES::
sage: G = Permutations(3)
sage: g = G([2, 1, 3])
sage: C = G.conjugacy_class(g)
sage: S = [[1, 3, 2], [2, 1, 3], [3, 2, 1]]
sage: C.set() == Set(G(x) for x in S)
True
"""
if not self._set:
self._set = Set(from_cycles(self._parent.n, x, self._parent)
for x in conjugacy_class_iterator(self._part, self._domain) )
return self._set