Affine crystals¶

class sage.combinat.crystals.kirillov_reshetikhin.KR_type_A(cartan_type, r, s)¶

Bases: sage.combinat.crystals.kirillov_reshetikhin.KirillovReshetikhinCrystalFromPromotion

Class of Kirillov-Reshetikhin crystals of type $A_n^{(1)}$ .

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]]

classical_decomposition()¶

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]]

dynkin_diagram_automorphism(i)¶

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

promotion()¶

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]]

promotion_inverse()¶

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]]

class sage.combinat.crystals.kirillov_reshetikhin.KR_type_C(cartan_type, r, s)¶

Bases: sage.combinat.crystals.kirillov_reshetikhin.KirillovReshetikhinGenericCrystal, sage.combinat.crystals.affine.AffineCrystalFromClassical

Class of Kirillov-Reshetikhin crystals $B^{r,s}$ of type $C_n^{(1)}$ for $r<n$ .

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]]

Element¶: alias of KR_type_CElement

ambient_crystal()¶

Returns the ambient crystal $'B^{r,s}$ of type $A_{2n+1}^{(2)}$ associated to the Kirillov-Reshetikhin crystal of type $C_n^{(1)}$ . 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)

ambient_dict_pm_diagrams()¶

Gives a dictionary of all self-dual $\pm$ 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]]}

ambient_highest_weight_dict()¶

Gives a dictionary of all ${2,...,n+1}$ -highest weight vectors in the ambient crystal. Their key is the inner shape of their corresponding $\pm$ diagram, or equivalently, their ${2,...,n+1}$ 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]]}

classical_decomposition()¶

Specifies the classical crystal underlying the Kirillov-Reshetikhin crystal of type $C_n^{(1)}$ . It is given by $B^{r,s} \cong \oplus_\Lambda B(\Lambda)$ where $\Lambda$ are weights obtained from a rectangle of width s and height r by removing horizontal dominoes. Here we identify the fundamental weight $\Lambda_i$ with a column of height $i$ .

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]]

from_ambient_crystal()¶

Provides a map from the ambient crystal of type $A_{2n+1}^{(2)}$ to the Kirillov-Reshetikhin crystal of type $C_n^{(1)}$ . Note that this map is only well-defined on elements that are in the image type $C_n^{(1)}$ elements 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]]

highest_weight_dict()¶

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]]}

to_ambient_crystal()¶

Provides a map from the Kirillov-Reshetikhin crystal of type $C_n^{(1)}$ to the ambient crystal of type $A_{2n+1}^{(2)}$ .

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() # Anne: please check this!!!!
Kirillov-Reshetikhin crystal of type ['B', 4, 1]^* with (r,s)=(2,2)

class sage.combinat.crystals.kirillov_reshetikhin.KR_type_CElement(value, parent)¶

Bases: sage.combinat.crystals.affine.AffineCrystalFromClassicalElement

Class for the elements in the Kirillov-Reshetikhin crystals $B^{r,s}$ of type $C_n^{(1)}$ for $r<n$ .

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'>

e0()¶

Gives $e_0$ on self by mapping self to the ambient crystal, calculating $e_1 e_0$ there and pulling the element back.

EXAMPLES:

sage: K=KirillovReshetikhinCrystal(['C',3,1],1,2)
sage: b = K(rows=[])
sage: b.e(0)
[[-1, -1]]

epsilon0()¶

Calculates $\epsilon_0$ of self by mapping the element to the ambient crystal and calculating $\epsilon_1$ there.

EXAMPLES:

sage: K = KirillovReshetikhinCrystal(['C',2,1], 1,2)
sage: b=K(rows=[[1,1]])
sage: b.epsilon(0)
2

f0()¶

Gives $f_0$ on self by mapping self to the ambient crystal, calculating $f_1 f_0$ there and pulling the element back.

EXAMPLES:

sage: K=KirillovReshetikhinCrystal(['C',3,1],1,2)
sage: b = K(rows=[])
sage: b.f(0)
[[1, 1]]

phi0()¶

Calculates $\phi_0$ of self by mapping the element to the ambient crystal and calculating $\phi_1$ there.

EXAMPLES:

sage: K = KirillovReshetikhinCrystal(['C',2,1], 1,2)
sage: b=K(rows=[[-1,-1]])
sage: b.phi(0)
2

class sage.combinat.crystals.kirillov_reshetikhin.KR_type_E6(cartan_type, r, s)¶

Bases: sage.combinat.crystals.kirillov_reshetikhin.KirillovReshetikhinCrystalFromPromotion

Class of Kirillov-Reshetikhin crystals of type $E_6^{(1)}$ for $r=1,2,6$ .

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 rectract().

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.affine.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)
True

affine_weight(b)¶

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)]

automorphism_on_affine_weight(weight)¶

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()]
[[(-1, 0, 0, 1, 0, 0, -1), (-1, -1, 0, 0, 0, 1, 0)], 
[(0, 0, 0, 0, 0, 1, -2), (-2, 0, 1, 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)], 
[(0, 0, 0, 0, 0, 0, 0), (0, 0, 0, 0, 0, 0, 0)]]

classical_decomposition()¶

Specifies the classical crystal underlying the KR crystal of type $E_6^{(1)}$ .

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],)

dynkin_diagram_automorphism(i)¶

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]

highest_weight_dict(*args, **kwds)¶

Returns a dictionary between ${1,2,3,4,5}$ 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()
{[[[3, -1, -6], [1]]]: ((-1, 0, 0, 1, 0, 0, -1), 1), 
[[[5, -2, -6], [-6, 2]]]: ((0, 0, 0, 0, 0, 1, -2), 1), 
[[[2, -1], [1]]]: ((-2, 0, 1, 0, 0, 0, 0), 1), 
[[[6, -2], [-6, 2]]]: ((0, 0, 0, 0, 0, 0, 0), 1), 
[]: ((0, 0, 0, 0, 0, 0, 0), 0)}

highest_weight_dict_inv(*args, **kwds)¶

Returns a dictionary between a tuple of affine weights and a classical component, and ${2,3,4,5,6}$ 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]]]}

hw_auxiliary(*args, **kwds)¶

Returns the ${2,3,4,5}$ 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]]]]

promotion()¶

Specifies the promotion operator used to construct the affine type $E_6^{(1)}$ 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

promotion_inverse()¶

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

promotion_on_highest_weight_vectors()¶

Gives a dictionary of the promotion map on ${1,2,3,4,5}$ highest weight elements to ${2,3,4,5,6}$ elements in self.

EXAMPLES:

sage: K = KirillovReshetikhinCrystal(['E',6,1],2,1)
sage: dic = K.promotion_on_highest_weight_vectors()
sage: dic
{[[[2, -1], [1]]]: [[[-1], [-1, 3]]], 
[[[5, -2, -6], [-6, 2]]]: [[[2, -1], [1]]], 
[[[3, -1, -6], [1]]]: [[[5, -3], [-1, 3]]], 
[[[6, -2], [-6, 2]]]: [], []: [[[1, -3], [-1, 3]]]}

promotion_on_highest_weight_vectors_function()¶

class sage.combinat.crystals.kirillov_reshetikhin.KR_type_box(cartan_type, r, s)¶

Bases: sage.combinat.crystals.kirillov_reshetikhin.KirillovReshetikhinGenericCrystal, sage.combinat.crystals.affine.AffineCrystalFromClassical

Class of Kirillov-Reshetikhin crystals $B^{r,s}$ of type $A_{2n}^{(2)}$ for $r\le n$ and type $D_{n+1}^{(2)}$ for $r<n$ .

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]]

Element¶: alias of KR_type_boxElement

ambient_crystal()¶

Returns the ambient crystal $'B^{r,2s}$ of type $C_n^{(1)}$ associated to the Kirillov-Reshetikhin. 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)

ambient_highest_weight_dict()¶

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]]}

classical_decomposition()¶

Specifies the classical crystal underlying the Kirillov-Reshetikhin crystal of type $A_{2n}^{(2)}$ and $D_{n+1}^{(2)}$ . It is given by $B^{r,s} \cong \oplus_\Lambda B(\Lambda)$ where $\Lambda$ are weights obtained from a rectangle of width s and height r by removing boxes. Here we identify the fundamental weight $\Lambda_i$ with a column of height $i$ .

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]]

from_ambient_crystal()¶

Provides a map from the ambient crystal of type $C_n^{(1)}$ 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)
[]

highest_weight_dict()¶

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]]}

similarity_factor()¶

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}

to_ambient_crystal()¶

Provides a map from self to the ambient crystal of type $C_n^{(1)}$ .

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]]]

class sage.combinat.crystals.kirillov_reshetikhin.KR_type_boxElement(value, parent)¶

Bases: sage.combinat.crystals.affine.AffineCrystalFromClassicalElement

Class for the elements in the Kirillov-Reshetikhin crystals $B^{r,s}$ of type $A_{2n}^{(2)}$ for $r\le n$ and type $D_{n+1}^{(2)}$ for $r<n$ .

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'>

e0()¶

Gives $e_0$ on self by mapping self to the ambient crystal, calculating $e_0$ there and pulling the element back.

EXAMPLES:

sage: K=KirillovReshetikhinCrystal(['A',4,2],1,1)
sage: b = K(rows=[])
sage: b.e(0)
[[-1]]

epsilon0()¶

Calculates $\epsilon_0$ of self by mapping the element to the ambient crystal and calculating $\epsilon_0$ there.

EXAMPLES:

sage: K = KirillovReshetikhinCrystal(['A',4,2], 1,1)
sage: b=K(rows=[[1]])
sage: b.epsilon(0)
2

f0()¶

Gives $f_0$ on self by mapping self to the ambient crystal, calculating $f_0$ there and pulling the element back.

EXAMPLES:

sage: K=KirillovReshetikhinCrystal(['A',4,2],1,1)
sage: b = K(rows=[])
sage: b.f(0)
[[1]]

phi0()¶

Calculates $\phi_0$ of self by mapping the element to the ambient crystal and calculating $\phi_0$ there.

EXAMPLES:

sage: K = KirillovReshetikhinCrystal(['D',3,2], 1,1)
sage: b=K(rows=[[-1]])
sage: b.phi(0)
2

class sage.combinat.crystals.kirillov_reshetikhin.KR_type_vertical(cartan_type, r, s)¶

Bases: sage.combinat.crystals.kirillov_reshetikhin.KirillovReshetikhinCrystalFromPromotion

Class of Kirillov-Reshetikhin crystals $B^{r,s}$ of type $D_n^{(1)}$ for $r\le n-2$ , $B_n^{(1)}$ for $r<n$ , and $A_{2n-1}^{(2)}$ for $r\le n$ .

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(['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]]]]

classical_decomposition()¶

Specifies the classical crystal underlying the Kirillov-Reshetikhin crystal of type $D_n^{(1)}$ , $B_n^{(1)}$ , and $A_{2n-1}^{(2)}$ . It is given by $B^{r,s} \cong \oplus_\Lambda B(\Lambda)$ where $\Lambda$ are weights obtained from a rectangle of width s and height r by removing verticle dominoes. Here we identify the fundamental weight $\Lambda_i$ with a column of height $i$ .

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]]

dynkin_diagram_automorphism(i)¶

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

from_highest_weight_vector_to_pm_diagram(b)¶

This gives the bijection between an element b in the classical decomposition of the KR crystal that is ${2,3,..,n}$ -highest weight and $\pm$ 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

from_pm_diagram_to_highest_weight_vector(pm)¶

This gives the bijection between a $\pm$ 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]]

promotion()¶

Specifies the promotion operator used to construct the affine type $D_n^{(1)}$ 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 $\pm$ 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]]

promotion_inverse()¶

Specifies the promotion operator used to construct the affine type $D_n^{(1)}$ 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 $\pm$ 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]]

promotion_on_highest_weight_vectors()¶

Calculates promotion on ${2,3,...,n}$ 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]]]

sage.combinat.crystals.kirillov_reshetikhin.KirillovReshetikhinCrystal(cartan_type, r, s)¶

Returns the Kirillov-Reshetikhin crystal $B^{r,s}$ 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 $p$ on the level of crystals which corresponds to a Dynkin diagram automorphism mapping node 0 to some other node i. The action of $f_0$ and $e_0$ is then constructed using $f_0 = p^{-1} \circ f_i \circ p$ .

For example, for type $A_n^{(1)}$ the Kirillov-Reshetikhin crystal $B^{r,s}$ is obtained from the classical crystal $B(s\omega_r)$ using the promotion operator. For other types, see

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)

A. Schilling, “Combinatorial structure of Kirillov-Reshetikhin crystals of type $D_n(1)$ , $B_n(1)$ , $A_{2n-1}(2)$ “, J. Algebra 319 (2008) 2938-2962 (arXiv:0704.2046 [math.QA])

B. Jones, A. Schilling, “”Affine structures and a tableau model for $E_6$ crystals”, preprint arXiv:0909.2442 [math.CO]

Other Kirillov-Reshetikhin crystals are constructed using similarity methods. See Section 4 of

G. Fourier, M. Okado, A. Schilling, “Kirillov-Reshetikhin crystals for nonexceptional types”, Advances in Mathematics 222 Issue 3 (2009) 1080-1116 (arXiv:0810.5067 [math.RT])

INPUT:

cartan_type Affine type and rank

r Label of finite Dynkin diagram

s Positive integer

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.affine.KR_type_vertical_with_category.element_class'>

class sage.combinat.crystals.kirillov_reshetikhin.KirillovReshetikhinCrystalFromPromotion(cartan_type, r, s)¶

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: - classical_decomposition - promotion - promotion_inverse - dynkin_diagram_automorphism

class sage.combinat.crystals.kirillov_reshetikhin.KirillovReshetikhinGenericCrystal(cartan_type, r, s)¶

Bases: sage.structure.parent.Parent

Generic class for Kirillov-Reshetikhin crystal $B^{r,s}$ of the given type.

Input is a Dynkin node $r$ , a positive integer $s$ , and a Cartan type $cartan_type$ .

R_matrix(K)¶

INPUT:

self – a crystal $L$

K – a Kirillov-Reshetikhin crystal of the same type as $L$ .

Returns the combinatorial `R`-matrix from $L \otimes K \to K \otimes L$ , where the combinatorial $R$ -matrix is the affine crystal isomorphism 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 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]]]

module_generator()¶

Yields the module generator of weight $s \Lambda_r$ of a Kirillov-Reshetikhin crystal $B^{r,s}$

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]]

r()¶

Returns r of the underlying Kirillov-Reshetikhin crystal $B^{r,s}$

EXAMPLE:

sage: K = KirillovReshetikhinCrystal(['D',4,1], 2, 1)
sage: K.r()
2

s()¶

Returns s of the underlying Kirillov-Reshetikhin crystal $B^{r,s}$

EXAMPLE:

sage: K = KirillovReshetikhinCrystal(['D',4,1], 2, 1)
sage: K.s()
1

class sage.combinat.crystals.kirillov_reshetikhin.PMDiagram(pm_diagram, from_shapes=None)¶

Bases: sage.combinat.combinat.CombinatorialObject

Class of $\pm$ diagrams. These diagrams are in one-to-one bijection with $X_{n-1}$ highest weight vectors in an $X_n$ highest weight crystal $X=B,C,D$ . See Section 4.1 of A. Schilling, “Combinatorial structure of Kirillov-Reshetikhin crystals of type $D_n(1)$ , $B_n(1)$ , $A_{2n-1}(2)$ “, J. Algebra 319 (2008) 2938-2962 (arXiv:0704.2046[math.QA]).

The input is a list $pm = [[a_0,b_0],[a_1,b_1],...,[a_{n-1},b_{n-1}],[b_n]]$ of 2-tuples and a last 1-tuple. The tuple $[a_i,b_i]$ specifies the number of $a_i$ + and $b_i$ - in the i-th row of the pm diagram if $n-i$ is odd and the number of $a_i$ +- pairs above row $i$ and $b_i$ columns of height $i$ not containing any + or - if $n-i$ is even.

Setting the option ‘from_shapes = True’ one can also input a $\pm$ diagram in terms of its outer, intermediate and inner shape by specifying a tuple [n, s, outer, intermediate, inner] where $s$ is the width of the $\pm$ 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

heights_of_addable_plus()¶

Returns a list with the heights of all addable plus in the $\pm$ 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]

heights_of_minus()¶

Returns a list with the heights of all minus in the $\pm$ 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]

inner_shape()¶

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]

intermediate_shape()¶

Returns the intermediate shape of the pm diagram (innner 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]

outer_shape()¶

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]

sigma()¶

Returns sigma on pm diagrams as needed for the analogue of the Dynkin diagram automorphism that interchanges nodes $0$ and $1$ for type $D_n(1)$ , $B_n(1)$ , $A_{2n-1}(2)$ 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]]

sage.combinat.crystals.kirillov_reshetikhin.horizontal_dominoes_removed(r, s)¶

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]]

sage.combinat.crystals.kirillov_reshetikhin.partitions_in_box(r, s)¶

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]]

sage.combinat.crystals.kirillov_reshetikhin.vertical_dominoes_removed(r, s)¶

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]]

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