def __classcall_private__(cls, shape, weight):
r"""
Straighten arguments before unique representation.
TESTS::
sage: LR = LittlewoodRichardsonTableaux([3,2,1],[[2,1],[2,1]])
sage: TestSuite(LR).run()
sage: LittlewoodRichardsonTableaux([3,2,1],[[2,1]])
Traceback (most recent call last):
...
ValueError: the sizes of shapes and sequence of weights do not match
"""
shape = Partition(shape)
weight = tuple(Partition(a) for a in weight)
if shape.size() != sum(a.size() for a in weight):
raise ValueError("the sizes of shapes and sequence of weights do not match")
return super(LittlewoodRichardsonTableaux, cls).__classcall__(cls, shape, weight)
def __classcall_private__(cls, shape, weight):
r"""
Straighten arguments before unique representation.
TESTS::
sage: LR = LittlewoodRichardsonTableaux([3,2,1],[[2,1],[2,1]])
sage: TestSuite(LR).run()
sage: LittlewoodRichardsonTableaux([3,2,1],[[2,1]])
Traceback (most recent call last):
...
ValueError: the sizes of shapes and sequence of weights do not match
"""
shape = Partition(shape)
weight = tuple(Partition(a) for a in weight)
if shape.size() != sum(a.size() for a in weight):
raise ValueError("the sizes of shapes and sequence of weights do not match")
return super(LittlewoodRichardsonTableaux, cls).__classcall__(cls, shape, weight)
def __classcall_private__(cls, deg, par):
r"""
Create a primary similarity class type.
EXAMPLES::
sage: PrimarySimilarityClassType(2, [3, 2, 1])
[2, [3, 2, 1]]
The parent class is the class of primary similarity class types of order
`d|\lambda\`::
sage: PT = PrimarySimilarityClassType(2, [3, 2, 1])
sage: PT.parent().size()
12
"""
par = Partition(par)
P = PrimarySimilarityClassTypes(par.size() * deg)
return P(deg, par)
def __classcall_private__(cls, deg, par):
r"""
Create a primary similarity class type.
EXAMPLES::
sage: PrimarySimilarityClassType(2, [3, 2, 1])
[2, [3, 2, 1]]
The parent class is the class of primary similarity class types of order
`d|\lambda\`::
sage: PT = PrimarySimilarityClassType(2, [3, 2, 1])
sage: PT.parent().size()
12
"""
par = Partition(par)
P = PrimarySimilarityClassTypes(par.size() * deg)
return P(deg, par)
def coefficient_cycle_type(self, t):
"""
Returns the coefficient of a cycle type ``t`` in ``self``.
EXAMPLES::
sage: from sage.combinat.species.generating_series import CycleIndexSeriesRing
sage: p = SymmetricFunctions(QQ).power()
sage: CIS = CycleIndexSeriesRing(QQ)
sage: f = CIS([0, p([1]), 2*p([1,1]),3*p([2,1])])
sage: f.coefficient_cycle_type([1])
1
sage: f.coefficient_cycle_type([1,1])
2
sage: f.coefficient_cycle_type([2,1])
3
"""
t = Partition(t)
p = self.coefficient(t.size())
return p.coefficient(t)
def coefficient_cycle_type(self, t):
"""
Returns the coefficient of a cycle type ``t`` in ``self``.
EXAMPLES::
sage: from sage.combinat.species.generating_series import CycleIndexSeriesRing
sage: p = SymmetricFunctions(QQ).power()
sage: CIS = CycleIndexSeriesRing(QQ)
sage: f = CIS([0, p([1]), 2*p([1,1]),3*p([2,1])])
sage: f.coefficient_cycle_type([1])
1
sage: f.coefficient_cycle_type([1,1])
2
sage: f.coefficient_cycle_type([2,1])
3
"""
t = Partition(t)
p = self.coefficient(t.size())
return p.coefficient(t)
class SymmetricGroupRepresentation_generic_class(SageObject):
r"""
Generic methods for a representation of the symmetric group.
"""
_default_ring = None
def __init__(self, partition, ring=None, cache_matrices=True):
r"""
An irreducible representation of the symmetric group corresponding
to ``partition``.
For more information, see the documentation for
:func:`SymmetricGroupRepresentation`.
EXAMPLES::
sage: spc = SymmetricGroupRepresentation([3])
sage: spc([3,2,1])
[1]
sage: spc == loads(dumps(spc))
True
sage: spc = SymmetricGroupRepresentation([3], cache_matrices=False)
sage: spc([3,2,1])
[1]
sage: spc == loads(dumps(spc))
True
"""
self._partition = Partition(partition)
self._ring = ring if not ring is None else self._default_ring
if cache_matrices is False:
self.representation_matrix = self._representation_matrix_uncached
def __eq__(self, other):
r"""
Test for equality.
EXAMPLES::
sage: spc1 = SymmetricGroupRepresentation([3], cache_matrices=True)
sage: spc1([3,1,2])
[1]
sage: spc2 = loads(dumps(spc1))
sage: spc1 == spc2
True
::
sage: spc3 = SymmetricGroupRepresentation([3], cache_matrices=False)
sage: spc3([3,1,2])
[1]
sage: spc4 = loads(dumps(spc3))
sage: spc3 == spc4
True
TESTS:
The following tests against some bug that was fixed in :trac:`8611`::
sage: spc = SymmetricGroupRepresentation([3])
sage: spc.important_info = 'Sage rules'
sage: spc == SymmetricGroupRepresentation([3])
True
"""
if not isinstance(other, type(other)):
return False
return (self._ring,self._partition)==(other._ring,other._partition)
# # both self and other must have caching enabled
# if 'representation_matrix' in self.__dict__:
# if 'representation_matrix' not in other.__dict__:
# return False
# else:
# for key in self.__dict__:
# if key != 'representation_matrix':
# if self.__dict__[key] != other.__dict__[key]:
# return False
# else:
# return True
# else:
# if 'representation_matrix' in other.__dict__:
# return False
# else:
# return self.__dict__.__eq__(other.__dict__)
def __call__(self, permutation):
r"""
Return the image of ``permutation`` in the representation.
EXAMPLES::
sage: spc = SymmetricGroupRepresentation([2,1])
sage: spc([1,3,2])
[ 1 0]
[ 1 -1]
"""
return self.representation_matrix(Permutation(permutation))
def __iter__(self):
r"""
Iterate over the matrices representing the elements of the
symmetric group.
EXAMPLES::
sage: spc = SymmetricGroupRepresentation([1,1,1])
sage: list(spc)
[[1], [-1], [-1], [1], [1], [-1]]
"""
for permutation in Permutations(self._partition.size()):
yield self.representation_matrix(permutation)
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._partition.size()
transpositions = []
for i in range(1,n):
si = Permutation(range(1,i) + [i+1,i] + range(i+2,n+1))
transpositions.append(si)
repn_matrices = map(self.representation_matrix, 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 to_character(self):
r"""
Return the character of the representation.
EXAMPLES:
The trivial character::
sage: rho = SymmetricGroupRepresentation([3])
sage: chi = rho.to_character(); chi
Character of Symmetric group of order 3! as a permutation group
sage: chi.values()
[1, 1, 1]
sage: all(chi(g) == 1 for g in SymmetricGroup(3))
True
The sign character::
sage: rho = SymmetricGroupRepresentation([1,1,1])
sage: chi = rho.to_character(); chi
Character of Symmetric group of order 3! as a permutation group
sage: chi.values()
[1, -1, 1]
sage: all(chi(g) == g.sign() for g in SymmetricGroup(3))
True
The defining representation::
sage: triv = SymmetricGroupRepresentation([4])
sage: hook = SymmetricGroupRepresentation([3,1])
sage: def_rep = lambda p : triv(p).block_sum(hook(p)).trace()
sage: map(def_rep, Permutations(4))
[4, 2, 2, 1, 1, 2, 2, 0, 1, 0, 0, 1, 1, 0, 2, 1, 0, 0, 0, 1, 1, 2, 0, 0]
sage: [p.to_matrix().trace() for p in Permutations(4)]
[4, 2, 2, 1, 1, 2, 2, 0, 1, 0, 0, 1, 1, 0, 2, 1, 0, 0, 0, 1, 1, 2, 0, 0]
"""
from sage.groups.perm_gps.permgroup_named import SymmetricGroup
Sym = SymmetricGroup(sum(self._partition))
values = [self(g).trace() for g in Sym.conjugacy_classes_representatives()]
return Sym.character(values)
class SymmetricGroupRepresentation_generic_class(SageObject):
r"""
Generic methods for a representation of the symmetric group.
"""
_default_ring = None
def __init__(self, partition, ring=None, cache_matrices=True):
r"""
An irreducible representation of the symmetric group corresponding
to ``partition``.
For more information, see the documentation for
:func:`SymmetricGroupRepresentation`.
EXAMPLES::
sage: spc = SymmetricGroupRepresentation([3])
sage: spc([3,2,1])
[1]
sage: spc == loads(dumps(spc))
True
sage: spc = SymmetricGroupRepresentation([3], cache_matrices=False)
sage: spc([3,2,1])
[1]
sage: spc == loads(dumps(spc))
True
"""
self._partition = Partition(partition)
self._ring = ring if not ring is None else self._default_ring
if cache_matrices is False:
self.representation_matrix = self._representation_matrix_uncached
def __eq__(self, other):
r"""
Test for equality.
EXAMPLES::
sage: spc1 = SymmetricGroupRepresentation([3], cache_matrices=True)
sage: spc1([3,1,2])
[1]
sage: spc2 = loads(dumps(spc1))
sage: spc1 == spc2
True
::
sage: spc3 = SymmetricGroupRepresentation([3], cache_matrices=False)
sage: spc3([3,1,2])
[1]
sage: spc4 = loads(dumps(spc3))
sage: spc3 == spc4
True
TESTS:
The following tests against some bug that was fixed in :trac:`8611`::
sage: spc = SymmetricGroupRepresentation([3])
sage: spc.important_info = 'Sage rules'
sage: spc == SymmetricGroupRepresentation([3])
True
"""
if not isinstance(other, type(other)):
return False
return (self._ring, self._partition) == (other._ring, other._partition)
# # both self and other must have caching enabled
# if self.__dict__.has_key('representation_matrix'):
# if not other.__dict__.has_key('representation_matrix'):
# return False
# else:
# for key in self.__dict__:
# if key != 'representation_matrix':
# if self.__dict__[key] != other.__dict__[key]:
# return False
# else:
# return True
# else:
# if other.__dict__.has_key('representation_matrix'):
# return False
# else:
# return self.__dict__.__eq__(other.__dict__)
def __call__(self, permutation):
r"""
Return the image of ``permutation`` in the representation.
EXAMPLES::
sage: spc = SymmetricGroupRepresentation([2,1])
sage: spc([1,3,2])
[ 1 0]
[ 1 -1]
"""
return self.representation_matrix(Permutation(permutation))
def __iter__(self):
r"""
Iterate over the matrices representing the elements of the
symmetric group.
EXAMPLES::
sage: spc = SymmetricGroupRepresentation([1,1,1])
sage: list(spc)
[[1], [-1], [-1], [1], [1], [-1]]
"""
for permutation in Permutations(self._partition.size()):
yield self.representation_matrix(permutation)
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._partition.size()
transpositions = []
for i in range(1, n):
si = Permutation(range(1, i) + [i + 1, i] + range(i + 2, n + 1))
transpositions.append(si)
repn_matrices = map(self.representation_matrix, 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 to_character(self):
r"""
Return the character of the representation.
EXAMPLES:
The trivial character::
sage: rho = SymmetricGroupRepresentation([3])
sage: chi = rho.to_character(); chi
Character of Symmetric group of order 3! as a permutation group
sage: chi.values()
[1, 1, 1]
sage: all(chi(g) == 1 for g in SymmetricGroup(3))
True
The sign character::
sage: rho = SymmetricGroupRepresentation([1,1,1])
sage: chi = rho.to_character(); chi
Character of Symmetric group of order 3! as a permutation group
sage: chi.values()
[1, -1, 1]
sage: all(chi(g) == g.sign() for g in SymmetricGroup(3))
True
The defining representation::
sage: triv = SymmetricGroupRepresentation([4])
sage: hook = SymmetricGroupRepresentation([3,1])
sage: def_rep = lambda p : triv(p).block_sum(hook(p)).trace()
sage: map(def_rep, Permutations(4))
[4, 2, 2, 1, 1, 2, 2, 0, 1, 0, 0, 1, 1, 0, 2, 1, 0, 0, 0, 1, 1, 2, 0, 0]
sage: [p.to_matrix().trace() for p in Permutations(4)]
[4, 2, 2, 1, 1, 2, 2, 0, 1, 0, 0, 1, 1, 0, 2, 1, 0, 0, 0, 1, 1, 2, 0, 0]
"""
from sage.groups.perm_gps.permgroup_named import SymmetricGroup
Sym = SymmetricGroup(sum(self._partition))
values = [
self(g).trace() for g in Sym.conjugacy_classes_representatives()
]
return Sym.character(values)
