def __init__(self, cartan_type, la, mu):
"""
Initialize ``self``.
TESTS::
sage: K = crystals.KacModule(['A', [2,1]], [2,1], [1])
sage: TestSuite(K).run()
"""
self._cartan_type = cartan_type
self._la = la
self._mu = mu
Parent.__init__(self, category=RegularSuperCrystals())
self._S = CrystalOfOddNegativeRoots(self._cartan_type)
self._dual = CrystalOfTableaux(['A', self._cartan_type.m], shape=la)
self._reg = CrystalOfTableaux(['A', self._cartan_type.n], shape=mu)
data = (self._S.module_generators[0], self._dual.module_generators[0],
self._reg.module_generators[0])
self.module_generators = (self.element_class(self, data), )
def example(self):
"""
Return an example of a quantum group representation as per
:meth:`Category.example <sage.categories.category.Category.example>`.
EXAMPLES::
sage: from sage.categories.quantum_group_representations import QuantumGroupRepresentations
sage: Cat = QuantumGroupRepresentations(ZZ['q'].fraction_field())
sage: Cat.example()
V((2, 1, 0))
"""
from sage.algebras.quantum_groups.representations import AdjointRepresentation
from sage.combinat.crystals.tensor_product import CrystalOfTableaux
T = CrystalOfTableaux(['A', 2], shape=[2, 1])
return AdjointRepresentation(self.base_ring(), T)
def _tableaux_isomorphism(self):
"""
Return the isomorphism from ``self`` to the tableaux model.
EXAMPLES::
sage: W = WeylGroup(['A',3,1], prefix='s')
sage: w = W.from_reduced_word([3,2,1])
sage: B = crystals.AffineFactorization(w,4)
sage: B._tableaux_isomorphism
['A', 3] Crystal morphism:
From: Crystal on affine factorizations of type A3 associated to s3*s2*s1
To: The crystal of tableaux of type ['A', 3] and shape(s) [[3]]
sage: W = WeylGroup(['A',3,1], prefix='s')
sage: w = W.from_reduced_word([2,1,3,2])
sage: B = crystals.AffineFactorization(w,3)
sage: B._tableaux_isomorphism
['A', 2] Crystal morphism:
From: Crystal on affine factorizations of type A2 associated to s2*s3*s1*s2
To: The crystal of tableaux of type ['A', 2] and shape(s) [[2, 2]]
"""
# Constructing the tableaux crystal
from sage.combinat.crystals.tensor_product import CrystalOfTableaux
def mg_to_shape(mg):
l = list(mg.weight().to_vector())
while l and l[-1] == 0:
l.pop()
return l
sh = [mg_to_shape(mg) for mg in self.highest_weight_vectors()]
C = CrystalOfTableaux(self.cartan_type(), shapes=sh)
phi = FactorizationToTableaux(Hom(self, C, category=self.category()))
phi.register_as_coercion()
return phi
def HighestWeightCrystal(dominant_weight, model=None):
r"""
Return the highest weight crystal of highest weight ``dominant_weight``
of the given ``model``.
INPUT:
- ``dominant_weight`` -- a dominant weight
- ``model`` -- (optional) if not specified, then we have the following
default models:
* types `A_n, B_n, C_n, D_n, G_2` - :class:`tableaux
<sage.combinat.crystals.tensor_product.CrystalOfTableaux>`
* types `E_{6,7}` - :class:`type E finite dimensional crystal
<FiniteDimensionalHighestWeightCrystal_TypeE>`
* all other types - :class:`LS paths
<sage.combinat.crystals.littelmann_path.CrystalOfLSPaths>`
otherwise can be one of the following:
* ``'Tableaux'`` - :class:`KN tableaux
<sage.combinat.crystals.tensor_product.CrystalOfTableaux>`
* ``'TypeE'`` - :class:`type E finite dimensional crystal
<FiniteDimensionalHighestWeightCrystal_TypeE>`
* ``'NakajimaMonomials'`` - :class:`Nakajima monomials
<sage.combinat.crystals.monomial_crystals.CrystalOfNakajimaMonomials>`
* ``'LSPaths'`` - :class:`LS paths
<sage.combinat.crystals.littelmann_path.CrystalOfLSPaths>`
* ``'AlcovePaths'`` - :class:`alcove paths
<sage.combinat.crystals.alcove_path.CrystalOfAlcovePaths>`
* ``'GeneralizedYoungWalls'`` - :class:`generalized Young walls
<sage.combinat.crystals.generalized_young_walls.CrystalOfGeneralizedYoungWalls>`
* ``'RiggedConfigurations'`` - :class:`rigged configurations
<sage.combinat.rigged_configurations.rc_crystal.CrystalOfRiggedConfigurations>`
EXAMPLES::
sage: La = RootSystem(['A',2]).weight_lattice().fundamental_weights()
sage: wt = La[1] + La[2]
sage: crystals.HighestWeight(wt)
The crystal of tableaux of type ['A', 2] and shape(s) [[2, 1]]
sage: La = RootSystem(['C',2]).weight_lattice().fundamental_weights()
sage: wt = 5*La[1] + La[2]
sage: crystals.HighestWeight(wt)
The crystal of tableaux of type ['C', 2] and shape(s) [[6, 1]]
Some type `E` examples::
sage: C = CartanType(['E',6])
sage: La = C.root_system().weight_lattice().fundamental_weights()
sage: T = crystals.HighestWeight(La[1])
sage: T.cardinality()
27
sage: T = crystals.HighestWeight(La[6])
sage: T.cardinality()
27
sage: T = crystals.HighestWeight(La[2])
sage: T.cardinality()
78
sage: T = crystals.HighestWeight(La[4])
sage: T.cardinality()
2925
sage: T = crystals.HighestWeight(La[3])
sage: T.cardinality()
351
sage: T = crystals.HighestWeight(La[5])
sage: T.cardinality()
351
sage: C = CartanType(['E',7])
sage: La = C.root_system().weight_lattice().fundamental_weights()
sage: T = crystals.HighestWeight(La[1])
sage: T.cardinality()
133
sage: T = crystals.HighestWeight(La[2])
sage: T.cardinality()
912
sage: T = crystals.HighestWeight(La[3])
sage: T.cardinality()
8645
sage: T = crystals.HighestWeight(La[4])
sage: T.cardinality()
365750
sage: T = crystals.HighestWeight(La[5])
sage: T.cardinality()
27664
sage: T = crystals.HighestWeight(La[6])
sage: T.cardinality()
1539
sage: T = crystals.HighestWeight(La[7])
sage: T.cardinality()
56
An example with an affine type::
sage: C = CartanType(['C',2,1])
sage: La = C.root_system().weight_lattice().fundamental_weights()
sage: T = crystals.HighestWeight(La[1])
sage: sorted(T.subcrystal(max_depth=3), key=str)
[(-Lambda[0] + 3*Lambda[1] - Lambda[2] - delta,),
(-Lambda[0] + Lambda[1] + Lambda[2] - delta,),
(-Lambda[1] + 2*Lambda[2] - delta,),
(2*Lambda[0] - Lambda[1],),
(Lambda[0] + Lambda[1] - Lambda[2],),
(Lambda[0] - Lambda[1] + Lambda[2],),
(Lambda[1],)]
Using the various models::
sage: La = RootSystem(['F',4]).weight_lattice().fundamental_weights()
sage: wt = La[1] + La[4]
sage: crystals.HighestWeight(wt)
The crystal of LS paths of type ['F', 4] and weight Lambda[1] + Lambda[4]
sage: crystals.HighestWeight(wt, model='NakajimaMonomials')
Highest weight crystal of modified Nakajima monomials of
Cartan type ['F', 4] and highest weight Lambda[1] + Lambda[4]
sage: crystals.HighestWeight(wt, model='AlcovePaths')
Highest weight crystal of alcove paths of type ['F', 4] and weight Lambda[1] + Lambda[4]
sage: crystals.HighestWeight(wt, model='RiggedConfigurations')
Crystal of rigged configurations of type ['F', 4] and weight Lambda[1] + Lambda[4]
sage: La = RootSystem(['A',3,1]).weight_lattice().fundamental_weights()
sage: wt = La[0] + La[2]
sage: crystals.HighestWeight(wt, model='GeneralizedYoungWalls')
Highest weight crystal of generalized Young walls of
Cartan type ['A', 3, 1] and highest weight Lambda[0] + Lambda[2]
"""
cartan_type = dominant_weight.parent().cartan_type()
if model is None:
if cartan_type.is_finite():
if cartan_type.type() == 'E':
model = 'TypeE'
elif cartan_type.type() in ['A','B','C','D','G']:
model = 'Tableaux'
else:
model = 'LSPaths'
else:
model = 'LSPaths'
if model == 'Tableaux':
sh = sum([[i]*c for i,c in dominant_weight], [])
sh = Partition(reversed(sh))
return CrystalOfTableaux(cartan_type, shape=sh.conjugate())
if model == 'TypeE':
if not cartan_type.is_finite() or cartan_type.type() != 'E':
raise ValueError("only for finite type E")
if cartan_type.rank() == 6:
return FiniteDimensionalHighestWeightCrystal_TypeE6(dominant_weight)
elif cartan_type.rank() == 7:
return FiniteDimensionalHighestWeightCrystal_TypeE7(dominant_weight)
raise NotImplementedError
if model == 'NakajimaMonomials':
# Make sure it's in the weight lattice
P = dominant_weight.parent().root_system.weight_lattice()
wt = P.sum_of_terms((i, c) for i,c in dominant_weight)
return CrystalOfNakajimaMonomials(cartan_type, wt)
if model == 'LSPaths':
# Make sure it's in the (extended) weight space
if cartan_type.is_affine():
P = dominant_weight.parent().root_system.weight_space(extended=True)
else:
P = dominant_weight.parent().root_system.weight_space()
wt = P.sum_of_terms((i, c) for i,c in dominant_weight)
return CrystalOfLSPaths(wt)
if model == 'AlcovePaths':
# Make sure it's in the weight space
P = dominant_weight.parent().root_system.weight_space()
wt = P.sum_of_terms((i, c) for i,c in dominant_weight)
return CrystalOfAlcovePaths(wt, highest_weight_crystal=True)
if model == 'GeneralizedYoungWalls':
if not cartan_type.is_affine():
raise ValueError("only for affine types")
if cartan_type.type() != 'A':
raise NotImplementedError("only for affine type A")
# Make sure it's in the weight lattice
P = dominant_weight.parent().root_system.weight_lattice(extended=True)
wt = P.sum_of_terms((i, c) for i,c in dominant_weight)
return CrystalOfGeneralizedYoungWalls(cartan_type.rank()-1, wt)
if model == 'RiggedConfigurations':
# Make sure it's in the weight lattice
P = dominant_weight.parent().root_system.weight_lattice()
wt = P.sum_of_terms((i, c) for i,c in dominant_weight)
return CrystalOfRiggedConfigurations(cartan_type, wt)
raise ValueError("invalid model")
def HighestWeightCrystal(dominant_weight, model=None):
r"""
Return the highest weight crystal of highest weight ``dominant_weight``
of the given ``model``.
INPUT:
- ``dominant_weight`` -- a dominant weight
- ``model`` -- (optional) if not specified, then we have the following
default models:
* types `A_n, B_n, C_n, D_n, G_2` - :class:`tableaux
<sage.combinat.crystals.tensor_product.CrystalOfTableaux>`
* types `E_{6,7}` - :class:`type E finite dimensional crystal
<FiniteDimensionalHighestWeightCrystal_TypeE>`
* all other types - :class:`LS paths
<sage.combinat.crystals.littelmann_path.CrystalOfLSPaths>`
otherwise can be one of the following:
* ``'Tableaux'`` - :class:`KN tableaux
<sage.combinat.crystals.tensor_product.CrystalOfTableaux>`
* ``'TypeE'`` - :class:`type E finite dimensional crystal
<FiniteDimensionalHighestWeightCrystal_TypeE>`
* ``'NakajimaMonomials'`` - :class:`Nakajima monomials
<sage.combinat.crystals.monomial_crystals.CrystalOfNakajimaMonomials>`
* ``'LSPaths'`` - :class:`LS paths
<sage.combinat.crystals.littelmann_path.CrystalOfLSPaths>`
* ``'AlcovePaths'`` - :class:`alcove paths
<sage.combinat.crystals.alcove_path.CrystalOfAlcovePaths>`
* ``'GeneralizedYoungWalls'`` - :class:`generalized Young walls
<sage.combinat.crystals.generalized_young_walls.CrystalOfGeneralizedYoungWalls>`
* ``'RiggedConfigurations'`` - :class:`rigged configurations
<sage.combinat.rigged_configurations.rc_crystal.CrystalOfRiggedConfigurations>`
EXAMPLES::
sage: La = RootSystem(['A',2]).weight_lattice().fundamental_weights()
sage: wt = La[1] + La[2]
sage: crystals.HighestWeight(wt)
The crystal of tableaux of type ['A', 2] and shape(s) [[2, 1]]
sage: La = RootSystem(['C',2]).weight_lattice().fundamental_weights()
sage: wt = 5*La[1] + La[2]
sage: crystals.HighestWeight(wt)
The crystal of tableaux of type ['C', 2] and shape(s) [[6, 1]]
sage: La = RootSystem(['B',2]).weight_lattice().fundamental_weights()
sage: wt = La[1] + La[2]
sage: crystals.HighestWeight(wt)
The crystal of tableaux of type ['B', 2] and shape(s) [[3/2, 1/2]]
Some type `E` examples::
sage: C = CartanType(['E',6])
sage: La = C.root_system().weight_lattice().fundamental_weights()
sage: T = crystals.HighestWeight(La[1])
sage: T.cardinality()
27
sage: T = crystals.HighestWeight(La[6])
sage: T.cardinality()
27
sage: T = crystals.HighestWeight(La[2])
sage: T.cardinality()
78
sage: T = crystals.HighestWeight(La[4])
sage: T.cardinality()
2925
sage: T = crystals.HighestWeight(La[3])
sage: T.cardinality()
351
sage: T = crystals.HighestWeight(La[5])
sage: T.cardinality()
351
sage: C = CartanType(['E',7])
sage: La = C.root_system().weight_lattice().fundamental_weights()
sage: T = crystals.HighestWeight(La[1])
sage: T.cardinality()
133
sage: T = crystals.HighestWeight(La[2])
sage: T.cardinality()
912
sage: T = crystals.HighestWeight(La[3])
sage: T.cardinality()
8645
sage: T = crystals.HighestWeight(La[4])
sage: T.cardinality()
365750
sage: T = crystals.HighestWeight(La[5])
sage: T.cardinality()
27664
sage: T = crystals.HighestWeight(La[6])
sage: T.cardinality()
1539
sage: T = crystals.HighestWeight(La[7])
sage: T.cardinality()
56
An example with an affine type::
sage: C = CartanType(['C',2,1])
sage: La = C.root_system().weight_lattice().fundamental_weights()
sage: T = crystals.HighestWeight(La[1])
sage: sorted(T.subcrystal(max_depth=3), key=str)
[(-Lambda[0] + 3*Lambda[1] - Lambda[2] - delta,),
(-Lambda[0] + Lambda[1] + Lambda[2] - delta,),
(-Lambda[1] + 2*Lambda[2] - delta,),
(2*Lambda[0] - Lambda[1],),
(Lambda[0] + Lambda[1] - Lambda[2],),
(Lambda[0] - Lambda[1] + Lambda[2],),
(Lambda[1],)]
Using the various models::
sage: La = RootSystem(['F',4]).weight_lattice().fundamental_weights()
sage: wt = La[1] + La[4]
sage: crystals.HighestWeight(wt)
The crystal of LS paths of type ['F', 4] and weight Lambda[1] + Lambda[4]
sage: crystals.HighestWeight(wt, model='NakajimaMonomials')
Highest weight crystal of modified Nakajima monomials of
Cartan type ['F', 4] and highest weight Lambda[1] + Lambda[4]
sage: crystals.HighestWeight(wt, model='AlcovePaths')
Highest weight crystal of alcove paths of type ['F', 4] and weight Lambda[1] + Lambda[4]
sage: crystals.HighestWeight(wt, model='RiggedConfigurations')
Crystal of rigged configurations of type ['F', 4] and weight Lambda[1] + Lambda[4]
sage: La = RootSystem(['A',3,1]).weight_lattice().fundamental_weights()
sage: wt = La[0] + La[2]
sage: crystals.HighestWeight(wt, model='GeneralizedYoungWalls')
Highest weight crystal of generalized Young walls of
Cartan type ['A', 3, 1] and highest weight Lambda[0] + Lambda[2]
TESTS:
Check that the correct crystal is constructed for the fundamental weights::
sage: for ct in CartanType.samples(finite=True, crystallographic=True):
....: L = ct.root_system().weight_lattice()
....: La = L.fundamental_weights()
....: for model in ['Tableaux', 'NakajimaMonomials', 'AlcovePaths', 'RiggedConfigurations']:
....: if model == 'Tableaux' and ct.type() in ["E", "F"]:
....: continue
....: for wt in La:
....: C = crystals.HighestWeight(wt, model=model)
....: assert L.weyl_dimension(wt) == C.cardinality(), "wrong cardinality in %s, weight %s" % (ct, wt)
....: assert C.highest_weight_vector().weight() == wt, "wrong weight in %s, weight %s" % (ct, wt)
Same thing for weights constructed from the simple roots::
sage: for ct in CartanType.samples(finite=True, crystallographic=True):
....: L = ct.root_system().root_space()
....: La = L.fundamental_weights_from_simple_roots()
....: for model in ['Tableaux', 'NakajimaMonomials', 'AlcovePaths', 'RiggedConfigurations']:
....: if model == 'Tableaux' and ct.type() in ["E", "F"]:
....: continue
....: for wt in La:
....: C1 = crystals.HighestWeight(wt.to_ambient().to_weight_space(ZZ), model=model)
....: C2 = crystals.HighestWeight(wt, model=model)
....: assert C1 == C2
"""
cartan_type = dominant_weight.parent().cartan_type()
if model is None:
if cartan_type.is_finite():
if cartan_type.type() == 'E':
model = 'TypeE'
elif cartan_type.type() in ['A', 'B', 'C', 'D', 'G']:
model = 'Tableaux'
else:
model = 'LSPaths'
else:
model = 'LSPaths'
# Make sure dominant_weight in the weight space
if cartan_type.is_finite():
dominant_weight = dominant_weight.to_ambient().to_weight_space(ZZ)
if model == 'Tableaux':
# we rely on the specific choice of positive roots here
# except in type G_2, the fundamental weights are realized by
# vectors with weakly decreasing nonnegative integer (or in
# type B_n and D_n, half-integer) entries
sh = dominant_weight.to_ambient().to_vector()
if cartan_type.type() == "G":
sh = (-sh)[2:0:-1]
return CrystalOfTableaux(cartan_type, shape=sh)
if model == 'TypeE':
if not cartan_type.is_finite() or cartan_type.type() != 'E':
raise ValueError("only for finite type E")
if cartan_type.rank() == 6:
return FiniteDimensionalHighestWeightCrystal_TypeE6(
dominant_weight)
elif cartan_type.rank() == 7:
return FiniteDimensionalHighestWeightCrystal_TypeE7(
dominant_weight)
raise NotImplementedError
if model == 'NakajimaMonomials':
return CrystalOfNakajimaMonomials(cartan_type, dominant_weight)
if model == 'LSPaths':
# Make sure it's in the (extended) weight space
if cartan_type.is_affine():
P = dominant_weight.parent().root_system.weight_space(
extended=True)
else:
P = dominant_weight.parent().root_system.weight_space()
wt = P.sum_of_terms((i, c) for i, c in dominant_weight)
return CrystalOfLSPaths(wt)
if model == 'AlcovePaths':
return CrystalOfAlcovePaths(dominant_weight,
highest_weight_crystal=True)
if model == 'GeneralizedYoungWalls':
if not cartan_type.is_affine():
raise ValueError("only for affine types")
if cartan_type.type() != 'A':
raise NotImplementedError("only for affine type A")
# Make sure it's in the weight lattice
P = dominant_weight.parent().root_system.weight_lattice(extended=True)
wt = P.sum_of_terms((i, c) for i, c in dominant_weight)
return CrystalOfGeneralizedYoungWalls(cartan_type.rank() - 1, wt)
if model == 'RiggedConfigurations':
return CrystalOfRiggedConfigurations(cartan_type, dominant_weight)
raise ValueError("invalid model")