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 map(self, f, name=None):
r"""
Return the image `\{f(x) | x \in \text{self}\}` of this
enumerated set by `f`, as an enumerated set.
`f` is supposed to be injective.
EXAMPLES::
sage: R = Compositions(4).map(attrcall('partial_sums')); R
Image of Compositions of 4 by *.partial_sums()
sage: R.cardinality()
8
sage: R.list()
[[1, 2, 3, 4], [1, 2, 4], [1, 3, 4], [1, 4], [2, 3, 4], [2, 4], [3, 4], [4]]
sage: [ r for r in R]
[[1, 2, 3, 4], [1, 2, 4], [1, 3, 4], [1, 4], [2, 3, 4], [2, 4], [3, 4], [4]]
.. warning::
If the function is not injective, then there may be
repeated elements::
sage: P = Compositions(4)
sage: P.list()
[[1, 1, 1, 1], [1, 1, 2], [1, 2, 1], [1, 3], [2, 1, 1], [2, 2], [3, 1], [4]]
sage: P.map(attrcall('major_index')).list()
[6, 3, 4, 1, 5, 2, 3, 0]
.. warning::
:class:`MapCombinatorialClass` needs to be refactored to use categories::
sage: R.category() # todo: not implemented
Category of enumerated sets
sage: TestSuite(R).run(skip=['_test_an_element', '_test_category', '_test_some_elements'])
"""
from sage.combinat.combinat import MapCombinatorialClass
return MapCombinatorialClass(self, f, name)
def map(self, f, name=None):
r"""
Returns the image `\{f(x) | x \in \text{self}\}` of this
enumerated set by `f`, as an enumerated set.
`f` is supposed to be injective.
EXAMPLES::
sage: R = SymmetricGroup(3).map(attrcall('reduced_word')); R
Image of Symmetric group of order 3! as a permutation group by *.reduced_word()
sage: R.cardinality()
6
sage: R.list()
[[], [2], [1], [2, 1], [1, 2], [1, 2, 1]]
sage: [ r for r in R]
[[], [2], [1], [2, 1], [1, 2], [1, 2, 1]]
.. warning::
If the function is not injective, then there may be
repeated elements::
sage: P = SymmetricGroup(3)
sage: P.list()
[(), (2,3), (1,2), (1,2,3), (1,3,2), (1,3)]
sage: P.map(attrcall('length')).list()
[0, 1, 1, 2, 2, 3]
.. warning::
:class:`MapCombinatorialClass` needs to be refactored to use categories::
sage: R.category() # todo: not implemented
Category of enumerated sets
sage: TestSuite(R).run(skip=['_test_an_element', '_test_category', '_test_some_elements'])
"""
from sage.combinat.combinat import MapCombinatorialClass
return MapCombinatorialClass(self, f, name)