def elements(self):
r"""
Returns the elements of ``self``
Those are constructed as the elements below the maximal
elements of ``self`` in Bruhat order.
OUTPUT: a :class:`RecursivelyEnumeratedSet_generic` object
EXAMPLES::
sage: PF = WeylGroup(['A',3]).pieri_factors()
sage: [w.reduced_word() for w in PF.elements()]
[[3, 2, 1], [2, 1], [3, 1], [3, 2], [2], [1], [3], []]
.. SEEALSO:: :meth:`maximal_elements`
.. TODO::
Possibly remove this method and instead have this class
inherit from :class:`RecursivelyEnumeratedSet_generic`.
"""
return RecursivelyEnumeratedSet(self.maximal_elements(),
attrcall('bruhat_lower_covers'), structure=None,
enumeration='naive')
def elements(self):
r"""
Returns the elements of ``self``
Those are constructed as the elements below the maximal
elements of ``self`` in Bruhat order.
OUTPUT: a :class:`RecursivelyEnumeratedSet_generic` object
EXAMPLES::
sage: PF = WeylGroup(['A',3]).pieri_factors()
sage: [w.reduced_word() for w in PF.elements()]
[[3, 2, 1], [2, 1], [1], [], [3, 1], [3], [3, 2], [2]]
.. SEEALSO:: :meth:`maximal_elements`
.. TODO::
Possibly remove this method and instead have this class
inherit from :class:`RecursivelyEnumeratedSet_generic`.
"""
return RecursivelyEnumeratedSet(self.maximal_elements(),
attrcall('bruhat_lower_covers'),
structure=None,
enumeration='naive')
def positive_roots(self):
r"""
Returns the positive roots of self.
EXAMPLES::
sage: L = RootSystem(['A',3]).root_lattice()
sage: sorted(L.positive_roots())
[alpha[1], alpha[1] + alpha[2], alpha[1] + alpha[2] + alpha[3], alpha[2], alpha[2] + alpha[3], alpha[3]]
Algorithm: generate them from the simple roots by applying
successive reflections toward the positive chamber.
"""
assert self.cartan_type().is_finite()
return TransitiveIdeal(attrcall('pred'), self.simple_roots())
def positive_roots(self):
r"""
Returns the positive roots of self.
EXAMPLES::
sage: L = RootSystem(['A',3]).root_lattice()
sage: sorted(L.positive_roots())
[alpha[1], alpha[1] + alpha[2], alpha[1] + alpha[2] + alpha[3], alpha[2], alpha[2] + alpha[3], alpha[3]]
Algorithm: generate them from the simple roots by applying
successive reflections toward the positive chamber.
"""
assert self.cartan_type().is_finite()
return TransitiveIdeal(attrcall('pred'), self.simple_roots())
def negative_roots(self):
r"""
Returns the negative roots of self.
EXAMPLES::
sage: L = RootSystem(['A', 2]).weight_lattice()
sage: sorted(L.negative_roots())
[-2*Lambda[1] + Lambda[2], -Lambda[1] - Lambda[2], Lambda[1] - 2*Lambda[2]]
Algorithm: negate the positive roots
"""
assert self.cartan_type().is_finite()
from sage.combinat.combinat import MapCombinatorialClass
return MapCombinatorialClass(self.positive_roots(), attrcall('__neg__'), "The negative roots of %s"%self)
def negative_roots(self):
r"""
Returns the negative roots of self.
EXAMPLES::
sage: L = RootSystem(['A', 2]).weight_lattice()
sage: sorted(L.negative_roots())
[-2*Lambda[1] + Lambda[2], -Lambda[1] - Lambda[2], Lambda[1] - 2*Lambda[2]]
Algorithm: negate the positive roots
"""
assert self.cartan_type().is_finite()
from sage.combinat.combinat import MapCombinatorialClass
return MapCombinatorialClass(self.positive_roots(),
attrcall('__neg__'),
"The negative roots of %s" % self)
def smaller(self):
r"""
Returns the elements in the orbit of self which are
smaller than self in the weak order.
EXAMPLES::
sage: L = RootSystem(['A',3]).ambient_lattice()
sage: e = L.basis()
sage: e[2].smaller()
[(0, 0, 1, 0), (0, 1, 0, 0), (1, 0, 0, 0)]
sage: len(L.rho().smaller())
1
sage: len((-L.rho()).smaller())
24
sage: sorted([len(x.smaller()) for x in L.rho().orbit()])
[1, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 5, 5, 6, 6, 6, 8, 8, 8, 8, 12, 12, 12, 24]
"""
return [x for x in TransitiveIdeal(attrcall('pred'), [self])]
def smaller(self):
r"""
Returns the elements in the orbit of self which are
smaller than self in the weak order.
EXAMPLES::
sage: L = RootSystem(['A',3]).ambient_lattice()
sage: e = L.basis()
sage: e[2].smaller()
[(0, 0, 1, 0), (0, 1, 0, 0), (1, 0, 0, 0)]
sage: len(L.rho().smaller())
1
sage: len((-L.rho()).smaller())
24
sage: sorted([len(x.smaller()) for x in L.rho().orbit()])
[1, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 5, 5, 6, 6, 6, 8, 8, 8, 8, 12, 12, 12, 24]
"""
return [x for x in TransitiveIdeal(attrcall('pred'), [self])]
def orbit(self):
r"""
The orbit of self under the action of the Weyl group
EXAMPLES::
`\rho` is a regular element whose orbit is in bijection with the Weyl group.
In particular, it as 6 elements for the symmetric group `S_3`::
sage: L = RootSystem(["A", 2]).ambient_lattice()
sage: sorted(L.rho().orbit()) # the output order is not specified
[(1, 2, 0), (1, 0, 2), (2, 1, 0), (2, 0, 1), (0, 1, 2), (0, 2, 1)]
sage: L = RootSystem(["A", 3]).weight_lattice()
sage: len(L.rho().orbit())
24
sage: len(L.fundamental_weights()[1].orbit())
4
sage: len(L.fundamental_weights()[2].orbit())
6
"""
return [x for x in TransitiveIdeal(attrcall('simple_reflections'), [self])]
def elements(self):
r"""
Returns the elements of ``self``
Those are constructed as the elements below the maximal
elements of ``self`` in Bruhat order.
OUTPUT: a :class:`TransitiveIdeal` object
EXAMPLES::
sage: PF = WeylGroup(['A',3]).pieri_factors()
sage: [w.reduced_word() for w in PF.elements()]
[[3, 2, 1], [2, 1], [1], [], [3, 1], [3], [3, 2], [2]]
.. SEEALSO:: :meth:`maximal_elements`
.. TODO::
Possibly remove this method and instead have this class
inherit from :class:`TransitiveIdeal`.
"""
return TransitiveIdeal(attrcall('bruhat_lower_covers'), self.maximal_elements())
def orbit(self):
r"""
The orbit of self under the action of the Weyl group
EXAMPLES::
`\rho` is a regular element whose orbit is in bijection with the Weyl group.
In particular, it as 6 elements for the symmetric group `S_3`::
sage: L = RootSystem(["A", 2]).ambient_lattice()
sage: sorted(L.rho().orbit()) # the output order is not specified
[(1, 2, 0), (1, 0, 2), (2, 1, 0), (2, 0, 1), (0, 1, 2), (0, 2, 1)]
sage: L = RootSystem(["A", 3]).weight_lattice()
sage: len(L.rho().orbit())
24
sage: len(L.fundamental_weights()[1].orbit())
4
sage: len(L.fundamental_weights()[2].orbit())
6
"""
return [
x for x in TransitiveIdeal(attrcall('simple_reflections'), [self])
]