def subcrystal(self, index_set=None, generators=None, max_depth=float("inf"),
direction="both"):
r"""
Construct the subcrystal from ``generators`` using `e_i` and `f_i`
for all `i` in ``index_set``.
INPUT:
- ``index_set`` -- (Default: ``None``) The index set; if ``None``
then use the index set of the crystal
- ``generators`` -- (Default: ``None``) The list of generators; if
``None`` then use the module generators of the crystal
- ``max_depth`` -- (Default: infinity) The maximum depth to build
- ``direction`` -- (Default: ``'both'``) The direction to build
the subcrystal. It can be one of the following:
- ``'both'`` - Using both `e_i` and `f_i`
- ``'upper'`` - Using `e_i`
- ``'lower'`` - Using `f_i`
EXAMPLES::
sage: C = crystals.KirillovReshetikhin(['A',3,1], 1, 2)
sage: S = list(C.subcrystal(index_set=[1,2])); S
[[[1, 1]], [[1, 2]], [[1, 3]], [[2, 2]], [[2, 3]], [[3, 3]]]
sage: C.cardinality()
10
sage: len(S)
6
sage: list(C.subcrystal(index_set=[1,3], generators=[C(1,4)]))
[[[1, 4]], [[2, 4]], [[1, 3]], [[2, 3]]]
sage: list(C.subcrystal(index_set=[1,3], generators=[C(1,4)], max_depth=1))
[[[1, 4]], [[2, 4]], [[1, 3]]]
sage: list(C.subcrystal(index_set=[1,3], generators=[C(1,4)], direction='upper'))
[[[1, 4]], [[1, 3]]]
sage: list(C.subcrystal(index_set=[1,3], generators=[C(1,4)], direction='lower'))
[[[1, 4]], [[2, 4]]]
"""
if index_set is None:
index_set = self.index_set()
if generators is None:
generators = self.module_generators
from sage.combinat.backtrack import TransitiveIdealGraded
if direction == 'both':
return TransitiveIdealGraded(lambda x: [x.f(i) for i in index_set]
+ [x.e(i) for i in index_set],
generators, max_depth)
if direction == 'upper':
return TransitiveIdealGraded(lambda x: [x.e(i) for i in index_set],
generators, max_depth)
if direction == 'lower':
return TransitiveIdealGraded(lambda x: [x.f(i) for i in index_set],
generators, max_depth)
raise ValueError("direction must be either 'both', 'upper', or 'lower'")
def __iter__(self, index_set=None, max_depth = float("inf")):
"""
Returns the iterator of ``self``.
INPUT:
- ``index_set`` -- (Default: ``None``) The index set; if ``None``
then use the index set of the crystal
- ``max_depth`` -- (Default: infinity) The maximum depth to build
EXAMPLES::
sage: C = CrystalOfLSPaths(['A',2,1],[0,1,0])
sage: [p for p in C.__iter__(max_depth=3)]
[(Lambda[1],), (Lambda[0] - Lambda[1] + Lambda[2],), (2*Lambda[0] - Lambda[2],),
(-Lambda[0] + 2*Lambda[2] - delta,),
(1/2*Lambda[0] + Lambda[1] - Lambda[2] - 1/2*delta, -1/2*Lambda[0] + Lambda[2] - 1/2*delta),
(-Lambda[0] + Lambda[1] + 1/2*Lambda[2] - delta, Lambda[0] - 1/2*Lambda[2])]
sage: [p for p in C.__iter__(index_set=[0, 1], max_depth=3)]
[(Lambda[1],), (Lambda[0] - Lambda[1] + Lambda[2],), (-Lambda[0] + 2*Lambda[2] - delta,)]
"""
if index_set is None:
index_set = self.index_set()
from sage.combinat.backtrack import TransitiveIdealGraded
return TransitiveIdealGraded(lambda x: [x.f(i) for i in index_set],
self.module_generators, max_depth).__iter__()
def __iter__(self, index_set=None, max_depth=float('inf')):
"""
Return an iterator over the elements of ``self``.
INPUT:
- ``index_set`` -- (Default: ``None``) The index set; if ``None``
then use the index set of the crystal
- ``max_depth`` -- (Default: infinity) The maximum depth to build
The iteration order is not specified except that, if
``max_depth`` is finite, then the iteration goes depth by
depth.
EXAMPLES::
sage: C = crystals.LSPaths(['A',2,1],[-1,0,1])
sage: C.__iter__.__module__
'sage.categories.crystals'
sage: g = C.__iter__()
sage: g.next()
(-Lambda[0] + Lambda[2],)
sage: g.next()
(Lambda[0] - Lambda[1] + delta,)
sage: g.next()
(Lambda[1] - Lambda[2] + delta,)
sage: g.next()
(-Lambda[0] + Lambda[2] + delta,)
sage: g.next()
(Lambda[1] - Lambda[2],)
sage: sorted(C.__iter__(index_set=[1,2]), key=str)
[(-Lambda[0] + Lambda[2],),
(Lambda[0] - Lambda[1],),
(Lambda[1] - Lambda[2],)]
sage: sorted(C.__iter__(max_depth=1), key=str)
[(-Lambda[0] + Lambda[2],),
(Lambda[0] - Lambda[1] + delta,),
(Lambda[1] - Lambda[2],)]
"""
if index_set is None:
index_set = self.index_set()
if max_depth < float('inf'):
from sage.combinat.backtrack import TransitiveIdealGraded
return TransitiveIdealGraded(lambda x: [x.f(i) for i in index_set]
+ [x.e(i) for i in index_set],
self.module_generators, max_depth).__iter__()
from sage.combinat.backtrack import TransitiveIdeal
return TransitiveIdeal(lambda x: [x.f(i) for i in index_set]
+ [x.e(i) for i in index_set],
self.module_generators).__iter__()
def __iter__(self, index_set=None, max_depth=float('inf')):
"""
Returns the iterator of ``self``.
INPUT:
- ``index_set`` -- (Default: ``None``) The index set; if ``None``
then use the index set of the crystal
- ``max_depth`` -- (Default: infinity) The maximum depth to build
EXAMPLES::
sage: C = CrystalOfLSPaths(['A',2,1],[-1,0,1])
sage: C.__iter__.__module__
'sage.categories.crystals'
sage: g = C.__iter__()
sage: g.next()
(-Lambda[0] + Lambda[2],)
sage: g.next()
(Lambda[0] - Lambda[1] + delta,)
sage: g.next()
(Lambda[1] - Lambda[2] + delta,)
sage: g.next()
(Lambda[1] - Lambda[2],)
sage: g.next()
(Lambda[0] - Lambda[1],)
sage: h = C.__iter__(index_set=[1,2])
sage: h.next()
(-Lambda[0] + Lambda[2],)
sage: h.next()
(Lambda[1] - Lambda[2],)
sage: h.next()
(Lambda[0] - Lambda[1],)
sage: h.next()
Traceback (most recent call last):
...
StopIteration
sage: g = C.__iter__(max_depth=1)
sage: g.next()
(-Lambda[0] + Lambda[2],)
sage: g.next()
(Lambda[1] - Lambda[2],)
sage: g.next()
(Lambda[0] - Lambda[1] + delta,)
sage: h.next()
Traceback (most recent call last):
...
StopIteration
"""
if index_set is None:
index_set = self.index_set()
if max_depth < float('inf'):
from sage.combinat.backtrack import TransitiveIdealGraded
return TransitiveIdealGraded(lambda x: [x.f(i) for i in index_set]
+ [x.e(i) for i in index_set],
self.module_generators, max_depth).__iter__()
from sage.combinat.backtrack import TransitiveIdeal
return TransitiveIdeal(lambda x: [x.f(i) for i in index_set]
+ [x.e(i) for i in index_set],
self.module_generators).__iter__()