class ClassicalCrystalOfAlcovePaths(UniqueRepresentation, Parent):
r"""
Implementation of crystal of alcove paths of the given classical type with
given highest weight, based on the Lenart--Postnikov model [LP2008].
These are highest weight crystals for classical types `A_n`, `B_n`, `C_n`,
`D_n` and the exceptional types `F_4`, `G_2`, `E_6`, `E_7`, `E_8`.
INPUT:
- ``cartan_type`` is the Cartan type of a classical Dynkin diagram
- ``highest_weight`` is a dominant weight as a list of coefficients of
the fundamental weights `Lambda_i`
In this model, a chain of roots is associated to the given highest_weight,
and then the elements of the crystal are indexed by "admissible subsets"
which indicate "folding positions" along the chain of roots. See [LP2008]
for details.
TODO:
- Resolve speed issues; `E_6(\Lambda_1)` takes just under 4 minutes to list().
To construct the highest-weight node takes 15 sec for `E_6(\Lambda_4)`.
The initial chain has 42 roots.
TESTS:
The following example appears in Figure 2 of [LP2008]::
sage: C = ClassicalCrystalOfAlcovePaths(['G',2],[0,1]);
sage: G = C.digraph()
sage: GG= DiGraph({
... () : {(0) : 2 },
... (0) : {(0,8) : 1 },
... (0,1) : {(0,1,7) : 2 },
... (0,1,2) : {(0,1,2,9) : 1 },
... (0,1,2,3) : {(0,1,2,3,4) : 2 },
... (0,1,2,6) : {(0,1,2,3) : 1 },
... (0,1,2,9) : {(0,1,2,6) : 1 },
... (0,1,7) : {(0,1,2) : 2 },
... (0,1,7,9) : {(0,1,2,9) : 2 },
... (0,5) : {(0,1) : 1, (0,5,7) : 2 },
... (0,5,7) : {(0,5,7,9) : 1 },
... (0,5,7,9) : {(0,1,7,9) : 1 },
... (0,8) : {(0,5) : 1 },
... })
sage: G.is_isomorphic(GG)
True
sage: for edge in sorted([(u.value, v.value, i) for (u,v,i) in G.edges()]):
... print edge
([], [0], 2)
([0], [0, 8], 1)
([0, 1], [0, 1, 7], 2)
([0, 1, 2], [0, 1, 2, 9], 1)
([0, 1, 2, 3], [0, 1, 2, 3, 4], 2)
([0, 1, 2, 6], [0, 1, 2, 3], 1)
([0, 1, 2, 9], [0, 1, 2, 6], 1)
([0, 1, 7], [0, 1, 2], 2)
([0, 1, 7, 9], [0, 1, 2, 9], 2)
([0, 5], [0, 1], 1)
([0, 5], [0, 5, 7], 2)
([0, 5, 7], [0, 5, 7, 9], 1)
([0, 5, 7, 9], [0, 1, 7, 9], 1)
([0, 8], [0, 5], 1)
REFERENCES:
.. [LP2008] C. Lenart and A. Postnikov. A combinatorial model for crystals of Kac-Moody algebras. Trans. Amer. Math. Soc. 360 (2008), 4349-4381.
"""
@staticmethod
def __classcall__(cls, cartan_type, highest_weight):
"""
cartan_type and heighest_weight are lists, which are mutable, this
causes a problem for class UniqueRepresentation, the following code
fixes this problem.
EXAMPLES::
sage: ClassicalCrystalOfAlcovePaths.__classcall__(ClassicalCrystalOfAlcovePaths,['A',3],[0,1,0])
<class 'sage.combinat.crystals.alcove_path.ClassicalCrystalOfAlcovePaths_with_category'>
"""
cartan_type = CartanType(cartan_type)
highest_weight = tuple(highest_weight)
return super(ClassicalCrystalOfAlcovePaths, cls).__classcall__(cls, cartan_type, highest_weight)
def __init__(self, cartan_type, highest_weight):
"""
EXAMPLES::
sage: C = ClassicalCrystalOfAlcovePaths(['A',3],[1,0,0])
sage: C.list()
[[], [0], [0, 1], [0, 1, 2]]
sage: TestSuite(C).run()
"""
Parent.__init__(self, category = ClassicalCrystals())
self._cartan_type = CartanType(cartan_type)
self._name = "The crystal of alcove paths for type %s"%cartan_type
self.chain_cache = {}
self.endweight_cache = {}
self.R = RootSystem(cartan_type)
alpha = self.R.root_space().simple_roots()
Lambda = self.R.weight_space().basis()
self.positive_roots = sorted(self.R.root_space().positive_roots());
self.weight = Lambda[Integer(1)] - Lambda[Integer(1)]
offset = self.R.index_set()[Integer(0)]
for j in self.R.index_set():
self.weight = self.weight + highest_weight[j-offset]*Lambda[j]
self.initial_element = self([])
self.initial_element.chain = self.get_initial_chain(self.weight)
rho = (Integer(1)/Integer(2))*sum(self.positive_roots)
self.initial_element.endweight = rho
self.chain_cache[ str([]) ] = self.initial_element.chain
self.endweight_cache[ str([]) ] = self.initial_element.endweight
self.module_generators = [self.initial_element]
self._list = super(ClassicalCrystalOfAlcovePaths, self).list()
self._digraph = super(ClassicalCrystalOfAlcovePaths, self).digraph()
self._digraph_closure = self.digraph().transitive_closure()
def get_initial_chain(self, highest_weight):
"""
Called internally by __init__() to construct the chain of roots
associated to the highest weight element.
EXAMPLES::
sage: C = ClassicalCrystalOfAlcovePaths(['A',3],[0,1,0])
sage: C.get_initial_chain(RootSystem(['A',3]).weight_space().basis()[1])
[[alpha[1], alpha[1]], [alpha[1] + alpha[2], alpha[1] + alpha[2]], [alpha[1] + alpha[2] + alpha[3], alpha[1] + alpha[2] + alpha[3]]]
"""
pos_roots = self.positive_roots
tis = self.R.index_set()
tis.reverse()
cv_to_pos_root = {}
to_sort = []
for r in pos_roots:
j = highest_weight.scalar( r.associated_coroot() )
if (int(math.floor(j)) - j == Integer(0)):
j = j - Integer(1)
j = int(math.floor(j))
for k in (ellipsis_range(Integer(0),Ellipsis,j)):
cv = []
cv.append((Integer(1)/highest_weight.scalar(r.associated_coroot()))*k)
for i in tis:
cv.append((Integer(1)/highest_weight.scalar(r.associated_coroot()))*r.associated_coroot().coefficient(i))
cv_to_pos_root[ str(cv) ] = r
to_sort.append( cv )
to_sort.sort() # Note: Python sorts nested lists lexicographically by default.
lambda_chain = []
for t in to_sort:
lambda_chain.append( [ cv_to_pos_root[str(t)], cv_to_pos_root[str(t)] ] )
return lambda_chain
def _element_constructor_(self, value):
"""
Coerces value into self.
EXAMPLES::
sage: C = ClassicalCrystalOfAlcovePaths(['A',3],[1,0,0])
sage: C.module_generators
[[]]
sage: C([]).e(1)
sage: C([]).f(1)
[0]
"""
return self.element_class(self, value)
def list(self):
"""
Returns a list of the elements of self.
.. warning::
This can be slow!
EXAMPLES::
sage: C = ClassicalCrystalOfAlcovePaths(['A',3],[0,1,0])
sage: C.list()
[[], [0], [0, 1], [0, 2], [0, 1, 2], [0, 1, 2, 3]]
"""
return self._list
def digraph(self):
"""
Returns the directed graph associated to self.
EXAMPLES::
sage: C = ClassicalCrystalOfAlcovePaths(['A',3],[0,1,0])
sage: C.digraph().vertices()
[[], [0], [0, 1], [0, 2], [0, 1, 2], [0, 1, 2, 3]]
sage: C.digraph().edges()
[([], [0], 2), ([0], [0, 1], 1), ([0], [0, 2], 3), ([0, 1], [0, 1, 2], 3), ([0, 2], [0, 1, 2], 1), ([0, 1, 2], [0, 1, 2, 3], 2)]
"""
return self._digraph
def lt_elements(self, x, y):
r"""
Returns True if and only if there is a path
from x to y in the crystal graph.
Because the crystal graph is classical, it is a directed
acyclic graph which can be interpreted as a poset. This
function implements the comparison function of this poset.
EXAMPLES::
sage: C = ClassicalCrystalOfAlcovePaths(['A',3],[0,1,0])
sage: x = C([])
sage: y = C([0,2])
sage: C.lt_elements(x,y)
True
sage: C.lt_elements(y,x)
False
sage: C.lt_elements(x,x)
False
"""
assert x.parent() == self and y.parent() == self
if self._digraph_closure.has_edge(x,y):
return True
return False
class ClassicalCrystalOfAlcovePaths(UniqueRepresentation, Parent):
r"""
Implementation of crystal of alcove paths of the given classical type with
given highest weight, based on the Lenart--Postnikov model [LP2008].
These are highest weight crystals for classical types `A_n`, `B_n`, `C_n`,
`D_n` and the exceptional types `F_4`, `G_2`, `E_6`, `E_7`, `E_8`.
INPUT:
- ``cartan_type`` is the Cartan type of a classical Dynkin diagram
- ``highest_weight`` is a dominant weight as a list of coefficients of
the fundamental weights `Lambda_i`
In this model, a chain of roots is associated to the given highest_weight,
and then the elements of the crystal are indexed by "admissible subsets"
which indicate "folding positions" along the chain of roots. See [LP2008]
for details.
TODO:
- Resolve speed issues; `E_6(\Lambda_1)` takes just under 4 minutes to list().
To construct the highest-weight node takes 15 sec for `E_6(\Lambda_4)`.
The initial chain has 42 roots.
TESTS:
The following example appears in Figure 2 of [LP2008]::
sage: C = ClassicalCrystalOfAlcovePaths(['G',2],[0,1]);
sage: G = C.digraph()
sage: GG= DiGraph({
... () : {(0) : 2 },
... (0) : {(0,8) : 1 },
... (0,1) : {(0,1,7) : 2 },
... (0,1,2) : {(0,1,2,9) : 1 },
... (0,1,2,3) : {(0,1,2,3,4) : 2 },
... (0,1,2,6) : {(0,1,2,3) : 1 },
... (0,1,2,9) : {(0,1,2,6) : 1 },
... (0,1,7) : {(0,1,2) : 2 },
... (0,1,7,9) : {(0,1,2,9) : 2 },
... (0,5) : {(0,1) : 1, (0,5,7) : 2 },
... (0,5,7) : {(0,5,7,9) : 1 },
... (0,5,7,9) : {(0,1,7,9) : 1 },
... (0,8) : {(0,5) : 1 },
... })
sage: G.is_isomorphic(GG)
True
sage: for edge in sorted([(u.value, v.value, i) for (u,v,i) in G.edges()]):
... print edge
([], [0], 2)
([0], [0, 8], 1)
([0, 1], [0, 1, 7], 2)
([0, 1, 2], [0, 1, 2, 9], 1)
([0, 1, 2, 3], [0, 1, 2, 3, 4], 2)
([0, 1, 2, 6], [0, 1, 2, 3], 1)
([0, 1, 2, 9], [0, 1, 2, 6], 1)
([0, 1, 7], [0, 1, 2], 2)
([0, 1, 7, 9], [0, 1, 2, 9], 2)
([0, 5], [0, 1], 1)
([0, 5], [0, 5, 7], 2)
([0, 5, 7], [0, 5, 7, 9], 1)
([0, 5, 7, 9], [0, 1, 7, 9], 1)
([0, 8], [0, 5], 1)
REFERENCES:
.. [LP2008] C. Lenart and A. Postnikov. A combinatorial model for crystals of Kac-Moody algebras. Trans. Amer. Math. Soc. 360 (2008), 4349-4381.
"""
@staticmethod
def __classcall__(cls, cartan_type, highest_weight):
"""
cartan_type and heighest_weight are lists, which are mutable, this
causes a problem for class UniqueRepresentation, the following code
fixes this problem.
EXAMPLES::
sage: ClassicalCrystalOfAlcovePaths.__classcall__(ClassicalCrystalOfAlcovePaths,['A',3],[0,1,0])
<class 'sage.combinat.crystals.alcove_path.ClassicalCrystalOfAlcovePaths_with_category'>
"""
cartan_type = CartanType(cartan_type)
highest_weight = tuple(highest_weight)
return super(ClassicalCrystalOfAlcovePaths,
cls).__classcall__(cls, cartan_type, highest_weight)
def __init__(self, cartan_type, highest_weight):
"""
EXAMPLES::
sage: C = ClassicalCrystalOfAlcovePaths(['A',3],[1,0,0])
sage: C.list()
[[], [0], [0, 1], [0, 1, 2]]
sage: TestSuite(C).run()
"""
Parent.__init__(self, category=ClassicalCrystals())
self._cartan_type = CartanType(cartan_type)
self._name = "The crystal of alcove paths for type %s" % cartan_type
self.chain_cache = {}
self.endweight_cache = {}
self.R = RootSystem(cartan_type)
alpha = self.R.root_space().simple_roots()
Lambda = self.R.weight_space().basis()
self.positive_roots = sorted(self.R.root_space().positive_roots())
self.weight = Lambda[Integer(1)] - Lambda[Integer(1)]
offset = self.R.index_set()[Integer(0)]
for j in self.R.index_set():
self.weight = self.weight + highest_weight[j - offset] * Lambda[j]
self.initial_element = self([])
self.initial_element.chain = self.get_initial_chain(self.weight)
rho = (Integer(1) / Integer(2)) * sum(self.positive_roots)
self.initial_element.endweight = rho
self.chain_cache[str([])] = self.initial_element.chain
self.endweight_cache[str([])] = self.initial_element.endweight
self.module_generators = [self.initial_element]
self._list = super(ClassicalCrystalOfAlcovePaths, self).list()
self._digraph = super(ClassicalCrystalOfAlcovePaths, self).digraph()
self._digraph_closure = self.digraph().transitive_closure()
def get_initial_chain(self, highest_weight):
"""
Called internally by __init__() to construct the chain of roots
associated to the highest weight element.
EXAMPLES::
sage: C = ClassicalCrystalOfAlcovePaths(['A',3],[0,1,0])
sage: C.get_initial_chain(RootSystem(['A',3]).weight_space().basis()[1])
[[alpha[1], alpha[1]], [alpha[1] + alpha[2], alpha[1] + alpha[2]], [alpha[1] + alpha[2] + alpha[3], alpha[1] + alpha[2] + alpha[3]]]
"""
pos_roots = self.positive_roots
tis = self.R.index_set()
tis.reverse()
cv_to_pos_root = {}
to_sort = []
for r in pos_roots:
j = highest_weight.scalar(r.associated_coroot())
if (int(math.floor(j)) - j == Integer(0)):
j = j - Integer(1)
j = int(math.floor(j))
for k in (ellipsis_range(Integer(0), Ellipsis, j)):
cv = []
cv.append(
(Integer(1) / highest_weight.scalar(r.associated_coroot()))
* k)
for i in tis:
cv.append((Integer(1) /
highest_weight.scalar(r.associated_coroot())) *
r.associated_coroot().coefficient(i))
cv_to_pos_root[str(cv)] = r
to_sort.append(cv)
to_sort.sort(
) # Note: Python sorts nested lists lexicographically by default.
lambda_chain = []
for t in to_sort:
lambda_chain.append(
[cv_to_pos_root[str(t)], cv_to_pos_root[str(t)]])
return lambda_chain
def _element_constructor_(self, value):
"""
Coerces value into self.
EXAMPLES::
sage: C = ClassicalCrystalOfAlcovePaths(['A',3],[1,0,0])
sage: C.module_generators
[[]]
sage: C([]).e(1)
sage: C([]).f(1)
[0]
"""
return self.element_class(self, value)
def list(self):
"""
Returns a list of the elements of self.
.. warning::
This can be slow!
EXAMPLES::
sage: C = ClassicalCrystalOfAlcovePaths(['A',3],[0,1,0])
sage: C.list()
[[], [0], [0, 1], [0, 2], [0, 1, 2], [0, 1, 2, 3]]
"""
return self._list
def digraph(self):
"""
Returns the directed graph associated to self.
EXAMPLES::
sage: C = ClassicalCrystalOfAlcovePaths(['A',3],[0,1,0])
sage: C.digraph().vertices()
[[], [0], [0, 1], [0, 2], [0, 1, 2], [0, 1, 2, 3]]
sage: C.digraph().edges()
[([], [0], 2), ([0], [0, 1], 1), ([0], [0, 2], 3), ([0, 1], [0, 1, 2], 3), ([0, 2], [0, 1, 2], 1), ([0, 1, 2], [0, 1, 2, 3], 2)]
"""
return self._digraph
def lt_elements(self, x, y):
r"""
Returns True if and only if there is a path
from x to y in the crystal graph.
Because the crystal graph is classical, it is a directed
acyclic graph which can be interpreted as a poset. This
function implements the comparison function of this poset.
EXAMPLES::
sage: C = ClassicalCrystalOfAlcovePaths(['A',3],[0,1,0])
sage: x = C([])
sage: y = C([0,2])
sage: C.lt_elements(x,y)
True
sage: C.lt_elements(y,x)
False
sage: C.lt_elements(x,x)
False
"""
assert x.parent() == self and y.parent() == self
if self._digraph_closure.has_edge(x, y):
return True
return False
