def maximal_elements(self):
r"""
Return the maximal elements of ``self`` with respect to Bruhat order.
The current implementation is via a conjectural type-free
formula. Use maximal_elements_combinatorial() for proven
type-specific implementations. To compare type-free and
type-specific (combinatorial) implementations, use method
:meth:`_test_maximal_elements`.
EXAMPLES::
sage: W = WeylGroup(['A',4,1])
sage: PF = W.pieri_factors()
sage: sorted([w.reduced_word() for w in PF.maximal_elements()], key=str)
[[0, 4, 3, 2], [1, 0, 4, 3], [2, 1, 0, 4], [3, 2, 1, 0], [4, 3, 2, 1]]
sage: W = WeylGroup(RootSystem(["C",3,1]).weight_space())
sage: PF = W.pieri_factors()
sage: sorted([w.reduced_word() for w in PF.maximal_elements()], key=str)
[[0, 1, 2, 3, 2, 1], [1, 0, 1, 2, 3, 2], [1, 2, 3, 2, 1, 0],
[2, 1, 0, 1, 2, 3], [2, 3, 2, 1, 0, 1], [3, 2, 1, 0, 1, 2]]
sage: W = WeylGroup(RootSystem(["B",3,1]).weight_space())
sage: PF = W.pieri_factors()
sage: sorted([w.reduced_word() for w in PF.maximal_elements()], key=str)
[[0, 2, 3, 2, 0], [1, 0, 2, 3, 2], [1, 2, 3, 2, 1],
[2, 1, 0, 2, 3], [2, 3, 2, 1, 0], [3, 2, 1, 0, 2]]
sage: W = WeylGroup(['D',4,1])
sage: PF = W.pieri_factors()
sage: sorted([w.reduced_word() for w in PF.maximal_elements()], key=str)
[[0, 2, 4, 3, 2, 0], [1, 0, 2, 4, 3, 2], [1, 2, 4, 3, 2, 1],
[2, 1, 0, 2, 4, 3], [2, 4, 3, 2, 1, 0], [3, 2, 1, 0, 2, 3],
[4, 2, 1, 0, 2, 4], [4, 3, 2, 1, 0, 2]]
"""
ct = self.W.cartan_type()
s = ct.translation_factors()[1]
R = RootSystem(ct).weight_space()
Lambda = R.fundamental_weights()
orbit = [
R.reduced_word_of_translation(x)
for x in (s * (Lambda[1] - Lambda[1].level() * Lambda[0])).orbit()
]
return [self.W.from_reduced_word(x) for x in orbit]
def maximal_elements(self):
r"""
Return the maximal elements of ``self`` with respect to Bruhat order.
The current implementation is via a conjectural type-free
formula. Use maximal_elements_combinatorial() for proven
type-specific implementations. To compare type-free and
type-specific (combinatorial) implementations, use method
:meth:`_test_maximal_elements`.
EXAMPLES::
sage: W = WeylGroup(['A',4,1])
sage: PF = W.pieri_factors()
sage: sorted([w.reduced_word() for w in PF.maximal_elements()], key=str)
[[0, 4, 3, 2], [1, 0, 4, 3], [2, 1, 0, 4], [3, 2, 1, 0], [4, 3, 2, 1]]
sage: W = WeylGroup(RootSystem(["C",3,1]).weight_space())
sage: PF = W.pieri_factors()
sage: sorted([w.reduced_word() for w in PF.maximal_elements()], key=str)
[[0, 1, 2, 3, 2, 1], [1, 0, 1, 2, 3, 2], [1, 2, 3, 2, 1, 0],
[2, 1, 0, 1, 2, 3], [2, 3, 2, 1, 0, 1], [3, 2, 1, 0, 1, 2]]
sage: W = WeylGroup(RootSystem(["B",3,1]).weight_space())
sage: PF = W.pieri_factors()
sage: sorted([w.reduced_word() for w in PF.maximal_elements()], key=str)
[[0, 2, 3, 2, 0], [1, 0, 2, 3, 2], [1, 2, 3, 2, 1],
[2, 1, 0, 2, 3], [2, 3, 2, 1, 0], [3, 2, 1, 0, 2]]
sage: W = WeylGroup(['D',4,1])
sage: PF = W.pieri_factors()
sage: sorted([w.reduced_word() for w in PF.maximal_elements()], key=str)
[[0, 2, 4, 3, 2, 0], [1, 0, 2, 4, 3, 2], [1, 2, 4, 3, 2, 1],
[2, 1, 0, 2, 4, 3], [2, 4, 3, 2, 1, 0], [3, 2, 1, 0, 2, 3],
[4, 2, 1, 0, 2, 4], [4, 3, 2, 1, 0, 2]]
"""
ct = self.W.cartan_type()
s = ct.translation_factors()[1]
R = RootSystem(ct).weight_space()
Lambda = R.fundamental_weights()
orbit = [R.reduced_word_of_translation(x)
for x in (s*(Lambda[1]-Lambda[1].level()*Lambda[0]))._orbit_iter()]
return [self.W.from_reduced_word(x) for x in orbit]
