Bases: sage.combinat.crystals.kirillov_reshetikhin.KirillovReshetikhinCrystalFromPromotion
Class of Kirillov-Reshetikhin crystals of type .
EXAMPLES:
sage: K = KirillovReshetikhinCrystal(['A',3,1], 2,2)
sage: b = K(rows=[[1,2],[2,4]])
sage: b.f(0)
[[1, 1], [2, 2]]
Specifies the classical crystal underlying the KR crystal of type A.
EXAMPLES:
sage: K = KirillovReshetikhinCrystal(['A',3,1], 2,2)
sage: K.classical_decomposition()
The crystal of tableaux of type ['A', 3] and shape(s) [[2, 2]]
Specifies the Dynkin diagram automorphism underlying the promotion action on the crystal elements. The automorphism needs to map node 0 to some other Dynkin node.
For type A we use the Dynkin diagram automorphism which maps i to i+1 mod n+1, where n is the rank.
EXAMPLES:
sage: K = KirillovReshetikhinCrystal(['A',3,1], 2,2)
sage: K.dynkin_diagram_automorphism(0)
1
sage: K.dynkin_diagram_automorphism(3)
0
Specifies the promotion operator used to construct the affine type A crystal. For type A this corresponds to the Dynkin diagram automorphism which maps i to i+1 mod n+1, where n is the rank.
EXAMPLES:
sage: K = KirillovReshetikhinCrystal(['A',3,1], 2,2)
sage: b = K.classical_decomposition()(rows=[[1,2],[3,4]])
sage: K.promotion()(b)
[[1, 3], [2, 4]]
Specifies the inverse promotion operator used to construct the affine type A crystal. For type A this corresponds to the Dynkin diagram automorphism which maps i to i-1 mod n+1, where n is the rank.
EXAMPLES:
sage: K = KirillovReshetikhinCrystal(['A',3,1], 2,2)
sage: b = K.classical_decomposition()(rows=[[1,3],[2,4]])
sage: K.promotion_inverse()(b)
[[1, 2], [3, 4]]
sage: b = K.classical_decomposition()(rows=[[1,2],[3,3]])
sage: K.promotion_inverse()(K.promotion()(b))
[[1, 2], [3, 3]]
Bases: sage.combinat.crystals.kirillov_reshetikhin.KirillovReshetikhinGenericCrystal
Class of Kirillov-Reshetikhin crystals of type
for
in the realization with classical subalgebra
. The cartan type in this case is inputted as
the dual of
.
This is an alternative implementation to KR_type_box which uses
the classical decomposition into type crystals.
EXAMPLES:
sage: C = CartanType(['A',4,2]).dual()
sage: K = sage.combinat.crystals.kirillov_reshetikhin.KR_type_A2(C, 1, 1)
sage: K
Kirillov-Reshetikhin crystal of type ['BC', 2, 2]^* with (r,s)=(1,1)
sage: b = K(rows=[[-1]])
sage: b.f(0)
[[1]]
sage: b.e(0)
We can now check whether the two KR crystals of type (namely the KR crystal and its dual
construction) are isomorphic up to relabelling of the edges:
sage: C = CartanType(['A',4,2])
sage: K = KirillovReshetikhinCrystal(C,1,1)
sage: Kdual = KirillovReshetikhinCrystal(C.dual(),1,1)
sage: G = K.digraph()
sage: Gdual = Kdual.digraph()
sage: f = {0:2, 1:1, 2:0}
sage: Gnew = DiGraph(); Gnew.add_vertices(Gdual.vertices()); Gnew.add_edges([(u,v,f[i]) for (u,v,i) in Gdual.edges()])
sage: G.is_isomorphic(Gnew, edge_labels = True)
True
alias of KR_type_A2Element
Returns the ambient crystal of type
associated to the Kirillov-Reshetikhin
crystal of type
dual. This ambient crystal is used to construct the zero arrows.
EXAMPLES:
sage: C = CartanType(['A',4,2]).dual()
sage: K = sage.combinat.crystals.kirillov_reshetikhin.KR_type_A2(C, 2, 3)
sage: K.ambient_crystal()
Kirillov-Reshetikhin crystal of type ['B', 3, 1] with (r,s)=(2,3)
Gives a dictionary of all self-dual diagrams for the ambient crystal.
Their key is their inner shape.
EXAMPLES:
sage: C = CartanType(['A',4,2]).dual()
sage: K = sage.combinat.crystals.kirillov_reshetikhin.KR_type_A2(C, 1, 1)
sage: K.ambient_dict_pm_diagrams()
{[1]: [[0, 0], [1]]}
sage: K = sage.combinat.crystals.kirillov_reshetikhin.KR_type_A2(C, 1, 2)
sage: K.ambient_dict_pm_diagrams()
{[]: [[1, 1], [0]], [2]: [[0, 0], [2]]}
sage: K = sage.combinat.crystals.kirillov_reshetikhin.KR_type_A2(C, 2, 2)
sage: K.ambient_dict_pm_diagrams()
{[2, 2]: [[0, 0], [0, 0], [2]], []: [[1, 1], [0, 0], [0]], [2]: [[0, 0], [1, 1], [0]]}
Gives a dictionary of all -highest weight vectors in the ambient crystal.
Their key is the inner shape of their corresponding
diagram, or equivalently, their
weight.
EXAMPLES:
sage: C = CartanType(['A',4,2]).dual()
sage: K = sage.combinat.crystals.kirillov_reshetikhin.KR_type_A2(C, 1, 2)
sage: K.ambient_highest_weight_dict()
{[]: [[1, -1]], [2]: [[2, 2]]}
Specifies the classical crystal underlying the Kirillov-Reshetikhin crystal of type
with
as classical subdiagram.
It is given by where
is a highest weight crystal of type
of highest weight
. The sum is over all weights
obtained from
a rectangle of width
and height
by removing horizontal dominoes. Here we identify the fundamental
weight
with a column of height
.
EXAMPLES:
sage: C = CartanType(['A',4,2]).dual()
sage: K = sage.combinat.crystals.kirillov_reshetikhin.KR_type_A2(C, 2, 2)
sage: K.classical_decomposition()
The crystal of tableaux of type ['B', 2] and shape(s) [[], [2], [2, 2]]
Provides a map from the ambient crystal of type to
the Kirillov-Reshetikhin crystal of type
.
Note that this map is only well-defined on type
elements that are in the image under to_ambient_crystal().
EXAMPLES:
sage: C = CartanType(['A',4,2]).dual()
sage: K = sage.combinat.crystals.kirillov_reshetikhin.KR_type_A2(C, 1, 2)
sage: b = K.ambient_crystal()(rows=[[2,2]])
sage: K.from_ambient_crystal()(b)
[[1, 1]]
Gives a dictionary of the classical highest weight vectors of self. Their key is their shape.
EXAMPLES:
sage: C = CartanType(['A',4,2]).dual()
sage: K = sage.combinat.crystals.kirillov_reshetikhin.KR_type_A2(C, 1, 2)
sage: K.highest_weight_dict()
{[]: [], [2]: [[1, 1]]}
Provides a map from the Kirillov-Reshetikhin crystal of type to the
ambient crystal of type
.
EXAMPLES:
sage: C = CartanType(['A',4,2]).dual()
sage: K = sage.combinat.crystals.kirillov_reshetikhin.KR_type_A2(C, 1, 2)
sage: b=K(rows=[[1,1]])
sage: K.to_ambient_crystal()(b)
[[2, 2]]
sage: K = sage.combinat.crystals.kirillov_reshetikhin.KR_type_A2(C, 2, 2)
sage: b=K(rows=[[1,1]])
sage: K.to_ambient_crystal()(b)
[[1, 2], [2, -1]]
sage: K.to_ambient_crystal()(b).parent()
Kirillov-Reshetikhin crystal of type ['B', 3, 1] with (r,s)=(2,2)
Bases: sage.combinat.crystals.kirillov_reshetikhin.KirillovReshetikhinGenericCrystalElement
Class for the elements in the Kirillov-Reshetikhin crystals of type
for
with underlying classcial algebra
.
EXAMPLES:
sage: C = CartanType(['A',4,2]).dual()
sage: K = sage.combinat.crystals.kirillov_reshetikhin.KR_type_A2(C, 1, 2)
sage: type(K.module_generators[0])
<class 'sage.combinat.crystals.kirillov_reshetikhin.KR_type_A2_with_category.element_class'>
Gives on self by mapping self to the ambient crystal, calculating
there and
pulling the element back.
EXAMPLES:
sage: C = CartanType(['A',4,2]).dual()
sage: K = sage.combinat.crystals.kirillov_reshetikhin.KR_type_A2(C, 1, 1)
sage: b = K(rows=[[1]])
sage: b.e(0) # indirect doctest
[[-1]]
Calculates of self by mapping the element to the ambient crystal
and calculating
there.
EXAMPLES:
sage: C = CartanType(['A',4,2]).dual()
sage: K = sage.combinat.crystals.kirillov_reshetikhin.KR_type_A2(C, 1, 1)
sage: b=K(rows=[[1]])
sage: b.epsilon(0) # indirect doctest
1
Gives on self by mapping self to the ambient crystal, calculating
there and
pulling the element back.
EXAMPLES:
sage: C = CartanType(['A',4,2]).dual()
sage: K = sage.combinat.crystals.kirillov_reshetikhin.KR_type_A2(C, 1, 1)
sage: b = K(rows=[[-1]])
sage: b.f(0) # indirect doctest
[[1]]
Calculates of self by mapping the element to the ambient crystal
and calculating
there.
EXAMPLES:
sage: C = CartanType(['A',4,2]).dual()
sage: K = sage.combinat.crystals.kirillov_reshetikhin.KR_type_A2(C, 1, 1)
sage: b=K(rows=[[-1]])
sage: b.phi(0) # indirect doctest
1
Bases: sage.combinat.crystals.kirillov_reshetikhin.KirillovReshetikhinGenericCrystal
Class of Kirillov-Reshetikhin crystals of type
.
EXAMPLES:
sage: K = KirillovReshetikhinCrystal(['B',3,1],3,2)
sage: K
Kirillov-Reshetikhin crystal of type ['B', 3, 1] with (r,s)=(3,2)
sage: b = K(rows=[[1],[2],[3]])
sage: b.f(0)
sage: b.e(0)
[[3]]
sage: K = KirillovReshetikhinCrystal(['B',3,1],3,2)
sage: [b.weight() for b in K if b.is_highest_weight([1,2,3])]
[-Lambda[0] + Lambda[1], -2*Lambda[0] + 2*Lambda[3]]
sage: [b.weight() for b in K if b.is_highest_weight([0,2,3])]
[Lambda[0] - Lambda[1], -2*Lambda[1] + 2*Lambda[3]]
alias of KR_type_BnElement
Returns the ambient crystal of type
associated to the Kirillov-Reshetikhin crystal;
see Lemma 4.2 of reference [4].
This ambient crystal is used to construct the zero arrows.
EXAMPLES:
sage: K = KirillovReshetikhinCrystal(['B',3,1],3,2)
sage: K.ambient_crystal()
Kirillov-Reshetikhin crystal of type ['B', 3, 1]^* with (r,s)=(3,2)
Gives a dictionary of the classical highest weight vectors of the ambient crystal of self. Their key is their shape.
EXAMPLES:
sage: K = KirillovReshetikhinCrystal(['B',3,1],3,2)
sage: K.ambient_highest_weight_dict()
{(2,): [[1, 1]], (2, 1, 1): [[1, 1], [2], [3]], (2, 2, 2): [[1, 1], [2, 2], [3, 3]]}
sage: K = KirillovReshetikhinCrystal(['B',3,1],3,3)
sage: K.ambient_highest_weight_dict()
{(3, 3, 3): [[1, 1, 1], [2, 2, 2], [3, 3, 3]], (3, 1, 1): [[1, 1, 1], [2], [3]],
(3, 2, 2): [[1, 1, 1], [2, 2], [3, 3]], (3,): [[1, 1, 1]]}
Specifies the classical crystal underlying the Kirillov-Reshetikhin crystal of type
.
It is the same as for , given by
where
are
weights obtained from a rectangle of width
and height
by removing horizontal dominoes.
Here we identify the fundamental weight
with a column of height
for
and
a column of width
for
.
EXAMPLES:
sage: K = KirillovReshetikhinCrystal(['B',3,1], 3, 2)
sage: K.classical_decomposition()
The crystal of tableaux of type ['B', 3] and shape(s) [[1], [1, 1, 1]]
sage: K = KirillovReshetikhinCrystal(['B',3,1], 3, 3)
sage: K.classical_decomposition()
The crystal of tableaux of type ['B', 3] and shape(s) [[3/2, 1/2, 1/2], [3/2, 3/2, 3/2]]
Provides a map from the ambient crystal of type to
the Kirillov-Reshetikhin crystal self.
Note that this map is only well-defined on elements that are in the image under to_ambient_crystal().
EXAMPLES:
sage: K = KirillovReshetikhinCrystal(['B',3,1],3,1)
sage: [b == K.from_ambient_crystal()(K.to_ambient_crystal()(b)) for b in K]
[True, True, True, True, True, True, True, True]
sage: b = K.ambient_crystal()(rows=[[1],[2],[-3]])
sage: K.from_ambient_crystal()(b)
[++-, []]
Gives a dictionary of the classical highest weight vectors of self. Their key is 2 times their shape.
EXAMPLES:
sage: K = KirillovReshetikhinCrystal(['B',3,1],3,2)
sage: K.highest_weight_dict()
{(2,): [[1]], (2, 2, 2): [[1], [2], [3]]}
sage: K = KirillovReshetikhinCrystal(['B',3,1],3,3)
sage: K.highest_weight_dict()
{(3, 3, 3): [+++, [[1], [2], [3]]], (3, 1, 1): [+++, [[1]]]}
Sets the similarity factor used to map to the ambient crystal.
EXAMPLES:
sage: K = KirillovReshetikhinCrystal(['B',3,1],3,2)
sage: K.similarity_factor()
{1: 2, 2: 2, 3: 1}
Provides a map from self to the ambient crystal of type .
EXAMPLES:
sage: K = KirillovReshetikhinCrystal(['B',3,1],3,1)
sage: [K.to_ambient_crystal()(b) for b in K]
[[[1], [2], [3]], [[1], [2], [-3]], [[1], [3], [-2]], [[2], [3], [-1]], [[1], [-3], [-2]],
[[2], [-3], [-1]], [[3], [-2], [-1]], [[-3], [-2], [-1]]]
Bases: sage.combinat.crystals.kirillov_reshetikhin.KirillovReshetikhinGenericCrystalElement
Class for the elements in the Kirillov-Reshetikhin crystals of type
.
EXAMPLES:
sage: K=KirillovReshetikhinCrystal(['B',3,1],3,2)
sage: type(K.module_generators[0])
<class 'sage.combinat.crystals.kirillov_reshetikhin.KR_type_Bn_with_category.element_class'>
Gives on self by mapping self to the ambient crystal, calculating
there and
pulling the element back.
EXAMPLES:
sage: K=KirillovReshetikhinCrystal(['B',3,1],3,1)
sage: b = K.module_generators[0]
sage: b.e(0) # indirect doctest
[--+, []]
Calculates of self by mapping the element to the ambient crystal
and calculating
there.
EXAMPLES:
sage: K=KirillovReshetikhinCrystal(['B',3,1],3,1)
sage: b = K.module_generators[0]
sage: b.epsilon(0) # indirect doctest
1
Gives on self by mapping self to the ambient crystal, calculating
there and
pulling the element back.
EXAMPLES:
sage: K=KirillovReshetikhinCrystal(['B',3,1],3,1)
sage: b = K.module_generators[0]
sage: b.f(0) # indirect doctest
Calculates of self by mapping the element to the ambient crystal
and calculating
there.
EXAMPLES:
sage: K=KirillovReshetikhinCrystal(['B',3,1],3,1)
sage: b = K.module_generators[0]
sage: b.phi(0) # indirect doctest
0
Bases: sage.combinat.crystals.kirillov_reshetikhin.KirillovReshetikhinGenericCrystal
Class of Kirillov-Reshetikhin crystals of type
for
.
EXAMPLES:
sage: K = KirillovReshetikhinCrystal(['C',2,1], 1,2)
sage: K
Kirillov-Reshetikhin crystal of type ['C', 2, 1] with (r,s)=(1,2)
sage: b = K(rows=[])
sage: b.f(0)
[[1, 1]]
sage: b.e(0)
[[-1, -1]]
alias of KR_type_CElement
Returns the ambient crystal of type
associated to the Kirillov-Reshetikhin
crystal of type
. This ambient crystal is used to construct the zero arrows.
EXAMPLES:
sage: K = KirillovReshetikhinCrystal(['C',3,1], 2,3)
sage: K.ambient_crystal()
Kirillov-Reshetikhin crystal of type ['B', 4, 1]^* with (r,s)=(2,3)
Gives a dictionary of all self-dual diagrams for the ambient crystal.
Their key is their inner shape.
EXAMPLES:
sage: K = KirillovReshetikhinCrystal(['C',2,1], 1,2)
sage: K.ambient_dict_pm_diagrams()
{[]: [[1, 1], [0]], [2]: [[0, 0], [2]]}
sage: K = KirillovReshetikhinCrystal(['C',3,1], 2,2)
sage: K.ambient_dict_pm_diagrams()
{[2, 2]: [[0, 0], [0, 0], [2]], []: [[1, 1], [0, 0], [0]], [2]: [[0, 0], [1, 1], [0]]}
sage: K = KirillovReshetikhinCrystal(['C',3,1], 2,3)
sage: K.ambient_dict_pm_diagrams()
{[3, 3]: [[0, 0], [0, 0], [3]], [3, 1]: [[0, 0], [1, 1], [1]], [1, 1]: [[1, 1], [0, 0], [1]]}
Gives a dictionary of all -highest weight vectors in the ambient crystal.
Their key is the inner shape of their corresponding
diagram, or equivalently, their
weight.
EXAMPLES:
sage: K = KirillovReshetikhinCrystal(['C',3,1], 2,2)
sage: K.ambient_highest_weight_dict()
{[]: [[2], [-2]], [2, 2]: [[2, 2], [3, 3]], [2]: [[1, 2], [2, -1]]}
Specifies the classical crystal underlying the Kirillov-Reshetikhin crystal of type .
It is given by where
are weights obtained from
a rectangle of width
and height
by removing horizontal dominoes. Here we identify the fundamental
weight
with a column of height
.
EXAMPLES:
sage: K = KirillovReshetikhinCrystal(['C',3,1], 2,2)
sage: K.classical_decomposition()
The crystal of tableaux of type ['C', 3] and shape(s) [[], [2], [2, 2]]
Provides a map from the ambient crystal of type to
the Kirillov-Reshetikhin crystal of type
.
Note that this map is only well-defined on type elements
that are in the image under to_ambient_crystal().
EXAMPLES:
sage: K = KirillovReshetikhinCrystal(['C',3,1], 2,2)
sage: b=K.ambient_crystal()(rows=[[2,2],[3,3]])
sage: K.from_ambient_crystal()(b)
[[1, 1], [2, 2]]
Gives a dictionary of the classical highest weight vectors of self. Their key is their shape.
EXAMPLES:
sage: K = KirillovReshetikhinCrystal(['C',3,1], 2,2)
sage: K.highest_weight_dict()
{[2, 2]: [[1, 1], [2, 2]], []: [], [2]: [[1, 1]]}
Provides a map from the Kirillov-Reshetikhin crystal of type to the
ambient crystal of type
.
EXAMPLES:
sage: K = KirillovReshetikhinCrystal(['C',3,1], 2,2)
sage: b=K(rows=[[1,1]])
sage: K.to_ambient_crystal()(b)
[[1, 2], [2, -1]]
sage: b=K(rows=[])
sage: K.to_ambient_crystal()(b)
[[2], [-2]]
sage: K.to_ambient_crystal()(b).parent()
Kirillov-Reshetikhin crystal of type ['B', 4, 1]^* with (r,s)=(2,2)
Bases: sage.combinat.crystals.kirillov_reshetikhin.KirillovReshetikhinGenericCrystalElement
Class for the elements in the Kirillov-Reshetikhin crystals of type
for
.
EXAMPLES:
sage: K=KirillovReshetikhinCrystal(['C',3,1],1,2)
sage: type(K.module_generators[0])
<class 'sage.combinat.crystals.kirillov_reshetikhin.KR_type_C_with_category.element_class'>
Gives on self by mapping self to the ambient crystal, calculating
there and
pulling the element back.
EXAMPLES:
sage: K=KirillovReshetikhinCrystal(['C',3,1],1,2)
sage: b = K(rows=[])
sage: b.e(0) # indirect doctest
[[-1, -1]]
Calculates of self by mapping the element to the ambient crystal
and calculating
there.
EXAMPLES:
sage: K = KirillovReshetikhinCrystal(['C',2,1], 1,2)
sage: b=K(rows=[[1,1]])
sage: b.epsilon(0) # indirect doctest
2
Gives on self by mapping self to the ambient crystal, calculating
there and
pulling the element back.
EXAMPLES:
sage: K=KirillovReshetikhinCrystal(['C',3,1],1,2)
sage: b = K(rows=[])
sage: b.f(0) # indirect doctest
[[1, 1]]
Calculates of self by mapping the element to the ambient crystal
and calculating
there.
EXAMPLES:
sage: K = KirillovReshetikhinCrystal(['C',2,1], 1,2)
sage: b=K(rows=[[-1,-1]])
sage: b.phi(0) # indirect doctest
2
Bases: sage.combinat.crystals.kirillov_reshetikhin.KirillovReshetikhinGenericCrystal
Class of Kirillov-Reshetikhin crystals of type
.
EXAMPLES:
sage: K = KirillovReshetikhinCrystal(['C',3,1],3,1)
sage: [[b,b.f(0)] for b in K]
[[[[1], [2], [3]], None], [[[1], [2], [-3]], None], [[[1], [3], [-3]], None],
[[[2], [3], [-3]], None], [[[1], [3], [-2]], None], [[[2], [3], [-2]], None],
[[[2], [3], [-1]], [[1], [2], [3]]], [[[1], [-3], [-2]], None], [[[2], [-3], [-2]], None],
[[[2], [-3], [-1]], [[1], [2], [-3]]], [[[3], [-3], [-2]], None], [[[3], [-3], [-1]],
[[1], [3], [-3]]], [[[3], [-2], [-1]], [[1], [3], [-2]]], [[[-3], [-2], [-1]], [[1], [-3], [-2]]]]
alias of KR_type_CnElement
Specifies the classical crystal underlying the Kirillov-Reshetikhin crystal
of type
. It is given by
.
EXAMPLES:
sage: K = KirillovReshetikhinCrystal(['C',3,1],3,2)
sage: K.classical_decomposition()
The crystal of tableaux of type ['C', 3] and shape(s) [[2, 2, 2]]
This gives the bijection between an element b in the classical decomposition
of the KR crystal that is -highest weight and
diagrams.
EXAMPLES:
sage: K = KirillovReshetikhinCrystal(['C',3,1],3,2)
sage: T = K.classical_decomposition()
sage: b = T(rows=[[2, 2], [3, 3], [-3, -1]])
sage: pm = K.from_highest_weight_vector_to_pm_diagram(b); pm
[[0, 0], [1, 0], [0, 1], [0]]
sage: pm.__repr__(pretty_printing=True)
. .
. +
- -
sage: hw = [ b for b in T if all(b.epsilon(i)==0 for i in [2,3]) ]
sage: all(K.from_pm_diagram_to_highest_weight_vector(K.from_highest_weight_vector_to_pm_diagram(b)) == b for b in hw)
True
This gives the bijection between a diagram and an element b in the classical
decomposition of the KR crystal that is {2,3,..,n}-highest weight.
EXAMPLES:
sage: K = KirillovReshetikhinCrystal(['C',3,1],3,2)
sage: pm = sage.combinat.crystals.kirillov_reshetikhin.PMDiagram([[0, 0], [1, 0], [0, 1], [0]])
sage: K.from_pm_diagram_to_highest_weight_vector(pm)
[[2, 2], [3, 3], [-3, -1]]
Bases: sage.combinat.crystals.kirillov_reshetikhin.KirillovReshetikhinGenericCrystalElement
Class for the elements in the Kirillov-Reshetikhin crystals of type
.
EXAMPLES:
sage: K=KirillovReshetikhinCrystal(['C',3,1],3,2)
sage: type(K.module_generators[0])
<class 'sage.combinat.crystals.kirillov_reshetikhin.KR_type_Cn_with_category.element_class'>
Gives on self by going to the
-diagram corresponding to the
-highest weight
vector in the component of
, then applying [Definition 6.1, 4], and pulling back from
-diagrams.
EXAMPLES:
sage: K=KirillovReshetikhinCrystal(['C',3,1],3,2)
sage: b = K.module_generators[0]
sage: b.e(0) # indirect doctest
[[1, 2], [2, 3], [3, -1]]
sage: b = K(rows=[[1,2],[2,3],[3,-1]])
sage: b.e(0)
[[2, 2], [3, 3], [-1, -1]]
sage: b=K(rows=[[1, -3], [3, -2], [-3, -1]])
sage: b.e(0)
[[3, -3], [-3, -2], [-1, -1]]
Calculates of self using Lemma 6.1 of [4].
EXAMPLES:
sage: K=KirillovReshetikhinCrystal(['C',3,1],3,1)
sage: b = K.module_generators[0]
sage: b.epsilon(0) # indirect doctest
1
Gives on self by going to the
-diagram corresponding to the
-highest weight
vector in the component of
, then applying [Definition 6.1, 4], and pulling back from
-diagrams.
EXAMPLES:
sage: K=KirillovReshetikhinCrystal(['C',3,1],3,1)
sage: b = K.module_generators[0]
sage: b.f(0) # indirect doctest
Calculates of self.
EXAMPLES:
sage: K=KirillovReshetikhinCrystal(['C',3,1],3,1)
sage: b = K.module_generators[0]
sage: b.phi(0) # indirect doctest
0
Bases: sage.combinat.crystals.kirillov_reshetikhin.KirillovReshetikhinGenericCrystal
Class of Kirillov-Reshetikhin crystals of type
.
EXAMPLES:
sage: K = KirillovReshetikhinCrystal(['D',4,2],3,1)
sage: [[b,b.f(0)] for b in K]
[[[+++, []], None], [[++-, []], None], [[+-+, []], None], [[-++, []],
[+++, []]], [[+--, []], None], [[-+-, []], [++-, []]], [[--+, []], [+-+, []]],
[[---, []], [+--, []]]]
alias of KR_type_Dn_twistedElement
Specifies the classical crystal underlying the Kirillov-Reshetikhin crystal
of type
. It is given by
.
EXAMPLES:
sage: K = KirillovReshetikhinCrystal(['D',4,2],3,1)
sage: K.classical_decomposition()
The crystal of tableaux of type ['B', 3] and shape(s) [[1/2, 1/2, 1/2]]
sage: K = KirillovReshetikhinCrystal(['D',4,2],3,2)
sage: K.classical_decomposition()
The crystal of tableaux of type ['B', 3] and shape(s) [[1, 1, 1]]
This gives the bijection between an element b in the classical decomposition
of the KR crystal that is -highest weight and
diagrams.
EXAMPLES:
sage: K = KirillovReshetikhinCrystal(['D',4,2],3,1)
sage: T = K.classical_decomposition()
sage: hw = [ b for b in T if all(b.epsilon(i)==0 for i in [2,3]) ]
sage: [K.from_highest_weight_vector_to_pm_diagram(b) for b in hw]
[[[0, 0], [0, 0], [1, 0], [0]], [[0, 0], [0, 0], [0, 1], [0]]]
sage: K = KirillovReshetikhinCrystal(['D',4,2],3,2)
sage: T = K.classical_decomposition()
sage: hw = [ b for b in T if all(b.epsilon(i)==0 for i in [2,3]) ]
sage: [K.from_highest_weight_vector_to_pm_diagram(b) for b in hw]
[[[0, 0], [0, 0], [2, 0], [0]], [[0, 0], [0, 0], [0, 0], [2]], [[0, 0], [2, 0], [0, 0], [0]],
[[0, 0], [0, 0], [0, 2], [0]]]
Note that, since the classical decomposition of this crystal is of type , there can
be (at most one) entry
in the
-highest weight elements at height
.
In the following implementation this is realized as an empty column of height
since
this uniquely specifies the existence of the
:
EXAMPLES:
sage: b = hw[1]
sage: pm = K.from_highest_weight_vector_to_pm_diagram(b)
sage: pm.__repr__(pretty_printing=True)
. .
. .
. .
TESTS:
sage: all(K.from_pm_diagram_to_highest_weight_vector(K.from_highest_weight_vector_to_pm_diagram(b)) == b for b in hw)
True
sage: K = KirillovReshetikhinCrystal(['D',4,2],3,2)
sage: T = K.classical_decomposition()
sage: hw = [ b for b in T if all(b.epsilon(i)==0 for i in [2,3]) ]
sage: all(K.from_pm_diagram_to_highest_weight_vector(K.from_highest_weight_vector_to_pm_diagram(b)) == b for b in hw)
True
sage: K = KirillovReshetikhinCrystal(['D',4,2],3,3)
sage: T = K.classical_decomposition()
sage: hw = [ b for b in T if all(b.epsilon(i)==0 for i in [2,3]) ]
sage: all(K.from_pm_diagram_to_highest_weight_vector(K.from_highest_weight_vector_to_pm_diagram(b)) == b for b in hw)
True
This gives the bijection between a diagram and an element b in the classical
decomposition of the KR crystal that is {2,3,..,n}-highest weight.
EXAMPLES:
sage: K = KirillovReshetikhinCrystal(['D',4,2],3,2)
sage: pm = sage.combinat.crystals.kirillov_reshetikhin.PMDiagram([[0, 0], [0, 0], [0, 0], [2]])
sage: K.from_pm_diagram_to_highest_weight_vector(pm)
[[2], [3], [0]]
Bases: sage.combinat.crystals.kirillov_reshetikhin.KirillovReshetikhinGenericCrystalElement
Class for the elements in the Kirillov-Reshetikhin crystals of type
.
EXAMPLES:
sage: K=KirillovReshetikhinCrystal(['D',4,2],3,2)
sage: type(K.module_generators[0])
<class 'sage.combinat.crystals.kirillov_reshetikhin.KR_type_Dn_twisted_with_category.element_class'>
Gives on self by going to the
-diagram corresponding to the
-highest weight
vector in the component of
, then applying [Definition 6.2, 4], and pulling back from
-diagrams.
EXAMPLES:
sage: K=KirillovReshetikhinCrystal(['D',4,2],3,3)
sage: b = K.module_generators[0]
sage: b.e(0) # indirect doctest
[+++, [[2], [3], [0]]]
Calculates of self using Lemma 6.2 of [4].
EXAMPLES:
sage: K=KirillovReshetikhinCrystal(['D',4,2],3,1)
sage: b = K.module_generators[0]
sage: b.epsilon(0) # indirect doctest
1
Gives on self by going to the
-diagram corresponding to the
-highest weight
vector in the component of
, then applying [Definition 6.2, 4], and pulling back from
-diagrams.
EXAMPLES:
sage: K=KirillovReshetikhinCrystal(['D',4,2],3,2)
sage: b = K.module_generators[0]
sage: b.f(0) # indirect doctest
Calculates of self.
EXAMPLES:
sage: K=KirillovReshetikhinCrystal(['D',4,2],3,1)
sage: b = K.module_generators[0]
sage: b.phi(0) # indirect doctest
0
Bases: sage.combinat.crystals.kirillov_reshetikhin.KirillovReshetikhinCrystalFromPromotion
Class of Kirillov-Reshetikhin crystals of type for
.
EXAMPLES:
sage: K = KirillovReshetikhinCrystal(['E',6,1],2,1)
sage: K.module_generator().e(0)
[]
sage: K.module_generator().e(0).f(0)
[[(2, -1), (1,)]]
sage: K = KirillovReshetikhinCrystal(['E',6,1], 1,1)
sage: b = K.module_generator()
sage: b
[(1,)]
sage: b.e(0)
[(-2, 1)]
sage: b = [t for t in K if t.epsilon(1) == 1 and t.phi(3) == 1 and t.phi(2) == 0 and t.epsilon(2) == 0][0]
sage: b
[(-1, 3)]
sage: b.e(0)
[(-1, -2, 3)]
The elements of the Kirillov-Reshetikhin crystals can be constructed from a classical crystal element using retract().
EXAMPLES:
sage: K = KirillovReshetikhinCrystal(['E',6,1],2,1)
sage: La = K.cartan_type().classical().root_system().weight_lattice().fundamental_weights()
sage: H = HighestWeightCrystal(La[2])
sage: t = H.module_generator()
sage: t
[[(2, -1), (1,)]]
sage: type(K.retract(t))
<class 'sage.combinat.crystals.kirillov_reshetikhin.KR_type_E6_with_category.element_class'>
sage: K.retract(t).e(0)
[]
TESTS:
sage: K = KirillovReshetikhinCrystal(['E',6,1], 2,1)
sage: La = K.weight_lattice_realization().fundamental_weights()
sage: all(b.weight() == sum( (K.affine_weight(b.lift())[i] * La[i] for i in K.index_set()), 0*La[0]) for b in K) # long time (26s on sage.math, 2011)
True
Returns the affine level zero weight corresponding to the element b of the classical crystal underlying self. For the coefficients to calculate the level, see Kac pg. 48.
EXAMPLES:
sage: K = KirillovReshetikhinCrystal(['E',6,1],2,1)
sage: [K.affine_weight(x.lift()) for x in K if all(x.epsilon(i) == 0 for i in [2,3,4,5])]
[(0, 0, 0, 0, 0, 0, 0),
(-2, 0, 1, 0, 0, 0, 0),
(-1, -1, 0, 0, 0, 1, 0),
(0, 0, 0, 0, 0, 0, 0),
(0, 0, 0, 0, 0, 1, -2),
(0, -1, 1, 0, 0, 0, -1),
(-1, 0, 0, 1, 0, 0, -1),
(-1, -1, 0, 0, 1, 0, -1),
(0, 0, 0, 0, 0, 0, 0),
(0, -2, 0, 1, 0, 0, 0)]
Acts with the Dynkin diagram automorphism on affine weights as outputted by the affine_weight method.
EXAMPLES:
sage: K = KirillovReshetikhinCrystal(['E',6,1],2,1)
sage: [[x[0], K.automorphism_on_affine_weight(x[0])] for x in K.highest_weight_dict().values()]
[[(0, 0, 0, 0, 0, 1, -2), (-2, 0, 1, 0, 0, 0, 0)],
[(-1, 0, 0, 1, 0, 0, -1), (-1, -1, 0, 0, 0, 1, 0)],
[(0, 0, 0, 0, 0, 0, 0), (0, 0, 0, 0, 0, 0, 0)],
[(-2, 0, 1, 0, 0, 0, 0), (0, -2, 0, 1, 0, 0, 0)],
[(0, 0, 0, 0, 0, 0, 0), (0, 0, 0, 0, 0, 0, 0)]]
Specifies the classical crystal underlying the KR crystal of type .
EXAMPLES:
sage: K = KirillovReshetikhinCrystal(['E',6,1], 2,2)
sage: K.classical_decomposition()
Direct sum of the crystals Family (Finite dimensional highest weight crystal of type ['E', 6] and highest weight 0, Finite dimensional highest weight crystal of type ['E', 6] and highest weight Lambda[2], Finite dimensional highest weight crystal of type ['E', 6] and highest weight 2*Lambda[2])
sage: K = KirillovReshetikhinCrystal(['E',6,1], 1,2)
sage: K.classical_decomposition()
Direct sum of the crystals Family (Finite dimensional highest weight crystal of type ['E', 6] and highest weight 2*Lambda[1],)
Specifies the Dynkin diagram automorphism underlying the promotion action on the crystal elements. The automorphism needs to map node 0 to some other Dynkin node.
Here we use the Dynkin diagram automorphism of order 3 which maps node 0 to node 1.
EXAMPLES:
sage: K = KirillovReshetikhinCrystal(['E',6,1],2,1)
sage: [K.dynkin_diagram_automorphism(i) for i in K.index_set()]
[1, 6, 3, 5, 4, 2, 0]
Returns a dictionary between highest weight elements, and a tuple of affine weights and its classical component.
EXAMPLES:
sage: K = KirillovReshetikhinCrystal(['E',6,1],2,1)
sage: K.highest_weight_dict()
{[[(5, -2, -6), (-6, 2)]]: ((0, 0, 0, 0, 0, 1, -2), 1),
[[(3, -1, -6), (1,)]]: ((-1, 0, 0, 1, 0, 0, -1), 1),
[[(6, -2), (-6, 2)]]: ((0, 0, 0, 0, 0, 0, 0), 1),
[[(2, -1), (1,)]]: ((-2, 0, 1, 0, 0, 0, 0), 1),
[]: ((0, 0, 0, 0, 0, 0, 0), 0)}
Returns a dictionary between a tuple of affine weights and a classical component, and
highest weight elements.
EXAMPLES:
sage: K = KirillovReshetikhinCrystal(['E',6,1],2,1)
sage: K.highest_weight_dict_inv()
{((0, 0, 0, 0, 0, 0, 0), 0): [],
((-1, -1, 0, 0, 0, 1, 0), 1): [[(5, -3), (-1, 3)]],
((0, 0, 0, 0, 0, 0, 0), 1): [[(1, -3), (-1, 3)]],
((0, -2, 0, 1, 0, 0, 0), 1): [[(-1,), (-1, 3)]],
((-2, 0, 1, 0, 0, 0, 0), 1): [[(2, -1), (1,)]]}
Returns the highest weight elements of self.
EXAMPLES:
sage: K = KirillovReshetikhinCrystal(['E',6,1],2,1)
sage: K.hw_auxiliary()
[[], [[(2, -1), (1,)]],
[[(5, -3), (-1, 3)]],
[[(6, -2), (-6, 2)]],
[[(5, -2, -6), (-6, 2)]],
[[(-1,), (-6, 2)]],
[[(3, -1, -6), (1,)]],
[[(4, -3, -6), (-1, 3)]],
[[(1, -3), (-1, 3)]],
[[(-1,), (-1, 3)]]]
Specifies the promotion operator used to construct the affine type crystal.
EXAMPLES:
sage: K = KirillovReshetikhinCrystal(['E',6,1], 2,1)
sage: promotion = K.promotion()
sage: all(promotion(promotion(promotion(b))) == b for b in K.classical_decomposition())
True
sage: K = KirillovReshetikhinCrystal(['E',6,1],1,1)
sage: promotion = K.promotion()
sage: all(promotion(promotion(promotion(b))) == b for b in K.classical_decomposition())
True
Returns the inverse promotion. Since promotion is of order 3, the inverse promotion is the same as promotion applied twice.
EXAMPLES:
sage: K = KirillovReshetikhinCrystal(['E',6,1], 2,1)
sage: p = K.promotion()
sage: p_inv = K.promotion_inverse()
sage: all(p_inv(p(b)) == b for b in K.classical_decomposition())
True
Gives a dictionary of the promotion map on highest weight elements to
elements in self.
EXAMPLES:
sage: K = KirillovReshetikhinCrystal(['E',6,1],2,1)
sage: dic = K.promotion_on_highest_weight_vectors()
sage: dic
{[[(5, -2, -6), (-6, 2)]]: [[(2, -1), (1,)]],
[[(3, -1, -6), (1,)]]: [[(5, -3), (-1, 3)]],
[[(6, -2), (-6, 2)]]: [],
[[(2, -1), (1,)]]: [[(-1,), (-1, 3)]],
[]: [[(1, -3), (-1, 3)]]}
Return a lambda function on x defined by self.promotion_on_highest_weight_vectors()[x].
EXAMPLES:
sage: K = KirillovReshetikhinCrystal(['E',6,1], 2,1)
sage: f = K.promotion_on_highest_weight_vectors_function()
sage: f(K.module_generator().lift())
[[(-1,), (-1, 3)]]
Bases: sage.combinat.crystals.kirillov_reshetikhin.KirillovReshetikhinGenericCrystal, sage.combinat.crystals.affine.AffineCrystalFromClassical
Class of Kirillov-Reshetikhin crystals of type
for
and type
for
.
EXAMPLES:
sage: K = KirillovReshetikhinCrystal(['A',4,2], 1,1)
sage: K
Kirillov-Reshetikhin crystal of type ['BC', 2, 2] with (r,s)=(1,1)
sage: b = K(rows=[])
sage: b.f(0)
[[1]]
sage: b.e(0)
[[-1]]
alias of KR_type_boxElement
Returns the ambient crystal of type
associated to the Kirillov-Reshetikhin crystal.
This ambient crystal is used to construct the zero arrows.
EXAMPLES:
sage: K = KirillovReshetikhinCrystal(['A',4,2], 2,2)
sage: K.ambient_crystal()
Kirillov-Reshetikhin crystal of type ['C', 2, 1] with (r,s)=(2,4)
Gives a dictionary of the classical highest weight vectors of the ambient crystal of self. Their key is their shape.
EXAMPLES:
sage: K = KirillovReshetikhinCrystal(['A',6,2], 2,2)
sage: K.ambient_highest_weight_dict()
{[4, 2]: [[1, 1, 1, 1], [2, 2]], [2, 2]: [[1, 1], [2, 2]], []: [], [4]: [[1, 1, 1, 1]], [4, 4]: [[1, 1, 1, 1], [2, 2, 2, 2]],
[2]: [[1, 1]]}
Specifies the classical crystal underlying the Kirillov-Reshetikhin crystal of type
and
.
It is given by where
are weights obtained from
a rectangle of width
and height
by removing boxes. Here we identify the fundamental
weight
with a column of height
.
EXAMPLES:
sage: K = KirillovReshetikhinCrystal(['A',4,2], 2,2)
sage: K.classical_decomposition()
The crystal of tableaux of type ['C', 2] and shape(s) [[], [1], [2], [1, 1], [2, 1], [2, 2]]
sage: K = KirillovReshetikhinCrystal(['D',4,2], 2,3)
sage: K.classical_decomposition()
The crystal of tableaux of type ['B', 3] and shape(s) [[], [1], [2], [1, 1], [3], [2, 1], [3, 1], [2, 2], [3, 2], [3, 3]]
Provides a map from the ambient crystal of type to the
Kirillov-Reshetikhin crystal self.
Note that this map is only well-defined on elements that are in the image under to_ambient_crystal().
EXAMPLES:
sage: K = KirillovReshetikhinCrystal(['D',4,2], 1,1)
sage: b = K.ambient_crystal()(rows=[[3,-3]])
sage: K.from_ambient_crystal()(b)
[[0]]
sage: K = KirillovReshetikhinCrystal(['A',4,2], 1,1)
sage: b = K.ambient_crystal()(rows=[])
sage: K.from_ambient_crystal()(b)
[]
Gives a dictionary of the classical highest weight vectors of self. Their key is 2 times their shape.
EXAMPLES:
sage: K = KirillovReshetikhinCrystal(['A',6,2], 2,2)
sage: K.highest_weight_dict()
{[4, 2]: [[1, 1], [2]], [2, 2]: [[1], [2]], []: [], [4]: [[1, 1]], [4, 4]: [[1, 1], [2, 2]], [2]: [[1]]}
Sets the similarity factor used to map to the ambient crystal.
EXAMPLES:
sage: K = KirillovReshetikhinCrystal(['A',6,2], 2,2)
sage: K.similarity_factor()
{1: 2, 2: 2, 3: 2}
sage: K = KirillovReshetikhinCrystal(['D',5,2], 1,1)
sage: K.similarity_factor()
{1: 2, 2: 2, 3: 2, 4: 1}
Provides a map from self to the ambient crystal of type .
EXAMPLES:
sage: K = KirillovReshetikhinCrystal(['D',4,2], 1,1)
sage: [K.to_ambient_crystal()(b) for b in K]
[[], [[1, 1]], [[2, 2]], [[3, 3]], [[3, -3]], [[-3, -3]], [[-2, -2]], [[-1, -1]]]
sage: K = KirillovReshetikhinCrystal(['A',4,2], 1,1)
sage: [K.to_ambient_crystal()(b) for b in K]
[[], [[1, 1]], [[2, 2]], [[-2, -2]], [[-1, -1]]]
Bases: sage.combinat.crystals.kirillov_reshetikhin.KirillovReshetikhinGenericCrystalElement
Class for the elements in the Kirillov-Reshetikhin crystals of type
for
and type
for
.
EXAMPLES:
sage: K=KirillovReshetikhinCrystal(['A',4,2],1,2)
sage: type(K.module_generators[0])
<class 'sage.combinat.crystals.kirillov_reshetikhin.KR_type_box_with_category.element_class'>
Gives on self by mapping self to the ambient crystal, calculating
there and
pulling the element back.
EXAMPLES:
sage: K=KirillovReshetikhinCrystal(['A',4,2],1,1)
sage: b = K(rows=[])
sage: b.e(0) # indirect doctest
[[-1]]
Calculates of self by mapping the element to the ambient crystal
and calculating
there.
EXAMPLES:
sage: K = KirillovReshetikhinCrystal(['A',4,2], 1,1)
sage: b=K(rows=[[1]])
sage: b.epsilon(0) # indirect doctest
2
Gives on self by mapping self to the ambient crystal, calculating
there and
pulling the element back.
EXAMPLES:
sage: K=KirillovReshetikhinCrystal(['A',4,2],1,1)
sage: b = K(rows=[])
sage: b.f(0) # indirect doctest
[[1]]
Calculates of self by mapping the element to the ambient crystal
and calculating
there.
EXAMPLES:
sage: K = KirillovReshetikhinCrystal(['D',3,2], 1,1)
sage: b=K(rows=[[-1]])
sage: b.phi(0) # indirect doctest
2
Bases: sage.combinat.crystals.kirillov_reshetikhin.KirillovReshetikhinCrystalFromPromotion
Class of Kirillov-Reshetikhin crystals of type
.
EXAMPLES:
sage: K = KirillovReshetikhinCrystal(['D',4,1],4,1); K
Kirillov-Reshetikhin crystal of type ['D', 4, 1] with (r,s)=(4,1)
sage: [[b,b.f(0)] for b in K]
[[[++++, []], None], [[++--, []], None], [[+-+-, []], None], [[-++-, []], None],
[[+--+, []], None], [[-+-+, []], None], [[--++, []], [++++, []]], [[----, []], [++--, []]]]
sage: K = KirillovReshetikhinCrystal(['D',4,1],4,2); K
Kirillov-Reshetikhin crystal of type ['D', 4, 1] with (r,s)=(4,2)
sage: [[b,b.f(0)] for b in K]
[[[[1], [2], [3], [4]], None], [[[1], [2], [-4], [4]], None], [[[1], [3], [-4], [4]], None],
[[[2], [3], [-4], [4]], None], [[[1], [4], [-4], [4]], None], [[[2], [4], [-4], [4]], None],
[[[3], [4], [-4], [4]], [[1], [2], [3], [4]]], [[[-4], [4], [-4], [4]], [[1], [2], [-4], [4]]],
[[[-4], [4], [-4], [-3]], [[1], [2], [-4], [-3]]], [[[-4], [4], [-4], [-2]], [[1], [3], [-4], [-3]]],
[[[-4], [4], [-4], [-1]], [[2], [3], [-4], [-3]]], [[[-4], [4], [-3], [-2]], [[1], [4], [-4], [-3]]],
[[[-4], [4], [-3], [-1]], [[2], [4], [-4], [-3]]], [[[-4], [4], [-2], [-1]], [[-4], [4], [-4], [4]]],
[[[-4], [-3], [-2], [-1]], [[-4], [4], [-4], [-3]]], [[[1], [2], [-4], [-3]], None], [[[1], [3], [-4], [-3]], None],
[[[2], [3], [-4], [-3]], None], [[[1], [3], [-4], [-2]], None], [[[2], [3], [-4], [-2]], None],
[[[2], [3], [-4], [-1]], None], [[[1], [4], [-4], [-3]], None], [[[2], [4], [-4], [-3]], None],
[[[3], [4], [-4], [-3]], None], [[[3], [4], [-4], [-2]], [[1], [3], [-4], [4]]],
[[[3], [4], [-4], [-1]], [[2], [3], [-4], [4]]], [[[1], [4], [-4], [-2]], None], [[[2], [4], [-4], [-2]], None],
[[[2], [4], [-4], [-1]], None], [[[1], [4], [-3], [-2]], None], [[[2], [4], [-3], [-2]], None],
[[[2], [4], [-3], [-1]], None], [[[3], [4], [-3], [-2]], [[1], [4], [-4], [4]]],
[[[3], [4], [-3], [-1]], [[2], [4], [-4], [4]]], [[[3], [4], [-2], [-1]], [[3], [4], [-4], [4]]]]
TESTS:
sage: K = KirillovReshetikhinCrystal(['D',4,1],3,1)
sage: all(b.e(0).f(0) == b for b in K if b.epsilon(0)>0)
True
sage: K = KirillovReshetikhinCrystal(['D',5,1],5,2)
sage: all(b.f(0).e(0) == b for b in K if b.phi(0)>0)
True
Returns the classical crystal underlying the Kirillov-Reshetikhin crystal
of type
for
. It is given by
.
EXAMPLES:
sage: K = KirillovReshetikhinCrystal(['D',4,1],4,1)
sage: K.classical_decomposition()
The crystal of tableaux of type ['D', 4] and shape(s) [[1/2, 1/2, 1/2, 1/2]]
sage: K = KirillovReshetikhinCrystal(['D',4,1],3,1)
sage: K.classical_decomposition()
The crystal of tableaux of type ['D', 4] and shape(s) [[1/2, 1/2, 1/2, -1/2]]
sage: K = KirillovReshetikhinCrystal(['D',4,1],3,2)
sage: K.classical_decomposition()
The crystal of tableaux of type ['D', 4] and shape(s) [[1, 1, 1, -1]]
Specifies the Dynkin diagram automorphism underlying the promotion action on the crystal elements. The automorphism needs to map node 0 to some other Dynkin node.
Here we use the Dynkin diagram automorphism which interchanges nodes 0 and 1 and leaves all other nodes unchanged.
EXAMPLES:
sage: K = KirillovReshetikhinCrystal(['D',4,1],4,1)
sage: K.dynkin_diagram_automorphism(0)
1
sage: K.dynkin_diagram_automorphism(1)
0
sage: K.dynkin_diagram_automorphism(4)
4
Returns the promotion operator on of type
for
.
EXAMPLES:
sage: K = KirillovReshetikhinCrystal(['D',4,1],3,1)
sage: T = K.classical_decomposition()
sage: promotion = K.promotion()
sage: for t in T:
... print t, promotion(t)
...
[+++-, []] [-++-, []]
[++-+, []] [-+-+, []]
[+-++, []] [--++, []]
[-+++, []] [++++, []]
[+---, []] [----, []]
[-+--, []] [++--, []]
[--+-, []] [+-+-, []]
[---+, []] [+--+, []]
Returns the inverse promotion operator on of type
for
.
EXAMPLES:
sage: K = KirillovReshetikhinCrystal(['D',4,1],3,1)
sage: T = K.classical_decomposition()
sage: promotion = K.promotion()
sage: promotion_inverse = K.promotion_inverse()
sage: all(promotion_inverse(promotion(t)) == t for t in T)
True
Returns the promotion operator on -highest weight vectors.
A -highest weight vector in
of weight
is mapped to a
-highest weight vector in
of weight
and vice versa.
See also promotion_on_highest_weight_vectors_inverse() and promotion().
EXAMPLES:
sage: KR = KirillovReshetikhinCrystal(['D',4,1],4,2)
sage: prom = KR.promotion_on_highest_weight_vectors()
sage: T = KR.classical_decomposition()
sage: HW = [t for t in T if t.is_highest_weight([2,3,4])]
sage: for t in HW:
... print t, prom[t]
...
[4, 3, 2, 1] [-1, 4, 3, 2]
[4, -4, 3, 2] [-4, 4, 3, 2]
[-1, -4, 3, 2] [-4, 3, 2, 1]
sage: KR = KirillovReshetikhinCrystal(['D',4,1],4,1)
sage: prom = KR.promotion_on_highest_weight_vectors()
sage: T = KR.classical_decomposition()
sage: HW = [t for t in T if t.is_highest_weight([2,3,4])]
sage: for t in HW:
... print t, prom[t]
...
[++++, []] [-+++, []]
[-++-, []] [+++-, []]
Returns the inverse promotion operator on -highest weight vectors.
See also promotion_on_highest_weight_vectors() and promotion_inverse().
EXAMPLES:
sage: KR = KirillovReshetikhinCrystal(['D',4,1],3,2)
sage: prom = KR.promotion_on_highest_weight_vectors()
sage: prom_inv = KR.promotion_on_highest_weight_vectors_inverse()
sage: T = KR.classical_decomposition()
sage: HW = [t for t in T if t.is_highest_weight([2,3,4])]
sage: all(prom_inv[prom[t]] == t for t in HW)
True
Bases: sage.combinat.crystals.kirillov_reshetikhin.KirillovReshetikhinCrystalFromPromotion
Class of Kirillov-Reshetikhin crystals of type
for
,
for
, and
for
.
EXAMPLES:
sage: K = KirillovReshetikhinCrystal(['D',4,1], 2,2)
sage: b = K(rows=[])
sage: b.f(0)
[[1], [2]]
sage: b.f(0).f(0)
[[1, 1], [2, 2]]
sage: b.e(0)
[[-2], [-1]]
sage: b.e(0).e(0)
[[-2, -2], [-1, -1]]
sage: K = KirillovReshetikhinCrystal(['D',5,1], 3,1)
sage: b = K(rows=[[1]])
sage: b.e(0)
[[3], [-3], [-2]]
sage: K = KirillovReshetikhinCrystal(['B',3,1], 1,1)
sage: [[b,b.f(0)] for b in K]
[[[[1]], None], [[[2]], None], [[[3]], None], [[[0]], None], [[[-3]], None], [[[-2]], [[1]]], [[[-1]], [[2]]]]
sage: K = KirillovReshetikhinCrystal(['A',5,2], 1,1)
sage: [[b,b.f(0)] for b in K]
[[[[1]], None], [[[2]], None], [[[3]], None], [[[-3]], None], [[[-2]], [[1]]], [[[-1]], [[2]]]]
Specifies the classical crystal underlying the Kirillov-Reshetikhin crystal of type ,
, and
.
It is given by where
are weights obtained from
a rectangle of width
and height
by removing verticle dominoes. Here we identify the fundamental
weight
with a column of height
.
EXAMPLES:
sage: K = KirillovReshetikhinCrystal(['D',4,1], 2,2)
sage: K.classical_decomposition()
The crystal of tableaux of type ['D', 4] and shape(s) [[], [1, 1], [2, 2]]
Specifies the Dynkin diagram automorphism underlying the promotion action on the crystal elements. The automorphism needs to map node 0 to some other Dynkin node.
Here we use the Dynkin diagram automorphism which interchanges nodes 0 and 1 and leaves all other nodes unchanged.
EXAMPLES:
sage: K = KirillovReshetikhinCrystal(['D',4,1],1,1)
sage: K.dynkin_diagram_automorphism(0)
1
sage: K.dynkin_diagram_automorphism(1)
0
sage: K.dynkin_diagram_automorphism(4)
4
This gives the bijection between an element b in the classical decomposition
of the KR crystal that is -highest weight and
diagrams.
EXAMPLES:
sage: K = KirillovReshetikhinCrystal(['D',4,1], 2,2)
sage: T = K.classical_decomposition()
sage: b = T(rows=[[2],[-2]])
sage: pm = K.from_highest_weight_vector_to_pm_diagram(b); pm
[[1, 1], [0, 0], [0]]
sage: pm.__repr__(pretty_printing=True)
+
-
sage: b = T(rows=[])
sage: pm=K.from_highest_weight_vector_to_pm_diagram(b); pm
[[0, 2], [0, 0], [0]]
sage: pm.__repr__(pretty_printing=True)
sage: hw = [ b for b in T if all(b.epsilon(i)==0 for i in [2,3,4]) ]
sage: all(K.from_pm_diagram_to_highest_weight_vector(K.from_highest_weight_vector_to_pm_diagram(b)) == b for b in hw)
True
This gives the bijection between a diagram and an element b in the classical
decomposition of the KR crystal that is {2,3,..,n}-highest weight.
EXAMPLES:
sage: K = KirillovReshetikhinCrystal(['D',4,1], 2,2)
sage: pm = sage.combinat.crystals.kirillov_reshetikhin.PMDiagram([[1, 1], [0, 0], [0]])
sage: K.from_pm_diagram_to_highest_weight_vector(pm)
[[2], [-2]]
Specifies the promotion operator used to construct the affine type etc. crystal.
This corresponds to the Dynkin diagram automorphism which interchanges nodes 0 and 1,
and leaves all other nodes unchanged. On the level of crystals it is constructed using
diagrams.
EXAMPLES:
sage: K = KirillovReshetikhinCrystal(['D',4,1], 2,2)
sage: promotion = K.promotion()
sage: b = K.classical_decomposition()(rows=[])
sage: promotion(b)
[[1, 2], [-2, -1]]
sage: b = K.classical_decomposition()(rows=[[1,3],[2,-1]])
sage: promotion(b)
[[1, 3], [2, -1]]
sage: b = K.classical_decomposition()(rows=[[1],[-3]])
sage: promotion(b)
[[2, -3], [-2, -1]]
Return inverse of promotion.
In this case promotion is an involution, so promotion inverse equals promotion.
EXAMPLES:
sage: K = KirillovReshetikhinCrystal(['D',4,1], 2,2)
sage: promotion = K.promotion()
sage: promotion_inverse = K.promotion_inverse()
sage: all( promotion_inverse(promotion(b.lift())) == b.lift() for b in K )
True
Calculates promotion on highest weight vectors.
EXAMPLES:
sage: K = KirillovReshetikhinCrystal(['D',4,1], 2,2)
sage: T = K.classical_decomposition()
sage: hw = [ b for b in T if all(b.epsilon(i)==0 for i in [2,3,4]) ]
sage: [K.promotion_on_highest_weight_vectors()(b) for b in hw]
[[[1, 2], [-2, -1]], [[2, 2], [-2, -1]], [[1, 2], [3, -1]], [[2], [-2]],
[[1, 2], [2, -2]], [[2, 2], [-1, -1]], [[2, 2], [3, -1]], [[2, 2], [3, 3]],
[], [[1], [2]], [[1, 1], [2, 2]], [[2], [-1]], [[1, 2], [2, -1]], [[2], [3]],
[[1, 2], [2, 3]]]
Return the Kirillov-Reshetikhin crystal of the given type.
For more information about general crystals see sage.combinat.crystals.
Many Kirillov-Reshetikhin crystals are constructed from a
classical crystal together with an automorphism on the level of crystals which
corresponds to a Dynkin diagram automorphism mapping node 0 to some other node
.
The action of
and
is then constructed using
.
For example, for type the Kirillov-Reshetikhin crystal
is obtained from the classical crystal
using the
promotion operator. For other types, see [Shimozono02], [Schilling08],
and [JS2010].
Other Kirillov-Reshetikhin crystals are constructed using similarity methods. See Section 4 of [FOS09].
For an implementation of Kirillov-Reshetikhin crystals for from
crystals of LS paths, see KirillovReshetikhinCrystalFromLSPaths().
INPUT:
EXAMPLES:
sage: K = KirillovReshetikhinCrystal(['A',3,1], 2, 1)
sage: K.index_set()
(0, 1, 2, 3)
sage: K.list()
[[[1], [2]], [[1], [3]], [[2], [3]], [[1], [4]], [[2], [4]], [[3], [4]]]
sage: b=K(rows=[[1],[2]])
sage: b.weight()
-Lambda[0] + Lambda[2]
sage: K = KirillovReshetikhinCrystal(['A',3,1], 2,2)
sage: K.automorphism(K.module_generators[0])
[[2, 2], [3, 3]]
sage: K.module_generators[0].e(0)
[[1, 2], [2, 4]]
sage: K.module_generators[0].f(2)
[[1, 1], [2, 3]]
sage: K.module_generators[0].f(1)
sage: K.module_generators[0].phi(0)
0
sage: K.module_generators[0].phi(1)
0
sage: K.module_generators[0].phi(2)
2
sage: K.module_generators[0].epsilon(0)
2
sage: K.module_generators[0].epsilon(1)
0
sage: K.module_generators[0].epsilon(2)
0
sage: b = K(rows=[[1,2],[2,3]])
sage: b
[[1, 2], [2, 3]]
sage: b.f(2)
[[1, 2], [3, 3]]
sage: K = KirillovReshetikhinCrystal(['D',4,1], 2, 1)
sage: K.cartan_type()
['D', 4, 1]
sage: type(K.module_generators[0])
<class 'sage.combinat.crystals.kirillov_reshetikhin.KR_type_vertical_with_category.element_class'>
The following gives some tests with regards to Lemma 3.11 in [LOS12].
TESTS:
sage: K = KirillovReshetikhinCrystal(['A',4,2],2,1)
sage: Lambda = K.weight_lattice_realization().fundamental_weights()
sage: [b for b in K if b.Epsilon() == Lambda[0]]
[[]]
sage: K = KirillovReshetikhinCrystal(['D',4,2],1,2)
sage: Lambda = K.weight_lattice_realization().fundamental_weights()
sage: [b for b in K if b.Epsilon() == 2*Lambda[0]]
[[]]
sage: [b for b in K if b.Epsilon() == 2*Lambda[3]]
[[[3, -3]]]
sage: K = KirillovReshetikhinCrystal(['D',4,2],1,1)
sage: [b for b in K if b.Epsilon() == Lambda[3]]
[[[0]]]
sage: K = KirillovReshetikhinCrystal(['B',3,1],2,1)
sage: Lambda = K.weight_lattice_realization().fundamental_weights()
sage: [b for b in K if b.Epsilon() == Lambda[0]]
[[]]
sage: [b for b in K if b.Epsilon() == Lambda[1]]
[[[2], [-2]]]
sage: K = KirillovReshetikhinCrystal(['B',3,1],2,2)
sage: [b for b in K if b.Epsilon() == 2*Lambda[0]]
[[]]
sage: [b for b in K if b.Epsilon() == 2*Lambda[1]]
[[[1, 2], [-2, -1]]]
sage: K = KirillovReshetikhinCrystal(['B',3,1],2,3)
sage: [b for b in K if b.Epsilon() == 3*Lambda[1]] # long time
[[[1, 2, 2], [-2, -2, -1]]]
sage: K = KirillovReshetikhinCrystal(['D',4,1],2,2)
sage: Lambda = K.weight_lattice_realization().fundamental_weights()
sage: [b for b in K if b.Epsilon() == 2*Lambda[0]] # long time
[[]]
sage: [b for b in K if b.Epsilon() == 2*Lambda[4]] # long time
[[[3, -4], [4, -3]]]
sage: K = KirillovReshetikhinCrystal(['B',3,1],3,1)
sage: Lambda = K.weight_lattice_realization().fundamental_weights()
sage: [b for b in K if b.Epsilon() == Lambda[0]]
[[+++, []]]
sage: [b for b in K if b.Epsilon() == Lambda[1]]
[[-++, []]]
sage: K = KirillovReshetikhinCrystal(['B',3,1],3,3)
sage: [b for b in K if b.Epsilon() == 2*Lambda[0]] # long time
[[+++, [[1]]]]
sage: [b for b in K if b.Epsilon() == 2*Lambda[1]] # long time
[[-++, [[-1]]]]
sage: K = KirillovReshetikhinCrystal(['B',4,1],4,1)
sage: Lambda = K.weight_lattice_realization().fundamental_weights()
sage: [b for b in K if b.Epsilon() == Lambda[0]]
[[++++, []]]
sage: [b for b in K if b.Epsilon() == Lambda[1]]
[[-+++, []]]
sage: K = KirillovReshetikhinCrystal(['C',3,1],1,1)
sage: Lambda = K.weight_lattice_realization().fundamental_weights()
sage: [b for b in K if b.Epsilon() == Lambda[0]]
[[[1]]]
sage: [b for b in K if b.Epsilon() == Lambda[3]]
[[[-3]]]
sage: K = KirillovReshetikhinCrystal(['C',3,1],1,3)
sage: [b for b in K if b.Epsilon() == 2*Lambda[3]] # long time
[[[3, -3, -3]]]
sage: [b for b in K if b.Epsilon() == 2*Lambda[0]] # long time
[[[1]]]
REFERENCES:
[Shimozono02] | M. Shimozono Affine type A crystal structure on tensor products of rectangles, Demazure characters, and nilpotent varieties, J. Algebraic Combin. 15 (2002). no. 2. 151-187. Arxiv math.QA/9804039. |
[Schilling08] | A. Schilling. “Combinatorial structure of
Kirillov-Reshetikhin crystals of type ![]() ![]() ![]() |
[JS2010] | B. Jones, A. Schilling.
“Affine structures and a tableau model for ![]() |
[FOS09] | G. Fourier, M. Okado, A. Schilling. Kirillov-Reshetikhin crystals for nonexceptional types. Advances in Mathematics. 222 (2009). Issue 3. 1080-1116. Arxiv 0810.5067. |
[LOS12] | C. Lecouvey, M. Okado, M. Shimozono.
“Affine crystals, one-dimensional sums and parabolic Lusztig
![]() |
Single column Kirillov-Reshetikhin crystals.
This yields the single column Kirillov-Reshetikhin crystals
from the projected level zero LS paths, see sage.combinat.crystals.littelmann_paths.CrystalOfLSPaths.
This works for all types (even exceptional types).
The weight of the canonical element in this crystal is .
For other implementation see KirillovReshetikhinCrystal().
EXAMPLES:
sage: from sage.combinat.crystals.kirillov_reshetikhin import KirillovReshetikhinCrystalFromLSPaths
sage: K = KirillovReshetikhinCrystalFromLSPaths(['A',2,1],2)
sage: KR = KirillovReshetikhinCrystal(['A',2,1],2,1)
sage: G = K.digraph()
sage: GR = KR.digraph()
sage: G.is_isomorphic(GR, edge_labels = True)
True
sage: K = KirillovReshetikhinCrystalFromLSPaths(['C',3,1],2)
sage: KR = KirillovReshetikhinCrystal(['C',3,1],2,1)
sage: G = K.digraph()
sage: GR = KR.digraph()
sage: G.is_isomorphic(GR, edge_labels = True)
True
sage: K = KirillovReshetikhinCrystalFromLSPaths(['E',6,1],1)
sage: KR = KirillovReshetikhinCrystal(['E',6,1],1,1)
sage: G = K.digraph()
sage: GR = KR.digraph()
sage: G.is_isomorphic(GR, edge_labels = True)
True
sage: K.cardinality()
27
sage: K = KirillovReshetikhinCrystalFromLSPaths(['G',2,1],1)
sage: K.cardinality()
7
sage: K = KirillovReshetikhinCrystalFromLSPaths(['B',3,1],2)
sage: KR = KirillovReshetikhinCrystal(['B',3,1],2,1)
sage: KR.cardinality()
22
sage: K.cardinality()
22
sage: G = K.digraph()
sage: GR = KR.digraph()
sage: G.is_isomorphic(GR, edge_labels = True)
True
TESTS:
sage: K = KirillovReshetikhinCrystalFromLSPaths(['G',2,1],2)
sage: K.cardinality()
15
For these crystals yield
-fold tensor products of Kirillov-Reshetikhin crystals:
sage: K = KirillovReshetikhinCrystalFromLSPaths(['A',1,1],1,3)
sage: B = KirillovReshetikhinCrystal(['A',1,1],1,1)
sage: T = TensorProductOfCrystals(B,B,B)
sage: G = K.digraph()
sage: GT = T.digraph()
sage: G.is_isomorphic(GT, edge_labels = True)
True
sage: K = KirillovReshetikhinCrystalFromLSPaths(['B',2,1],1,2)
sage: B = KirillovReshetikhinCrystal(['B',2,1],1,1)
sage: T = TensorProductOfCrystals(B,B)
sage: G = K.digraph()
sage: GT = T.digraph()
sage: G.is_isomorphic(GT, edge_labels = True)
True
sage: K = KirillovReshetikhinCrystalFromLSPaths(['B',2,1],2,3)
sage: B = KirillovReshetikhinCrystal(['B',2,1],2,1)
sage: T = TensorProductOfCrystals(B,B,B)
sage: GT = T.digraph()
sage: G = K.digraph()
sage: G.is_isomorphic(GT, edge_labels = True)
True
Bases: sage.combinat.crystals.kirillov_reshetikhin.KirillovReshetikhinGenericCrystal, sage.combinat.crystals.affine.AffineCrystalFromClassicalAndPromotion
This generic class assumes that the Kirillov-Reshetikhin crystal is constructed from a classical crystal ‘classical_decomposition’ and an automorphism ‘promotion’ and its inverse which corresponds to a Dynkin diagram automorphism ‘dynkin_diagram_automorphism’.
Each instance using this class needs to implement the methods:
Bases: sage.combinat.crystals.affine.AffineCrystalFromClassicalAndPromotionElement, sage.combinat.crystals.kirillov_reshetikhin.KirillovReshetikhinGenericCrystalElement
Element for a Kirillov-Reshetikhin crystal from promotion.
Bases: sage.combinat.crystals.affine.AffineCrystalFromClassical
Generic class for Kirillov-Reshetikhin crystal of the given type.
Input is a Dynkin node r, a positive integer s, and a Cartan type cartan_type.
INPUT:
Returns the combinatorial `R`-matrix from , where the combinatorial
-matrix is the affine
crystal isomorphism which maps
to
, where
is the unique element in
of weight
(see
module_generator).
EXAMPLES:
sage: K = KirillovReshetikhinCrystal(['A',2,1],1,1)
sage: L = KirillovReshetikhinCrystal(['A',2,1],1,2)
sage: f = K.R_matrix(L)
sage: [[b,f(b)] for b in TensorProductOfCrystals(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 = KirillovReshetikhinCrystal(['D',4,1],1,1)
sage: L = KirillovReshetikhinCrystal(['D',4,1],2,1)
sage: f = K.R_matrix(L)
sage: T = TensorProductOfCrystals(K,L)
sage: b = T( K(rows=[[1]]), L(rows=[]) )
sage: f(b)
[[[2], [-2]], [[1]]]
Alternatively, one can compute the combinatorial -matrix using the isomorphism method
of digraphs:
sage: K1 = KirillovReshetikhinCrystal(['A',2,1],1,1)
sage: K2 = KirillovReshetikhinCrystal(['A',2,1],2,1)
sage: T1 = TensorProductOfCrystals(K1,K2)
sage: T2 = TensorProductOfCrystals(K2,K1)
sage: T1.digraph().is_isomorphic(T2.digraph(), edge_labels = True, certify = 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]]]})
Return the classical decomposition of self.
EXAMPLES:
sage: K = KirillovReshetikhinCrystal(['A',3,1], 2,2)
sage: K.classical_decomposition()
The crystal of tableaux of type ['A', 3] and shape(s) [[2, 2]]
Returns True or False depending on whether self is a perfect crystal or not, respectively.
If self is the Kirillov-Reshetikhin crystal , then it was proven in [FOS2010]
that it is perfect if and only if
is an integer (where
is a constant related to the
type of the crystal).
REFERENCES:
EXAMPLES:
sage: K = KirillovReshetikhinCrystal(['A',2,1], 1, 1)
sage: K.is_perfect()
True
sage: K = KirillovReshetikhinCrystal(['C',2,1], 1, 1)
sage: K.is_perfect()
False
sage: K = KirillovReshetikhinCrystal(['C',2,1], 1, 2)
sage: K.is_perfect()
True
Return the corresponding set of KirillovReshetikhinTableaux.
EXAMPLES:
sage: KRC = KirillovReshetikhinCrystal(['D', 4, 1], 2, 2)
sage: KRC.kirillov_reshetikhin_tableaux()
Kirillov-Reshetikhin tableaux of type ['D', 4, 1] and shape (2, 2)
Returns the level of self assuming that it is a perfect crystal.
If self is the Kirillov-Reshetikhin crystal , then it was proven in [FOS2010]
that its level is
which is an integer if self is perfect
(here
is a constant related to the type of the crystal).
EXAMPLES:
sage: K = KirillovReshetikhinCrystal(['A',2,1], 1, 1)
sage: K.level()
1
sage: K = KirillovReshetikhinCrystal(['C',2,1], 1, 2)
sage: K.level()
1
sage: K = KirillovReshetikhinCrystal(['D',4,1], 1, 3)
sage: K.level()
3
sage: K = KirillovReshetikhinCrystal(['C',2,1], 1, 1)
sage: K.level()
Traceback (most recent call last):
...
AssertionError: This crystal is not perfect!
Returns the unique module generator of classical weight of a Kirillov-Reshetikhin crystal
EXAMPLES:
sage: K = KirillovReshetikhinCrystal(['C',2,1],1,2)
sage: K.module_generator()
[[1, 1]]
sage: K = KirillovReshetikhinCrystal(['E',6,1],1,1)
sage: K.module_generator()
[(1,)]
sage: K = KirillovReshetikhinCrystal(['D',4,1],2,1)
sage: K.module_generator()
[[1], [2]]
Returns r of the underlying Kirillov-Reshetikhin crystal
EXAMPLES:
sage: K = KirillovReshetikhinCrystal(['D',4,1], 2, 1)
sage: K.r()
2
Returns s of the underlying Kirillov-Reshetikhin crystal
EXAMPLES:
sage: K = KirillovReshetikhinCrystal(['D',4,1], 2, 1)
sage: K.s()
1
Bases: sage.combinat.crystals.affine.AffineCrystalFromClassicalElement
Abstract class for all Kirillov-Reshetikhin crystal elements.
Construct the corresponding KirillovReshetikhinTableauxElement from self.
We construct the Kirillov-Reshetikhin tableau element as follows:
EXAMPLES:
sage: KRC = KirillovReshetikhinCrystal(['A', 4, 1], 2, 1)
sage: KRC(columns=[[2,1]]).to_kirillov_reshetikhin_tableau()
[[1], [2]]
sage: KRC = KirillovReshetikhinCrystal(['D', 4, 1], 2, 1)
sage: KRC(rows=[]).to_kirillov_reshetikhin_tableau()
[[1], [-1]]
Bases: sage.combinat.combinat.CombinatorialObject
Class of diagrams. These diagrams are in one-to-one bijection with
highest weight vectors
in an
highest weight crystal
. See Section 4.1 of A. Schilling, “Combinatorial structure of
Kirillov-Reshetikhin crystals of type
,
,
”, J. Algebra 319 (2008) 2938-2962
(arXiv:0704.2046[math.QA]).
The input is a list of 2-tuples and a last 1-tuple.
The tuple
specifies the number of
+ and
- in the i-th row of the pm diagram
if
is odd and the number of
+- pairs above row
and
columns of height
not containing
any + or - if
is even.
Setting the option ‘from_shapes = True’ one can also input a diagram in terms of its
outer, intermediate and inner shape by specifying a tuple [n, s, outer, intermediate, inner]
where
is the width of the
diagram, and ‘outer’ , ‘intermediate’,
and ‘inner’ are the outer, intermediate and inner shape, respectively.
EXAMPLES:
sage: pm=sage.combinat.crystals.kirillov_reshetikhin.PMDiagram([[0,1],[1,2],[1]])
sage: pm.pm_diagram
[[0, 1], [1, 2], [1]]
sage: pm._list
[1, 1, 2, 0, 1]
sage: pm.n
2
sage: pm.width
5
sage: pm.__repr__(pretty_printing=True)
. . . .
. + - -
sage: sage.combinat.crystals.kirillov_reshetikhin.PMDiagram([2,5,[4,4],[4,2],[4,1]], from_shapes=True)
[[0, 1], [1, 2], [1]]
TESTS:
sage: pm=sage.combinat.crystals.kirillov_reshetikhin.PMDiagram([[1,2],[1,1],[1,1],[1,1],[1]])
sage: sage.combinat.crystals.kirillov_reshetikhin.PMDiagram([pm.n, pm.width, pm.outer_shape(), pm.intermediate_shape(), pm.inner_shape()], from_shapes=True) == pm
True
sage: pm=sage.combinat.crystals.kirillov_reshetikhin.PMDiagram([[1,2],[1,2],[1,1],[1,1],[1,1],[1]])
sage: sage.combinat.crystals.kirillov_reshetikhin.PMDiagram([pm.n, pm.width, pm.outer_shape(), pm.intermediate_shape(), pm.inner_shape()], from_shapes=True) == pm
True
Returns a list with the heights of all addable plus in the diagram.
EXAMPLES:
sage: pm=sage.combinat.crystals.kirillov_reshetikhin.PMDiagram([[1,2],[1,2],[1,1],[1,1],[1,1],[1]])
sage: pm.heights_of_addable_plus()
[1, 1, 2, 3, 4, 5]
sage: pm=sage.combinat.crystals.kirillov_reshetikhin.PMDiagram([[1,2],[1,1],[1,1],[1,1],[1]])
sage: pm.heights_of_addable_plus()
[1, 2, 3, 4]
Returns a list with the heights of all minus in the diagram.
EXAMPLES:
sage: pm=sage.combinat.crystals.kirillov_reshetikhin.PMDiagram([[1,2],[1,2],[1,1],[1,1],[1,1],[1]])
sage: pm.heights_of_minus()
[5, 5, 3, 3, 1, 1]
sage: pm=sage.combinat.crystals.kirillov_reshetikhin.PMDiagram([[1,2],[1,1],[1,1],[1,1],[1]])
sage: pm.heights_of_minus()
[4, 4, 2, 2]
Returns the inner shape of the pm diagram
EXAMPLES:
sage: pm=sage.combinat.crystals.kirillov_reshetikhin.PMDiagram([[0,1],[1,2],[1]])
sage: pm.inner_shape()
[4, 1]
sage: pm=sage.combinat.crystals.kirillov_reshetikhin.PMDiagram([[1,2],[1,1],[1,1],[1,1],[1]])
sage: pm.inner_shape()
[7, 5, 3, 1]
sage: pm=sage.combinat.crystals.kirillov_reshetikhin.PMDiagram([[1,2],[1,2],[1,1],[1,1],[1,1],[1]])
sage: pm.inner_shape()
[10, 7, 5, 3, 1]
Returns the intermediate shape of the pm diagram (inner shape plus positions of plusses)
EXAMPLES:
sage: pm=sage.combinat.crystals.kirillov_reshetikhin.PMDiagram([[0,1],[1,2],[1]])
sage: pm.intermediate_shape()
[4, 2]
sage: pm=sage.combinat.crystals.kirillov_reshetikhin.PMDiagram([[1,2],[1,1],[1,1],[1,1],[1]])
sage: pm.intermediate_shape()
[8, 6, 4, 2]
sage: pm=sage.combinat.crystals.kirillov_reshetikhin.PMDiagram([[1,2],[1,2],[1,1],[1,1],[1,1],[1]])
sage: pm.intermediate_shape()
[11, 8, 6, 4, 2]
sage: pm=sage.combinat.crystals.kirillov_reshetikhin.PMDiagram([[1,0],[0,1],[2,0],[0,0],[0]])
sage: pm.intermediate_shape()
[4, 2, 2]
sage: pm=sage.combinat.crystals.kirillov_reshetikhin.PMDiagram([[1, 0], [0, 0], [0, 0], [0, 0], [0]])
sage: pm.intermediate_shape()
[1]
Returns the outer shape of the pm diagram
EXAMPLES:
sage: pm=sage.combinat.crystals.kirillov_reshetikhin.PMDiagram([[0,1],[1,2],[1]])
sage: pm.outer_shape()
[4, 4]
sage: pm=sage.combinat.crystals.kirillov_reshetikhin.PMDiagram([[1,2],[1,1],[1,1],[1,1],[1]])
sage: pm.outer_shape()
[8, 8, 4, 4]
sage: pm=sage.combinat.crystals.kirillov_reshetikhin.PMDiagram([[1,2],[1,2],[1,1],[1,1],[1,1],[1]])
sage: pm.outer_shape()
[13, 8, 8, 4, 4]
Returns sigma on pm diagrams as needed for the analogue of the Dynkin diagram automorphism
that interchanges nodes and
for type
,
,
for
Kirillov-Reshetikhin crystals.
EXAMPLES:
sage: pm=sage.combinat.crystals.kirillov_reshetikhin.PMDiagram([[0,1],[1,2],[1]])
sage: pm.sigma().pm_diagram
[[1, 0], [2, 1], [1]]
Returns all partitions obtained from a rectangle of width s and height r by removing horizontal dominoes.
EXAMPLES:
sage: sage.combinat.crystals.kirillov_reshetikhin.horizontal_dominoes_removed(2,2)
[[], [2], [2, 2]]
sage: sage.combinat.crystals.kirillov_reshetikhin.horizontal_dominoes_removed(3,2)
[[], [2], [2, 2], [2, 2, 2]]
Returns all partitions in a box of width s and height r.
EXAMPLES:
sage: sage.combinat.crystals.kirillov_reshetikhin.partitions_in_box(3,2)
[[], [1], [2], [1, 1], [2, 1], [1, 1, 1], [2, 2], [2, 1, 1],
[2, 2, 1], [2, 2, 2]]
Returns all partitions obtained from a rectangle of width s and height r by removing vertical dominoes.
EXAMPLES:
sage: sage.combinat.crystals.kirillov_reshetikhin.vertical_dominoes_removed(2,2)
[[], [1, 1], [2, 2]]
sage: sage.combinat.crystals.kirillov_reshetikhin.vertical_dominoes_removed(3,2)
[[2], [2, 1, 1], [2, 2, 2]]
sage: sage.combinat.crystals.kirillov_reshetikhin.vertical_dominoes_removed(4,2)
[[], [1, 1], [1, 1, 1, 1], [2, 2], [2, 2, 1, 1], [2, 2, 2, 2]]