def __init__(self, parent, on_gens, cartan_type=None,
virtualization=None, scaling_factors=None,
gens=None, check=True):
"""
Construct a crystal morphism.
TESTS::
sage: B = crystals.infinity.Tableaux(['B',2])
sage: C = crystals.infinity.NakajimaMonomials(['B',2])
sage: psi = B.crystal_morphism(C.module_generators)
sage: B = crystals.Tableaux(['B',3], shape=[1])
sage: C = crystals.Tableaux(['D',4], shape=[2])
sage: H = Hom(B, C)
sage: psi = H(C.module_generators)
"""
if cartan_type is None:
cartan_type = parent.domain().cartan_type()
if isinstance(on_gens, dict):
gens = on_gens.keys()
I = cartan_type.index_set()
if gens is None:
if cartan_type == parent.domain().cartan_type():
gens = parent.domain().highest_weight_vectors()
else:
gens = tuple(x for x in parent.domain() if x.is_highest_weight(I))
self._hw_gens = True
elif check:
self._hw_gens = all(x.is_highest_weight(I) for x in gens)
else:
self._hw_gens = False
CrystalMorphismByGenerators.__init__(self, parent, on_gens, cartan_type,
virtualization, scaling_factors,
gens, check)
def __init__(self, parent, on_gens, cartan_type=None,
virtualization=None, scaling_factors=None,
gens=None, check=True):
"""
Construct a crystal morphism.
TESTS::
sage: B = crystals.infinity.Tableaux(['B',2])
sage: C = crystals.infinity.NakajimaMonomials(['B',2])
sage: psi = B.crystal_morphism(C.module_generators)
sage: B = crystals.Tableaux(['B',3], shape=[1])
sage: C = crystals.Tableaux(['D',4], shape=[2])
sage: H = Hom(B, C)
sage: psi = H(C.module_generators)
"""
if cartan_type is None:
cartan_type = parent.domain().cartan_type()
if isinstance(on_gens, dict):
gens = on_gens.keys()
I = cartan_type.index_set()
if gens is None:
if cartan_type == parent.domain().cartan_type():
gens = parent.domain().highest_weight_vectors()
else:
gens = tuple(x for x in parent.domain() if x.is_highest_weight(I))
self._hw_gens = True
elif check:
self._hw_gens = all(x.is_highest_weight(I) for x in gens)
else:
self._hw_gens = False
CrystalMorphismByGenerators.__init__(self, parent, on_gens, cartan_type,
virtualization, scaling_factors,
gens, check)
def _call_(self, x):
"""
Return the image of ``x`` under ``self``.
TESTS::
sage: B = crystals.infinity.Tableaux(['B',2])
sage: C = crystals.infinity.NakajimaMonomials(['B',2])
sage: psi = B.crystal_morphism(C.module_generators)
sage: b = B.highest_weight_vector()
sage: psi(b)
1
sage: c = psi(b.f_string([1,1,1,2,2,1,2,2])); c
Y(1,0)^-4 Y(2,0)^4 Y(2,1)^-4
sage: c == C.highest_weight_vector().f_string([1,1,1,2,2,1,2,2])
True
sage: B = crystals.Tableaux(['B',3], shape=[1])
sage: C = crystals.Tableaux(['D',4], shape=[2])
sage: H = Hom(B, C)
sage: psi = H(C.module_generators)
sage: psi(B.module_generators[0])
[[1, 1]]
We check with the morphism defined on the lowest weight vector::
sage: B = crystals.Tableaux(['A',2], shape=[1])
sage: La = RootSystem(['A',2]).weight_lattice().fundamental_weights()
sage: T = crystals.elementary.T(['A',2], La[2])
sage: Bp = T.tensor(B)
sage: C = crystals.Tableaux(['A',2], shape=[2,1])
sage: H = Hom(Bp, C)
sage: x = C.module_generators[0].f_string([1,2])
sage: psi = H({Bp.lowest_weight_vectors()[0]: x})
sage: psi
['A', 2] Crystal morphism:
From: Full tensor product of the crystals
[The T crystal of type ['A', 2] and weight Lambda[2],
The crystal of tableaux of type ['A', 2] and shape(s) [[1]]]
To: The crystal of tableaux of type ['A', 2] and shape(s) [[2, 1]]
Defn: [Lambda[2], [[3]]] |--> [[1, 3], [2]]
sage: psi(Bp.highest_weight_vector())
[[1, 1], [2]]
"""
if not self._hw_gens:
return CrystalMorphismByGenerators._call_(self, x)
mg, path = x.to_highest_weight(self._cartan_type.index_set())
cur = self._on_gens(mg)
for i in reversed(path):
if cur is None:
return None
s = []
sf = self._scaling_factors[i]
for j in self._virtualization[i]:
s += [j] * sf
cur = cur.f_string(s)
return cur
def _call_(self, x):
"""
Return the image of ``x`` under ``self``.
TESTS::
sage: B = crystals.infinity.Tableaux(['B',2])
sage: C = crystals.infinity.NakajimaMonomials(['B',2])
sage: psi = B.crystal_morphism(C.module_generators)
sage: b = B.highest_weight_vector()
sage: psi(b)
1
sage: c = psi(b.f_string([1,1,1,2,2,1,2,2])); c
Y(1,0)^-4 Y(2,0)^4 Y(2,1)^-4
sage: c == C.highest_weight_vector().f_string([1,1,1,2,2,1,2,2])
True
sage: B = crystals.Tableaux(['B',3], shape=[1])
sage: C = crystals.Tableaux(['D',4], shape=[2])
sage: H = Hom(B, C)
sage: psi = H(C.module_generators)
sage: psi(B.module_generators[0])
[[1, 1]]
We check with the morphism defined on the lowest weight vector::
sage: B = crystals.Tableaux(['A',2], shape=[1])
sage: La = RootSystem(['A',2]).weight_lattice().fundamental_weights()
sage: T = crystals.elementary.T(['A',2], La[2])
sage: Bp = T.tensor(B)
sage: C = crystals.Tableaux(['A',2], shape=[2,1])
sage: H = Hom(Bp, C)
sage: x = C.module_generators[0].f_string([1,2])
sage: psi = H({Bp.lowest_weight_vectors()[0]: x})
sage: psi
['A', 2] Crystal morphism:
From: Full tensor product of the crystals
[The T crystal of type ['A', 2] and weight Lambda[2],
The crystal of tableaux of type ['A', 2] and shape(s) [[1]]]
To: The crystal of tableaux of type ['A', 2] and shape(s) [[2, 1]]
Defn: [Lambda[2], [[3]]] |--> [2, 1, 3]
sage: psi(Bp.highest_weight_vector())
[[1, 1], [2]]
"""
if not self._hw_gens:
return CrystalMorphismByGenerators._call_(self, x)
mg, path = x.to_highest_weight(self._cartan_type.index_set())
cur = self._on_gens(mg)
for i in reversed(path):
if cur is None:
return None
s = []
sf = self._scaling_factors[i]
for j in self._virtualization[i]:
s += [j]*sf
cur = cur.f_string(s)
return cur
