def __init__(self, dominant_weight):
"""
EXAMPLES::
sage: C=CartanType(['E',6])
sage: La=C.root_system().weight_lattice().fundamental_weights()
sage: p2=2*La[2]
sage: p1=La[2]
sage: p0=0*La[2]
sage: T = crystals.HighestWeight(0*La[2])
sage: T.cardinality()
1
sage: T = crystals.HighestWeight(La[2])
sage: T.cardinality()
78
sage: T = crystals.HighestWeight(2*La[2])
sage: T.cardinality()
2430
"""
B1 = CrystalOfLetters(['E',6])
B6 = CrystalOfLetters(['E',6], dual = True)
self.column_crystal = {1 : B1, 6 : B6,
4 : TensorProductOfCrystals(B1,B1,B1,generators=[[B1([-3,4]),B1([-1,3]),B1([1])]]),
3 : TensorProductOfCrystals(B1,B1,generators=[[B1([-1,3]),B1([1])]]),
5 : TensorProductOfCrystals(B6,B6,generators=[[B6([5,-6]),B6([6])]]),
2 : TensorProductOfCrystals(B6,B1,generators=[[B6([2,-1]),B1([1])]])}
FiniteDimensionalHighestWeightCrystal_TypeE.__init__(self, dominant_weight)
def __init__(self, dominant_weight):
"""
EXAMPLES::
sage: C=CartanType(['E',7])
sage: La=C.root_system().weight_lattice().fundamental_weights()
sage: T = crystals.HighestWeight(0*La[1])
sage: T.cardinality()
1
sage: T = crystals.HighestWeight(La[1])
sage: T.cardinality()
133
sage: T = crystals.HighestWeight(2*La[1])
sage: T.cardinality()
7371
"""
B = CrystalOfLetters(['E',7])
self.column_crystal = {7 : B,
1 : TensorProductOfCrystals(B,B,generators=[[B([-7,1]),B([7])]]),
2 : TensorProductOfCrystals(B,B,B,generators=[[B([-1,2]),B([-7,1]),B([7])]]),
3 : TensorProductOfCrystals(B,B,B,B,generators=[[B([-2,3]),B([-1,2]),B([-7,1]),B([7])]]),
4 : TensorProductOfCrystals(B,B,B,B,generators=[[B([-5,4]),B([-6,5]),B([-7,6]),B([7])]]),
5 : TensorProductOfCrystals(B,B,B,generators=[[B([-6,5]),B([-7,6]),B([7])]]),
6 : TensorProductOfCrystals(B,B,generators=[[B([-7,6]),B([7])]])}
FiniteDimensionalHighestWeightCrystal_TypeE.__init__(self, dominant_weight)
def finite_tensor_product(self, k):
"""
Return the finite tensor product of crystals of length ``k``
by truncating ``self``.
EXAMPLES::
sage: B = crystals.infinity.PathModel(['A',2])
sage: B.finite_tensor_product(5)
Full tensor product of the crystals
[The 1-elementary crystal of type ['A', 2],
The 2-elementary crystal of type ['A', 2],
The 1-elementary crystal of type ['A', 2],
The 2-elementary crystal of type ['A', 2],
The 1-elementary crystal of type ['A', 2]]
"""
N = len(self._factors)
crystals = [self._factors[i % N] for i in range(k)]
return TensorProductOfCrystals(*crystals)
def finite_tensor_product(self, k):
"""
Return the finite tensor product of crystals of length ``k``
from truncating ``self``.
EXAMPLES::
sage: B1 = crystals.KirillovReshetikhin(['A',2,1], 1,1)
sage: B2 = crystals.KirillovReshetikhin(['A',2,1], 2,1)
sage: La = RootSystem(['A',2,1]).weight_lattice().fundamental_weights()
sage: C = crystals.KyotoPathModel([B1,B2,B1], La[0])
sage: C.finite_tensor_product(5)
Full tensor product of the crystals
[Kirillov-Reshetikhin crystal of type ['A', 2, 1] with (r,s)=(1,1),
Kirillov-Reshetikhin crystal of type ['A', 2, 1] with (r,s)=(2,1),
Kirillov-Reshetikhin crystal of type ['A', 2, 1] with (r,s)=(1,1),
Kirillov-Reshetikhin crystal of type ['A', 2, 1] with (r,s)=(1,1),
Kirillov-Reshetikhin crystal of type ['A', 2, 1] with (r,s)=(2,1)]
"""
N = len(self.crystals)
crystals = [self.crystals[i % N] for i in range(k)]
return TensorProductOfCrystals(*crystals)
def R_matrix(self, K):
r"""
Return the combinatorial `R`-matrix of ``self`` to ``K``.
The *combinatorial* `R`-*matrix* is the affine crystal
isomorphism `R : L \otimes K \to K \otimes L` which maps
`u_{L} \otimes u_K` to `u_K \otimes u_{L}`, where `u_K`
is the unique element in `K = B^{r,s}` of weight
`s\Lambda_r - s c \Lambda_0` (see :meth:`maximal_vector`).
INPUT:
- ``self`` -- a crystal `L`
- ``K`` -- a Kirillov-Reshetikhin crystal of the same type as `L`
EXAMPLES::
sage: K = crystals.KirillovReshetikhin(['A',2,1],1,1)
sage: L = crystals.KirillovReshetikhin(['A',2,1],1,2)
sage: f = K.R_matrix(L)
sage: [[b,f(b)] for b in crystals.TensorProduct(K,L)]
[[[[[1]], [[1, 1]]], [[[1, 1]], [[1]]]],
[[[[1]], [[1, 2]]], [[[1, 1]], [[2]]]],
[[[[1]], [[2, 2]]], [[[1, 2]], [[2]]]],
[[[[1]], [[1, 3]]], [[[1, 1]], [[3]]]],
[[[[1]], [[2, 3]]], [[[1, 2]], [[3]]]],
[[[[1]], [[3, 3]]], [[[1, 3]], [[3]]]],
[[[[2]], [[1, 1]]], [[[1, 2]], [[1]]]],
[[[[2]], [[1, 2]]], [[[2, 2]], [[1]]]],
[[[[2]], [[2, 2]]], [[[2, 2]], [[2]]]],
[[[[2]], [[1, 3]]], [[[2, 3]], [[1]]]],
[[[[2]], [[2, 3]]], [[[2, 2]], [[3]]]],
[[[[2]], [[3, 3]]], [[[2, 3]], [[3]]]],
[[[[3]], [[1, 1]]], [[[1, 3]], [[1]]]],
[[[[3]], [[1, 2]]], [[[1, 3]], [[2]]]],
[[[[3]], [[2, 2]]], [[[2, 3]], [[2]]]],
[[[[3]], [[1, 3]]], [[[3, 3]], [[1]]]],
[[[[3]], [[2, 3]]], [[[3, 3]], [[2]]]],
[[[[3]], [[3, 3]]], [[[3, 3]], [[3]]]]]
sage: K = crystals.KirillovReshetikhin(['D',4,1],1,1)
sage: L = crystals.KirillovReshetikhin(['D',4,1],2,1)
sage: f = K.R_matrix(L)
sage: T = crystals.TensorProduct(K,L)
sage: b = T( K(rows=[[1]]), L(rows=[]) )
sage: f(b)
[[[2], [-2]], [[1]]]
Alternatively, one can compute the combinatorial `R`-matrix
using the isomorphism method of digraphs::
sage: K1 = crystals.KirillovReshetikhin(['A',2,1],1,1)
sage: K2 = crystals.KirillovReshetikhin(['A',2,1],2,1)
sage: T1 = crystals.TensorProduct(K1,K2)
sage: T2 = crystals.TensorProduct(K2,K1)
sage: T1.digraph().is_isomorphic(T2.digraph(), edge_labels=True, certificate=True) #todo: not implemented (see #10904 and #10549)
(True, {[[[1]], [[2], [3]]]: [[[1], [3]], [[2]]], [[[3]], [[2], [3]]]: [[[2], [3]], [[3]]],
[[[3]], [[1], [3]]]: [[[1], [3]], [[3]]], [[[1]], [[1], [3]]]: [[[1], [3]], [[1]]], [[[1]],
[[1], [2]]]: [[[1], [2]], [[1]]], [[[2]], [[1], [2]]]: [[[1], [2]], [[2]]], [[[3]],
[[1], [2]]]: [[[2], [3]], [[1]]], [[[2]], [[1], [3]]]: [[[1], [2]], [[3]]], [[[2]], [[2], [3]]]: [[[2], [3]], [[2]]]})
"""
from sage.combinat.crystals.tensor_product import TensorProductOfCrystals
T1 = TensorProductOfCrystals(self, K)
T2 = TensorProductOfCrystals(K, self)
gen1 = T1(self.maximal_vector(), K.maximal_vector())
gen2 = T2(K.maximal_vector(), self.maximal_vector())
return T1.crystal_morphism({gen1: gen2}, check=False)
def R_matrix(self, K):
r"""
Return the combinatorial `R`-matrix of ``self`` to ``K``.
The *combinatorial* `R`-*matrix* is the affine crystal
isomorphism `R : L \otimes K \to K \otimes L` which maps
`u_{L} \otimes u_K` to `u_K \otimes u_{L}`, where `u_K`
is the unique element in `K = B^{r,s}` of weight
`s\Lambda_r - s c \Lambda_0` (see :meth:`maximal_vector`).
INPUT:
- ``self`` -- a crystal `L`
- ``K`` -- a Kirillov-Reshetikhin crystal of the same type as `L`
EXAMPLES::
sage: K = crystals.KirillovReshetikhin(['A',2,1],1,1)
sage: L = crystals.KirillovReshetikhin(['A',2,1],1,2)
sage: f = K.R_matrix(L)
sage: [[b,f(b)] for b in crystals.TensorProduct(K,L)]
[[[[[1]], [[1, 1]]], [[[1, 1]], [[1]]]],
[[[[1]], [[1, 2]]], [[[1, 1]], [[2]]]],
[[[[1]], [[2, 2]]], [[[1, 2]], [[2]]]],
[[[[1]], [[1, 3]]], [[[1, 1]], [[3]]]],
[[[[1]], [[2, 3]]], [[[1, 2]], [[3]]]],
[[[[1]], [[3, 3]]], [[[1, 3]], [[3]]]],
[[[[2]], [[1, 1]]], [[[1, 2]], [[1]]]],
[[[[2]], [[1, 2]]], [[[2, 2]], [[1]]]],
[[[[2]], [[2, 2]]], [[[2, 2]], [[2]]]],
[[[[2]], [[1, 3]]], [[[2, 3]], [[1]]]],
[[[[2]], [[2, 3]]], [[[2, 2]], [[3]]]],
[[[[2]], [[3, 3]]], [[[2, 3]], [[3]]]],
[[[[3]], [[1, 1]]], [[[1, 3]], [[1]]]],
[[[[3]], [[1, 2]]], [[[1, 3]], [[2]]]],
[[[[3]], [[2, 2]]], [[[2, 3]], [[2]]]],
[[[[3]], [[1, 3]]], [[[3, 3]], [[1]]]],
[[[[3]], [[2, 3]]], [[[3, 3]], [[2]]]],
[[[[3]], [[3, 3]]], [[[3, 3]], [[3]]]]]
sage: K = crystals.KirillovReshetikhin(['D',4,1],1,1)
sage: L = crystals.KirillovReshetikhin(['D',4,1],2,1)
sage: f = K.R_matrix(L)
sage: T = crystals.TensorProduct(K,L)
sage: b = T( K(rows=[[1]]), L(rows=[]) )
sage: f(b)
[[[2], [-2]], [[1]]]
Alternatively, one can compute the combinatorial `R`-matrix
using the isomorphism method of digraphs::
sage: K1 = crystals.KirillovReshetikhin(['A',2,1],1,1)
sage: K2 = crystals.KirillovReshetikhin(['A',2,1],2,1)
sage: T1 = crystals.TensorProduct(K1,K2)
sage: T2 = crystals.TensorProduct(K2,K1)
sage: T1.digraph().is_isomorphic(T2.digraph(), edge_labels=True, certificate=True) #todo: not implemented (see #10904 and #10549)
(True, {[[[1]], [[2], [3]]]: [[[1], [3]], [[2]]], [[[3]], [[2], [3]]]: [[[2], [3]], [[3]]],
[[[3]], [[1], [3]]]: [[[1], [3]], [[3]]], [[[1]], [[1], [3]]]: [[[1], [3]], [[1]]], [[[1]],
[[1], [2]]]: [[[1], [2]], [[1]]], [[[2]], [[1], [2]]]: [[[1], [2]], [[2]]], [[[3]],
[[1], [2]]]: [[[2], [3]], [[1]]], [[[2]], [[1], [3]]]: [[[1], [2]], [[3]]], [[[2]], [[2], [3]]]: [[[2], [3]], [[2]]]})
"""
from sage.combinat.crystals.tensor_product import TensorProductOfCrystals
T1 = TensorProductOfCrystals(self, K)
T2 = TensorProductOfCrystals(K, self)
gen1 = T1(self.maximal_vector(), K.maximal_vector())
gen2 = T2(K.maximal_vector(), self.maximal_vector())
return T1.crystal_morphism({gen1: gen2}, check=False)
