r"""
Finite Crystals
"""
from sage.misc.cachefunc import cached_method
from sage.categories.category import Category
from sage.categories.crystals import Crystals
from sage.categories.finite_enumerated_sets import FiniteEnumeratedSets
class FiniteCrystals(Category):
"""
The category of finite crystals.
EXAMPLES::
sage: C = FiniteCrystals()
sage: C
Category of finite crystals
sage: C.super_categories()
[Category of crystals, Category of finite enumerated sets]
sage: C.example()
Highest weight crystal of type A_3 of highest weight omega_1
TESTS::
sage: TestSuite(C).run()
sage: B = FiniteCrystals().example()
sage: TestSuite(B).run(verbose = True)
running ._test_an_element() . . . pass
running ._test_category() . . . pass
running ._test_elements() . . .
Running the test suite of self.an_element()
running ._test_category() . . . pass
running ._test_eq() . . . pass
running ._test_not_implemented_methods() . . . pass
running ._test_pickling() . . . pass
running ._test_stembridge_local_axioms() . . . pass
pass
running ._test_elements_eq() . . . pass
running ._test_enumerated_set_contains() . . . pass
running ._test_enumerated_set_iter_cardinality() . . . pass
running ._test_enumerated_set_iter_list() . . . pass
running ._test_eq() . . . pass
running ._test_fast_iter() . . . pass
running ._test_not_implemented_methods() . . . pass
running ._test_pickling() . . . pass
running ._test_some_elements() . . . pass
running ._test_stembridge_local_axioms() . . . pass
"""
@cached_method
def super_categories(self):
r"""
EXAMPLES::
sage: FiniteCrystals().super_categories()
[Category of crystals, Category of finite enumerated sets]
"""
return [Crystals(), FiniteEnumeratedSets()]
def example(self, n = 3):
"""
Returns an example of highest weight crystals, as per
:meth:`Category.example`.
EXAMPLES::
sage: B = FiniteCrystals().example(); B
Highest weight crystal of type A_3 of highest weight omega_1
"""
from sage.categories.crystals import Crystals
return Crystals().example(n)
class ParentMethods:
def _test_stembridge_local_axioms(self, index_set=None, verbose=False, complete=False, **options):
r"""
This implements tests for the Stembridge local characterization on the finite crystal ``self``.
The current implementation only uses the rules for simply-laced types. Crystals
of other types should still pass the test, but expansion of this test to
non-simply laced type would be desirable.
One can specify an index set smaller than the full index set of the crystal,
using the option ``index_set``.
Running with ``verbose=True`` will print each node for which a local axiom
test applies.
Running with ``complete=True`` will continue to run the test past the first
failure of the local axioms. This is probably only useful in conjunction
with the verbose option, to see all places where the local axioms fail.
EXAMPLES::
sage: T = CrystalOfTableaux(['A',3], shape=[2,1])
sage: T._test_stembridge_local_axioms()
True
sage: T._test_stembridge_local_axioms(verbose=True)
True
sage: T._test_stembridge_local_axioms(index_set=[1,3])
True
sage: B=Crystals().example(choice='naive')
sage: B._test_stembridge_local_axioms()
Traceback (most recent call last):
...
AssertionError: None
"""
tester = self._tester(**options)
goodness=True
for x in self:
goodness=x._test_stembridge_local_axioms(index_set, verbose)
if goodness==False and not complete:
tester.fail()
tester.assertTrue(goodness)
return goodness
def list(self):
"""
Returns a list of the elements of ``self`` obtained by
repeatedly applying the `f_i` operators to the module
generators of ``self``.
EXAMPLES::
sage: C = FiniteCrystals().example(5)
sage: l = C._list_brute_force()
sage: l.sort(); l
[1, 2, 3, 4, 5, 6]
"""
result = set(self.module_generators)
todo = result.copy()
while len(todo) > 0:
x = todo.pop()
for i in self.index_set():
y = x.f(i)
if y == None or y in result:
continue
todo.add(y)
result.add(y)
return list(result)
_list_brute_force = list
