Kirillov-Reshetikhin Crystals

class sage.combinat.crystals.kirillov_reshetikhin.AmbientRetractMap(base, ambient, pdict_inv, index_set, similarity_factor_domain=None, automorphism=None)

Bases: Map

The retraction map from the ambient crystal.

Consider a crystal embedding \(\phi : X \to Y\), then the elements \(X\) can be considered as a subcrystal of the ambient crystal \(Y\). The ambient retract is the partial map \(\tilde{\phi} : Y \to X\) such that \(\tilde{\phi} \circ \phi\) is the identity on \(X\).

class sage.combinat.crystals.kirillov_reshetikhin.CrystalDiagramAutomorphism(C, on_hw, index_set=None, automorphism=None, cache=True)

Bases: CrystalMorphism

The crystal automorphism induced from the diagram automorphism.

For example, in type \(A_n^{(1)}\) this is the promotion operator and in type \(D_n^{(1)}\), this corresponds to the automorphism induced from interchanging the \(0\) and \(1\) nodes in the Dynkin diagram.

INPUT:

• C – a crystal

• on_hw – a function for the images of the index_set-highest weight elements

• index_set – (default: the empty set) the index set

• automorphism – (default: the identity) the twisting automorphism

• cache – (default: True) cache the result

is_embedding()

Return True as self is a crystal isomorphism.

EXAMPLES:

sage: K = crystals.KirillovReshetikhin(['A',3,1], 2,2)
sage: K.promotion().is_isomorphism()
True

is_isomorphism()

Return True as self is a crystal isomorphism.

EXAMPLES:

sage: K = crystals.KirillovReshetikhin(['A',3,1], 2,2)
sage: K.promotion().is_isomorphism()
True

is_strict()

Return True as self is a crystal isomorphism.

EXAMPLES:

sage: K = crystals.KirillovReshetikhin(['A',3,1], 2,2)
sage: K.promotion().is_isomorphism()
True

is_surjective()

Return True as self is a crystal isomorphism.

EXAMPLES:

sage: K = crystals.KirillovReshetikhin(['A',3,1], 2,2)
sage: K.promotion().is_isomorphism()
True

class sage.combinat.crystals.kirillov_reshetikhin.CrystalOfTableaux_E7(cartan_type, shapes)

Bases: CrystalOfTableaux

The type \(E_7\) crystal \(B(s\Lambda_7)\).

This is a helper class for the corresponding:class:\(KR crystal <sage.combinat.crystals.kirillov_reshetikhin.KR_type_E7>\) \(B^{7,s}\).

module_generator(shape)

Return the module generator of self with shape shape.

Note

Only implemented for single rows (i.e., highest weight \(s\Lambda_7\)).

EXAMPLES:

sage: from sage.combinat.crystals.kirillov_reshetikhin import CrystalOfTableaux_E7
sage: T = CrystalOfTableaux_E7(CartanType(['E',7]), shapes=(Partition([5]),))
sage: T.module_generator([5])
[[(7,), (7,), (7,), (7,), (7,)]]

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

Bases: KirillovReshetikhinCrystalFromPromotion

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

EXAMPLES:

sage: K = crystals.KirillovReshetikhin(['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 = crystals.KirillovReshetikhin(['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 \(i \mapsto i+1 \mod n+1\), where \(n\) is the rank.

EXAMPLES:

sage: K = crystals.KirillovReshetikhin(['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 \(i \mapsto i+1 \mod n+1\), where \(n\) is the rank.

EXAMPLES:

sage: K = crystals.KirillovReshetikhin(['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 \(i \mapsto i-1 \mod n+1\), where \(n\) is the rank.

EXAMPLES:

sage: K = crystals.KirillovReshetikhin(['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_A2(cartan_type, r, s, dual=None)

Bases: KirillovReshetikhinGenericCrystal

Class of Kirillov-Reshetikhin crystals \(B^{r,s}\) of type \(A_{2n}^{(2)}\) for \(1 \leq r \leq n\) in the realization with classical subalgebra \(B_n\). The Cartan type in this case is inputted as the dual of \(A_{2n}^{(2)}\).

This is an alternative implementation to KR_type_box that uses the classical decomposition into type \(C_n\) 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 \(A_4^{(2)}\) (namely the KR crystal and its dual construction) are isomorphic up to relabelling of the edges:

sage: C = CartanType(['A',4,2])
sage: K = crystals.KirillovReshetikhin(C,1,1)
sage: Kdual = crystals.KirillovReshetikhin(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(sort=True)); Gnew.add_edges([(u,v,f[i]) for (u,v,i) in Gdual.edges(sort=True)])
sage: G.is_isomorphic(Gnew, edge_labels = True)
True

Element

alias of KR_type_A2Element

ambient_crystal()

Return the ambient crystal \(B^{r,s}\) of type \(B_{n+1}^{(1)}\) associated to the Kirillov-Reshetikhin crystal of type \(A_{2n}^{(2)}\) 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)

ambient_dict_pm_diagrams()

Return a dictionary of all self-dual \(\pm\) diagrams for the ambient crystal whose keys are 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()
{[]: [[1, 1], [0, 0], [0]],
 [2]: [[0, 0], [1, 1], [0]],
 [2, 2]: [[0, 0], [0, 0], [2]]}

ambient_highest_weight_dict()

Return a dictionary of all \(\{2,\ldots,n+1\}\)-highest weight vectors in the ambient crystal.

The key is the inner shape of their corresponding \(\pm\) diagram, or equivalently, their \(\{2,\ldots,n+1\}\) 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]]}

classical_decomposition()

Return the classical crystal underlying the Kirillov-Reshetikhin crystal of type \(A_{2n}^{(2)}\) with \(B_n\) as classical subdiagram.

It is given by \(B^{r,s} \cong \bigoplus_{\Lambda} B(\Lambda)\), where \(B(\Lambda)\) is a highest weight crystal of type \(B_n\) of highest weight \(\Lambda\). The sum is over all weights \(\Lambda\) 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: 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]]

from_ambient_crystal()

Return a map from the ambient crystal of type \(B_{n+1}^{(1)}\) to the Kirillov-Reshetikhin crystal of type \(A_{2n}^{(2)}\).

Note that this map is only well-defined on type \(A_{2n}^{(2)}\) 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]]

highest_weight_dict()

Return a dictionary of the classical highest weight vectors of self whose keys are 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]]}

module_generator()

Return the unique module generator of classical weight \(s \Lambda_r\) of a Kirillov-Reshetikhin crystal \(B^{r,s}\).

EXAMPLES:

sage: ct = CartanType(['A',8,2]).dual()
sage: K = crystals.KirillovReshetikhin(ct, 3, 5)
sage: K.module_generator()
[[1, 1, 1, 1, 1], [2, 2, 2, 2, 2], [3, 3, 3, 3, 3]]

to_ambient_crystal()

Return a map from the Kirillov-Reshetikhin crystal of type \(A_{2n}^{(2)}\) to the ambient crystal of type \(B_{n+1}^{(1)}\).

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)

class sage.combinat.crystals.kirillov_reshetikhin.KR_type_A2Element

Bases: KirillovReshetikhinGenericCrystalElement

Class for the elements in the Kirillov-Reshetikhin crystals \(B^{r,s}\) of type \(A_{2n}^{(2)}\) for \(r<n\) with underlying classical algebra \(B_n\).

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'>

e0()

Return \(e_0\) on self by mapping self to the ambient crystal, calculating \(e_1 e_0\) 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]]

epsilon0()

Calculate \(\varepsilon_0\) of self by mapping the element to the ambient crystal and calculating \varepsilon_1 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

f0()

Return \(f_0\) on self by mapping self to the ambient crystal, calculating \(f_1 f_0\) 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]]

phi0()

Calculate \(\varphi_0\) of self by mapping the element to the ambient crystal and calculating \(\varphi_1\) 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

class sage.combinat.crystals.kirillov_reshetikhin.KR_type_Bn(cartan_type, r, s, dual=None)

Bases: KirillovReshetikhinGenericCrystal

Class of Kirillov-Reshetikhin crystals \(B^{n,s}\) of type \(B_{n}^{(1)}\).

EXAMPLES:

sage: K = crystals.KirillovReshetikhin(['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 = crystals.KirillovReshetikhin(['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]]

Element

alias of KR_type_BnElement

ambient_crystal()

Return the ambient crystal \(B^{n,s}\) of type \(A_{2n-1}^{(2)}\) associated to the Kirillov-Reshetikhin crystal.

The ambient crystal is used to construct the zero arrows.

EXAMPLES:

sage: K = crystals.KirillovReshetikhin(['B',3,1],3,2)
sage: K.ambient_crystal()
Kirillov-Reshetikhin crystal of type ['B', 3, 1]^* with (r,s)=(3,2)

ambient_highest_weight_dict()

Return a dictionary of the classical highest weight vectors of the ambient crystal of self whose keys are their shape.

EXAMPLES:

sage: K = crystals.KirillovReshetikhin(['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 = crystals.KirillovReshetikhin(['B',3,1],3,3)
sage: K.ambient_highest_weight_dict()
{(3,): [[1, 1, 1]],
 (3, 1, 1): [[1, 1, 1], [2], [3]],
 (3, 2, 2): [[1, 1, 1], [2, 2], [3, 3]],
 (3, 3, 3): [[1, 1, 1], [2, 2, 2], [3, 3, 3]]}

classical_decomposition()

Return the classical crystal underlying the Kirillov-Reshetikhin crystal \(B^{n,s}\) of type \(B_n^{(1)}\).

It is the same as for \(r < n\), given by \(B^{n,s} \cong \bigoplus_{\Lambda} B(\Lambda)\), where \(\Lambda\) are weights obtained from a rectangle of width \(s/2\) and height \(n\) by removing horizontal dominoes. Here we identify the fundamental weight \(\Lambda_i\) with a column of height \(i\) for \(i<n\) and a column of width \(1/2\) for \(i=n\).

EXAMPLES:

sage: K = crystals.KirillovReshetikhin(['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 = crystals.KirillovReshetikhin(['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]]

from_ambient_crystal()

Return a map from the ambient crystal of type \(A_{2n-1}^{(2)}\) 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 = crystals.KirillovReshetikhin(['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)
[++-, []]

highest_weight_dict()

Return a dictionary of the classical highest weight vectors of self whose keys are 2 times their shape.

EXAMPLES:

sage: K = crystals.KirillovReshetikhin(['B',3,1],3,2)
sage: K.highest_weight_dict()
{(2,): [[1]], (2, 2, 2): [[1], [2], [3]]}
sage: K = crystals.KirillovReshetikhin(['B',3,1],3,3)
sage: K.highest_weight_dict()
{(3, 1, 1): [+++, [[1]]], (3, 3, 3): [+++, [[1], [2], [3]]]}

similarity_factor()

Sets the similarity factor used to map to the ambient crystal.

EXAMPLES:

sage: K = crystals.KirillovReshetikhin(['B',3,1],3,2)
sage: K.similarity_factor()
{1: 2, 2: 2, 3: 1}

to_ambient_crystal()

Return a map from self to the ambient crystal of type \(A_{2n-1}^{(2)}\).

EXAMPLES:

sage: K = crystals.KirillovReshetikhin(['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]]]

class sage.combinat.crystals.kirillov_reshetikhin.KR_type_BnElement

Bases: KirillovReshetikhinGenericCrystalElement

Class for the elements in the Kirillov-Reshetikhin crystals \(B^{n,s}\) of type \(B_n^{(1)}\).

EXAMPLES:

sage: K=crystals.KirillovReshetikhin(['B',3,1],3,2)
sage: type(K.module_generators[0])
<class 'sage.combinat.crystals.kirillov_reshetikhin.KR_type_Bn_with_category.element_class'>

e0()

Return \(e_0\) on self by mapping self to the ambient crystal, calculating \(e_0\) there and pulling the element back.

EXAMPLES:

sage: K = crystals.KirillovReshetikhin(['B',3,1],3,1)
sage: b = K.module_generators[0]
sage: b.e(0) # indirect doctest
[--+, []]

epsilon0()

Calculate \(\varepsilon_0\) of self by mapping the element to the ambient crystal and calculating \(\varepsilon_0\) there.

EXAMPLES:

sage: K=crystals.KirillovReshetikhin(['B',3,1],3,1)
sage: b = K.module_generators[0]
sage: b.epsilon(0) # indirect doctest
1

f0()

Return \(f_0\) on self by mapping self to the ambient crystal, calculating \(f_0\) there and pulling the element back.

EXAMPLES:

sage: K=crystals.KirillovReshetikhin(['B',3,1],3,1)
sage: b = K.module_generators[0]
sage: b.f(0) # indirect doctest

phi0()

Calculate \(\varphi_0\) of self by mapping the element to the ambient crystal and calculating \(\varphi_0\) there.

EXAMPLES:

sage: K=crystals.KirillovReshetikhin(['B',3,1],3,1)
sage: b = K.module_generators[0]
sage: b.phi(0) # indirect doctest
0

class sage.combinat.crystals.kirillov_reshetikhin.KR_type_C(cartan_type, r, s, dual=None)

Bases: KirillovReshetikhinGenericCrystal

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

EXAMPLES:

sage: K = crystals.KirillovReshetikhin(['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()

Return 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 = crystals.KirillovReshetikhin(['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()

Return a dictionary of all self-dual \(\pm\) diagrams for the ambient crystal whose keys are their inner shape.

EXAMPLES:

sage: K = crystals.KirillovReshetikhin(['C',2,1], 1,2)
sage: K.ambient_dict_pm_diagrams()
{[]: [[1, 1], [0]], [2]: [[0, 0], [2]]}
sage: K = crystals.KirillovReshetikhin(['C',3,1], 2,2)
sage: K.ambient_dict_pm_diagrams()
{[]: [[1, 1], [0, 0], [0]],
 [2]: [[0, 0], [1, 1], [0]],
 [2, 2]: [[0, 0], [0, 0], [2]]}
sage: K = crystals.KirillovReshetikhin(['C',3,1], 2,3)
sage: K.ambient_dict_pm_diagrams()
{[1, 1]: [[1, 1], [0, 0], [1]],
 [3, 1]: [[0, 0], [1, 1], [1]],
 [3, 3]: [[0, 0], [0, 0], [3]]}

ambient_highest_weight_dict()

Return a dictionary of all \(\{2,\ldots,n+1\}\)-highest weight vectors in the ambient crystal.

The key is the inner shape of their corresponding \(\pm\) diagram, or equivalently, their \(\{2,\ldots,n+1\}\) weight.

EXAMPLES:

sage: K = crystals.KirillovReshetikhin(['C',3,1], 2,2)
sage: K.ambient_highest_weight_dict()
{[]: [[2], [-2]], [2]: [[1, 2], [2, -1]], [2, 2]: [[2, 2], [3, 3]]}

classical_decomposition()

Return the classical crystal underlying the Kirillov-Reshetikhin crystal of type \(C_n^{(1)}\).

It is given by \(B^{r,s} \cong \bigoplus_{\Lambda} B(\Lambda)\), 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 = crystals.KirillovReshetikhin(['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()

Return 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 type \(C_n^{(1)}\) elements that are in the image under to_ambient_crystal().

EXAMPLES:

sage: K = crystals.KirillovReshetikhin(['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()

Return a dictionary of the classical highest weight vectors of self whose keys are their shape.

EXAMPLES:

sage: K = crystals.KirillovReshetikhin(['C',3,1], 2,2)
sage: K.highest_weight_dict()
{[]: [], [2]: [[1, 1]], [2, 2]: [[1, 1], [2, 2]]}

to_ambient_crystal()

Return 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 = crystals.KirillovReshetikhin(['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)

class sage.combinat.crystals.kirillov_reshetikhin.KR_type_CElement

Bases: KirillovReshetikhinGenericCrystalElement

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

EXAMPLES:

sage: K=crystals.KirillovReshetikhin(['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()

Return \(e_0\) on self by mapping self to the ambient crystal, calculating \(e_1 e_0\) there and pulling the element back.

EXAMPLES:

sage: K=crystals.KirillovReshetikhin(['C',3,1],1,2)
sage: b = K(rows=[])
sage: b.e(0) # indirect doctest
[[-1, -1]]

epsilon0()

Calculate \(\varepsilon_0\) of self by mapping the element to the ambient crystal and calculating \(\varepsilon_1\) there.

EXAMPLES:

sage: K = crystals.KirillovReshetikhin(['C',2,1], 1,2)
sage: b=K(rows=[[1,1]])
sage: b.epsilon(0) # indirect doctest
2

f0()

Return \(f_0\) on self by mapping self to the ambient crystal, calculating \(f_1 f_0\) there and pulling the element back.

EXAMPLES:

sage: K=crystals.KirillovReshetikhin(['C',3,1],1,2)
sage: b = K(rows=[])
sage: b.f(0) # indirect doctest
[[1, 1]]

phi0()

Calculate \(\varphi_0\) of self by mapping the element to the ambient crystal and calculating \(\varphi_1\) there.

EXAMPLES:

sage: K = crystals.KirillovReshetikhin(['C',2,1], 1,2)
sage: b=K(rows=[[-1,-1]])
sage: b.phi(0) # indirect doctest
2

class sage.combinat.crystals.kirillov_reshetikhin.KR_type_Cn(cartan_type, r, s, dual=None)

Bases: KirillovReshetikhinGenericCrystal

Class of Kirillov-Reshetikhin crystals \(B^{n,s}\) of type \(C_n^{(1)}\).

EXAMPLES:

sage: K = crystals.KirillovReshetikhin(['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]]]]

Element

alias of KR_type_CnElement

classical_decomposition()

Specifies the classical crystal underlying the Kirillov-Reshetikhin crystal \(B^{n,s}\) of type \(C_n^{(1)}\).

The classical decomposition is given by \(B^{n,s} \cong B(s \Lambda_n)\).

EXAMPLES:

sage: K = crystals.KirillovReshetikhin(['C',3,1],3,2)
sage: K.classical_decomposition()
The crystal of tableaux of type ['C', 3] and shape(s) [[2, 2, 2]]

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 = crystals.KirillovReshetikhin(['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.pp()
.  .
.  +
-  -

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

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 = crystals.KirillovReshetikhin(['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]]

class sage.combinat.crystals.kirillov_reshetikhin.KR_type_CnElement

Bases: KirillovReshetikhinGenericCrystalElement

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

EXAMPLES:

sage: K=crystals.KirillovReshetikhin(['C',3,1],3,2)
sage: type(K.module_generators[0])
<class 'sage.combinat.crystals.kirillov_reshetikhin.KR_type_Cn_with_category.element_class'>

e0()

Return \(e_0\) on self by going to the \(\pm\)-diagram corresponding to the \(\{2,...,n\}\)-highest weight vector in the component of self, then applying [Definition 6.1, 4], and pulling back from \(\pm\)-diagrams.

EXAMPLES:

sage: K=crystals.KirillovReshetikhin(['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]]

epsilon0()

Calculate \(\varepsilon_0\) of self using Lemma 6.1 of [4].

EXAMPLES:

sage: K=crystals.KirillovReshetikhin(['C',3,1],3,1)
sage: b = K.module_generators[0]
sage: b.epsilon(0) # indirect doctest
1

f0()

Return \(e_0\) on self by going to the \(\pm\)-diagram corresponding to the \(\{2,...,n\}\)-highest weight vector in the component of self, then applying [Definition 6.1, 4], and pulling back from \(\pm\)-diagrams.

EXAMPLES:

sage: K = crystals.KirillovReshetikhin(['C',3,1],3,1)
sage: b = K.module_generators[0]
sage: b.f(0) # indirect doctest

phi0()

Calculate \(\varphi_0\) of self.

EXAMPLES:

sage: K=crystals.KirillovReshetikhin(['C',3,1],3,1)
sage: b = K.module_generators[0]
sage: b.phi(0) # indirect doctest
0

class sage.combinat.crystals.kirillov_reshetikhin.KR_type_D_tri1(ct, s)

Bases: KirillovReshetikhinGenericCrystal

Class of Kirillov-Reshetikhin crystals \(B^{1,s}\) of type \(D_4^{(3)}\).

The crystal structure was defined in Section 4 of [KMOY2007] using the coordinate representation.

class Element

Bases: KirillovReshetikhinGenericCrystalElement

coordinates()

Return self as coordinates.

EXAMPLES:

sage: K = crystals.KirillovReshetikhin(['D',4,3], 1, 3)
sage: all(K.from_coordinates(x.coordinates()) == x for x in K)
True

e0()

Return the action of \(e_0\) on self.

EXAMPLES:

sage: K = crystals.KirillovReshetikhin(['D',4,3], 1,1)
sage: [x.e0() for x in K]
[[[-1]], [], [[-3]], [[-2]], None, None, None, None]

epsilon0()

Return \(\varepsilon_0\) of self.

EXAMPLES:

sage: K = crystals.KirillovReshetikhin(['D',4,3], 1, 5)
sage: [mg.epsilon0() for mg in K.module_generators]
[5, 6, 7, 8, 9, 10]

f0()

Return the action of \(f_0\) on self.

EXAMPLES:

sage: K = crystals.KirillovReshetikhin(['D',4,3], 1,1)
sage: [x.f0() for x in K]
[[[1]], None, None, None, None, [[2]], [[3]], []]

phi0()

Return \(\varphi_0\) of self.

EXAMPLES:

sage: K = crystals.KirillovReshetikhin(['D',4,3], 1, 5)
sage: [mg.phi0() for mg in K.module_generators]
[5, 4, 3, 2, 1, 0]

classical_decomposition()

Return the classical decomposition of self.

EXAMPLES:

sage: K = crystals.KirillovReshetikhin(['D',4,3], 1, 5)
sage: K.classical_decomposition()
The crystal of tableaux of type ['G', 2]
 and shape(s) [[], [1], [2], [3], [4], [5]]

from_coordinates(coords)

Return an element of self from the coordinates coords.

EXAMPLES:

sage: K = crystals.KirillovReshetikhin(['D',4,3], 1, 5)
sage: K.from_coordinates((0, 2, 3, 1, 0, 1))
[[2, 2, 3, 0, -1]]

class sage.combinat.crystals.kirillov_reshetikhin.KR_type_Dn_twisted(cartan_type, r, s, dual=None)

Bases: KirillovReshetikhinGenericCrystal

Class of Kirillov-Reshetikhin crystals \(B^{n,s}\) of type \(D_{n+1}^{(2)}\).

EXAMPLES:

sage: K = crystals.KirillovReshetikhin(['D',4,2],3,1)
sage: [[b,b.f(0)] for b in K]
[[[+++, []], None], [[++-, []], None], [[+-+, []], None], [[-++, []],
[+++, []]], [[+--, []], None], [[-+-, []], [++-, []]], [[--+, []], [+-+, []]],
[[---, []], [+--, []]]]

Element

alias of KR_type_Dn_twistedElement

classical_decomposition()

Return the classical crystal underlying the Kirillov-Reshetikhin crystal \(B^{n,s}\) of type \(D_{n+1}^{(2)}\).

The classical decomposition is given by \(B^{n,s} \cong B(s \Lambda_n)\).

EXAMPLES:

sage: K = crystals.KirillovReshetikhin(['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 = crystals.KirillovReshetikhin(['D',4,2],3,2)
sage: K.classical_decomposition()
The crystal of tableaux of type ['B', 3] and shape(s) [[1, 1, 1]]

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 \(\{2,3,\ldots,n\}\)-highest weight and \(\pm\) diagrams.

EXAMPLES:

sage: K = crystals.KirillovReshetikhin(['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 = crystals.KirillovReshetikhin(['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 \(B_n\), there can be (at most one) entry \(0\) in the \(\{2,3,\ldots,n\}\)-highest weight elements at height \(n\). In the following implementation this is realized as an empty column of height \(n\) since this uniquely specifies the existence of the \(0\).

EXAMPLES:

sage: b = hw[1]
sage: pm = K.from_highest_weight_vector_to_pm_diagram(b)
sage: pm.pp()
.  .
.  .
.  .

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 \(\{2,3,\ldots,n\}\)-highest weight.

EXAMPLES:

sage: K = crystals.KirillovReshetikhin(['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]]

class sage.combinat.crystals.kirillov_reshetikhin.KR_type_Dn_twistedElement

Bases: KirillovReshetikhinGenericCrystalElement

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

EXAMPLES:

sage: K=crystals.KirillovReshetikhin(['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'>

e0()

Return \(e_0\) on self by going to the \(\pm\)-diagram corresponding to the \(\{2,\ldots,n\}\)-highest weight vector in the component of self, then applying [Definition 6.2, 4], and pulling back from \(\pm\)-diagrams.

EXAMPLES:

sage: K=crystals.KirillovReshetikhin(['D',4,2],3,3)
sage: b = K.module_generators[0]
sage: b.e(0) # indirect doctest
[+++, [[2], [3], [0]]]

epsilon0()

Calculate \(\varepsilon_0\) of self using Lemma 6.2 of [4].

EXAMPLES:

sage: K=crystals.KirillovReshetikhin(['D',4,2],3,1)
sage: b = K.module_generators[0]
sage: b.epsilon(0) # indirect doctest
1

f0()

Return \(e_0\) on self by going to the \(\pm\)-diagram corresponding to the \(\{2,\ldots,n\}\)-highest weight vector in the component of self, then applying [Definition 6.2, 4], and pulling back from \(\pm\)-diagrams.

EXAMPLES:

sage: K = crystals.KirillovReshetikhin(['D',4,2],3,2)
sage: b = K.module_generators[0]
sage: b.f(0) # indirect doctest

phi0()

Calculate \(\varphi_0\) of self.

EXAMPLES:

sage: K = crystals.KirillovReshetikhin(['D',4,2],3,1)
sage: b = K.module_generators[0]
sage: b.phi(0) # indirect doctest
0

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

Bases: KirillovReshetikhinCrystalFromPromotion

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

EXAMPLES:

sage: K = crystals.KirillovReshetikhin(['E',6,1],2,1)
sage: K.module_generator().e(0)
[]
sage: K.module_generator().e(0).f(0)
[[(2, -1), (1,)]]
sage: K = crystals.KirillovReshetikhin(['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 = crystals.KirillovReshetikhin(['E',6,1],2,1)
sage: La = K.cartan_type().classical().root_system().weight_lattice().fundamental_weights()
sage: H = crystals.HighestWeight(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)
[]

affine_weight(b)

Return the affine level zero weight corresponding to the element b of the classical crystal underlying self.

For the coefficients to calculate the level, see Table Aff 1 in [Ka1990].

EXAMPLES:

sage: K = crystals.KirillovReshetikhin(['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)

Act with the Dynkin diagram automorphism on affine weights as outputted by the affine_weight method.

EXAMPLES:

sage: K = crystals.KirillovReshetikhin(['E',6,1],2,1)
sage: sorted([x[0], K.automorphism_on_affine_weight(x[0])]
....:  for x in K.highest_weight_dict().values())
[[(-2, 0, 1, 0, 0, 0, 0), (0, -2, 0, 1, 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)],
 [(0, 0, 0, 0, 0, 0, 0), (0, 0, 0, 0, 0, 0, 0)],
 [(0, 0, 0, 0, 0, 1, -2), (-2, 0, 1, 0, 0, 0, 0)]]

classical_decomposition()

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

EXAMPLES:

sage: K = crystals.KirillovReshetikhin(['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 = crystals.KirillovReshetikhin(['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.

Here we use the Dynkin diagram automorphism of order 3 which maps node 0 to node 1.

EXAMPLES:

sage: K = crystals.KirillovReshetikhin(['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()

Return a dictionary between \(\{1,2,3,4,5\}\)-highest weight elements, and a tuple of affine weights and its classical component.

EXAMPLES:

sage: K = crystals.KirillovReshetikhin(['E',6,1],2,1)
sage: sorted(K.highest_weight_dict().items(), key=str)
[([[(2, -1), (1,)]], ((-2, 0, 1, 0, 0, 0, 0), 1)),
 ([[(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)),
 ([[(6, -2), (-6, 2)]], ((0, 0, 0, 0, 0, 0, 0), 1)),
 ([], ((0, 0, 0, 0, 0, 0, 0), 0))]

highest_weight_dict_inv()

Return a dictionary between a tuple of affine weights and a classical component, and \(\{2,3,4,5,6\}\)-highest weight elements.

EXAMPLES:

sage: K = crystals.KirillovReshetikhin(['E',6,1],2,1)
sage: K.highest_weight_dict_inv()
{((-2, 0, 1, 0, 0, 0, 0), 1): [[(2, -1), (1,)]],
 ((-1, -1, 0, 0, 0, 1, 0), 1): [[(5, -3), (-1, 3)]],
 ((0, -2, 0, 1, 0, 0, 0), 1): [[(-1,), (-1, 3)]],
 ((0, 0, 0, 0, 0, 0, 0), 0): [],
 ((0, 0, 0, 0, 0, 0, 0), 1): [[(1, -3), (-1, 3)]]}

hw_auxiliary()

Return the \({2,3,4,5}\) highest weight elements of self.

EXAMPLES:

sage: K = crystals.KirillovReshetikhin(['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 = crystals.KirillovReshetikhin(['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 = crystals.KirillovReshetikhin(['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()

Return the inverse promotion. Since promotion is of order 3, the inverse promotion is the same as promotion applied twice.

EXAMPLES:

sage: K = crystals.KirillovReshetikhin(['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()

Return a dictionary of the promotion map on \(\{1,2,3,4,5\}\)-highest weight elements to \(\{2,3,4,5,6\}\)-highest weight elements in self.

EXAMPLES:

sage: K = crystals.KirillovReshetikhin(['E',6,1], 2, 1)
sage: dic = K.promotion_on_highest_weight_vectors()
sage: sorted(dic.items(), key=str)
[([[(2, -1), (1,)]], [[(-1,), (-1, 3)]]),
 ([[(3, -1, -6), (1,)]], [[(5, -3), (-1, 3)]]),
 ([[(5, -2, -6), (-6, 2)]], [[(2, -1), (1,)]]),
 ([[(6, -2), (-6, 2)]], []),
 ([], [[(1, -3), (-1, 3)]])]

promotion_on_highest_weight_vectors_function()

Return a lambda function on x defined by self.promotion_on_highest_weight_vectors()[x].

EXAMPLES:

sage: K = crystals.KirillovReshetikhin(['E',6,1], 2, 1)
sage: f = K.promotion_on_highest_weight_vectors_function()
sage: f(K.module_generator().lift())
[[(-1,), (-1, 3)]]

class sage.combinat.crystals.kirillov_reshetikhin.KR_type_E7(ct, r, s)

Bases: KirillovReshetikhinGenericCrystal

The Kirillov-Reshetikhin crystal \(B^{7,s}\) of type \(E_7^{(1)}\).

A7_decomposition()

Return the decomposition of self into \(A_7\) highest weight crystals.

The \(A_7\) decomposition of \(B^{7,s}\) is given by the parameters \(m_4, m_5, m_6, m_7 \geq 0\) such that \(m_4 + m_5 \leq m_7\) and \(s = m_4 + m_5 + m_6 + m_7\). The corresponding \(A_7\) highest weight crystal has highest weight \(\lambda = (m_7 - m_4 - m_5) \Lambda_6 + m_5 \Lambda_4 + m_6 \Lambda_2\).

EXAMPLES:

sage: K = crystals.KirillovReshetikhin(['E',7,1], 7, 3)
sage: K.A7_decomposition()
The crystal of tableaux of type ['A', 7] and shape(s)
 [[3, 3, 3, 3, 3, 3], [3, 3, 2, 2, 2, 2], [3, 3, 1, 1, 1, 1], [3, 3],
  [2, 2, 2, 2, 1, 1], [2, 2, 1, 1], [1, 1, 1, 1, 1, 1], [1, 1]]

class Element

Bases: KirillovReshetikhinGenericCrystalElement

e0()

Return the action of \(e_0\) on self.

EXAMPLES:

sage: K = crystals.KirillovReshetikhin(['E',7,1], 7, 2)
sage: mg = K.module_generator()
sage: mg.e0()
[[(7,), (-1, 7)]]
sage: mg.e0().e0()
[[(-1, 7), (-1, 7)]]
sage: mg.e_string([0,0,0]) is None
True

f0()

Return the action of \(f_0\) on self.

EXAMPLES:

sage: K = crystals.KirillovReshetikhin(['E',7,1], 7, 2)
sage: mg = K.module_generator()
sage: x = mg.f_string([7,6,5,4,3,2,4,5,6,1,3,4,5,2,4,3,1])
sage: x.f0()
[[(7,), (7,)]]
sage: mg.f0() is None
True

classical_decomposition()

Return the classical decomposition of self.

EXAMPLES:

sage: K = crystals.KirillovReshetikhin(['E',7,1], 7, 4)
sage: K.classical_decomposition()
The crystal of tableaux of type ['E', 7] and shape(s) [[4]]

from_A7_crystal()

Return the inclusion of the KR crystal \(B^{7,s}\) of type \(E_7^{(1)}\) into type \(A_7\) highest weight crystals.

EXAMPLES:

sage: K = crystals.KirillovReshetikhin(['E',7,1], 7, 2)
sage: K.from_A7_crystal()
['A', 6] -> ['E', 7, 1] Virtual Crystal morphism:
  From: The crystal of tableaux of type ['A', 7] and shape(s)
    [[2, 2, 2, 2, 2, 2], [2, 2, 1, 1, 1, 1], [2, 2], [1, 1, 1, 1], []]
  To:   Kirillov-Reshetikhin crystal of type ['E', 7, 1] with (r,s)=(7,2)
  Defn: ...

to_A7_crystal()

Return the map decomposing the KR crystal \(B^{7,s}\) of type \(E_7^{(1)}\) into type \(A_7\) highest weight crystals.

EXAMPLES:

sage: K = crystals.KirillovReshetikhin(['E',7,1], 7, 2)
sage: K.to_A7_crystal()
['A', 6] relabelled by {1: 1, 2: 3, 3: 4, 4: 5, 5: 6, 6: 7} -> ['A', 7] Virtual Crystal morphism:
  From: Kirillov-Reshetikhin crystal of type ['E', 7, 1] with (r,s)=(7,2)
  To:   The crystal of tableaux of type ['A', 7] and shape(s)
    [[2, 2, 2, 2, 2, 2], [2, 2, 1, 1, 1, 1], [2, 2], [1, 1, 1, 1], []]
  Defn: ...

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

Bases: KirillovReshetikhinGenericCrystal, 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 = crystals.KirillovReshetikhin(['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()

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

The ambient crystal is used to construct the zero arrows.

EXAMPLES:

sage: K = crystals.KirillovReshetikhin(['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()

Return a dictionary of the classical highest weight vectors of the ambient crystal of self whose keys are their shape.

EXAMPLES:

sage: K = crystals.KirillovReshetikhin(['A',6,2], 2,2)
sage: K.ambient_highest_weight_dict()
{[]: [],
 [2]: [[1, 1]],
 [2, 2]: [[1, 1], [2, 2]],
 [4]: [[1, 1, 1, 1]],
 [4, 2]: [[1, 1, 1, 1], [2, 2]],
 [4, 4]: [[1, 1, 1, 1], [2, 2, 2, 2]]}

classical_decomposition()

Return the classical crystal underlying the Kirillov-Reshetikhin crystal of type \(A_{2n}^{(2)}\) and \(D_{n+1}^{(2)}\).

It is given by \(B^{r,s} \cong \bigoplus_{\Lambda} B(\Lambda)\), 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 = crystals.KirillovReshetikhin(['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 = crystals.KirillovReshetikhin(['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()

Return 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 = crystals.KirillovReshetikhin(['D',4,2], 1,1)
sage: b = K.ambient_crystal()(rows=[[3,-3]])
sage: K.from_ambient_crystal()(b)
[[0]]
sage: K = crystals.KirillovReshetikhin(['A',4,2], 1,1)
sage: b = K.ambient_crystal()(rows=[])
sage: K.from_ambient_crystal()(b)
[]

highest_weight_dict()

Return a dictionary of the classical highest weight vectors of self whose keys are 2 times their shape.

EXAMPLES:

sage: K = crystals.KirillovReshetikhin(['A',6,2], 2,2)
sage: K.highest_weight_dict()
{[]: [],
 [2]: [[1]],
 [2, 2]: [[1], [2]],
 [4]: [[1, 1]],
 [4, 2]: [[1, 1], [2]],
 [4, 4]: [[1, 1], [2, 2]]}

similarity_factor()

Sets the similarity factor used to map to the ambient crystal.

EXAMPLES:

sage: K = crystals.KirillovReshetikhin(['A',6,2], 2,2)
sage: K.similarity_factor()
{1: 2, 2: 2, 3: 2}
sage: K = crystals.KirillovReshetikhin(['D',5,2], 1,1)
sage: K.similarity_factor()
{1: 2, 2: 2, 3: 2, 4: 1}

to_ambient_crystal()

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

EXAMPLES:

sage: K = crystals.KirillovReshetikhin(['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 = crystals.KirillovReshetikhin(['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

Bases: KirillovReshetikhinGenericCrystalElement

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

EXAMPLES:

sage: K = crystals.KirillovReshetikhin(['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()

Return \(e_0\) on self by mapping self to the ambient crystal, calculating \(e_0\) there and pulling the element back.

EXAMPLES:

sage: K = crystals.KirillovReshetikhin(['A',4,2],1,1)
sage: b = K(rows=[])
sage: b.e(0) # indirect doctest
[[-1]]

epsilon0()

Return \(\varepsilon_0\) of self by mapping the element to the ambient crystal and calculating \(\varepsilon_0\) there.

EXAMPLES:

sage: K = crystals.KirillovReshetikhin(['A',4,2], 1,1)
sage: b = K(rows=[[1]])
sage: b.epsilon(0) # indirect doctest
2

f0()

Return \(f_0\) on self by mapping self to the ambient crystal, calculating \(f_0\) there and pulling the element back.

EXAMPLES:

sage: K = crystals.KirillovReshetikhin(['A',4,2],1,1)
sage: b = K(rows=[])
sage: b.f(0) # indirect doctest
[[1]]

phi0()

Return \(\varphi_0\) of self by mapping the element to the ambient crystal and calculating \(\varphi_0\) there.

EXAMPLES:

sage: K = crystals.KirillovReshetikhin(['D',3,2], 1,1)
sage: b = K(rows=[[-1]])
sage: b.phi(0) # indirect doctest
2

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

Bases: KirillovReshetikhinCrystalFromPromotion

Class of Kirillov-Reshetikhin crystals \(B^{n,s}\) of type \(D_n^{(1)}\).

EXAMPLES:

sage: K = crystals.KirillovReshetikhin(['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 = crystals.KirillovReshetikhin(['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]]]]

classical_decomposition()

Return the classical crystal underlying the Kirillov-Reshetikhin crystal \(B^{r,s}\) of type \(D_n^{(1)}\) for \(r=n-1,n\).

The classical decomposition is given by \(B^{n,s} \cong B(s \Lambda_r)\).

EXAMPLES:

sage: K = crystals.KirillovReshetikhin(['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 = crystals.KirillovReshetikhin(['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 = crystals.KirillovReshetikhin(['D',4,1],3,2)
sage: K.classical_decomposition()
The crystal of tableaux of type ['D', 4] and shape(s) [[1, 1, 1, -1]]

dynkin_diagram_automorphism(i)

Specifies the Dynkin diagram automorphism underlying the promotion action on the crystal elements.

Here we use the Dynkin diagram automorphism which interchanges nodes 0 and 1 and leaves all other nodes unchanged.

EXAMPLES:

sage: K = crystals.KirillovReshetikhin(['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

promotion()

Return the promotion operator on \(B^{r,s}\) of type \(D_n^{(1)}\) for \(r = n-1,n\).

EXAMPLES:

sage: K = crystals.KirillovReshetikhin(['D',4,1],3,1)
sage: T = K.classical_decomposition()
sage: promotion = K.promotion()
sage: for t in T:
....:     print("{} {}".format(t, promotion(t)))
[+++-, []] [-++-, []]
[++-+, []] [-+-+, []]
[+-++, []] [--++, []]
[-+++, []] [++++, []]
[+---, []] [----, []]
[-+--, []] [++--, []]
[--+-, []] [+-+-, []]
[---+, []] [+--+, []]

promotion_inverse()

Return the inverse promotion operator on \(B^{r,s}\) of type \(D_n^{(1)}\) for \(r=n-1,n\).

EXAMPLES:

sage: K = crystals.KirillovReshetikhin(['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

promotion_on_highest_weight_vectors()

Return the promotion operator on \(\{2,3,\ldots,n\}\)-highest weight vectors.

A \(\{2,3,\ldots,n\}\)-highest weight vector in \(B(s\Lambda_n)\) of weight \(w = (w_1,\ldots,w_n)\) is mapped to a \(\{2,3,\ldots,n\}\)-highest weight vector in \(B(s\Lambda_{n-1})\) of weight \((-w_1,w_2,\ldots,w_n)\) and vice versa.

See also

• promotion_on_highest_weight_vectors_inverse()

• promotion()

EXAMPLES:

sage: KR = crystals.KirillovReshetikhin(['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("{} {}".format(t, prom[t]))
[[1], [2], [3], [4]] [[2], [3], [4], [-1]]
[[2], [3], [-4], [4]] [[2], [3], [4], [-4]]
[[2], [3], [-4], [-1]] [[1], [2], [3], [-4]]

sage: KR = crystals.KirillovReshetikhin(['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("{} {}".format(t, prom[t]))
[++++, []] [-+++, []]
[-++-, []] [+++-, []]

promotion_on_highest_weight_vectors_inverse()

Return the inverse promotion operator on \(\{2,3,\ldots,n\}\)-highest weight vectors.

See also

• promotion_on_highest_weight_vectors()

• promotion_inverse()

EXAMPLES:

sage: KR = crystals.KirillovReshetikhin(['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

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

Bases: 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 = crystals.KirillovReshetikhin(['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 = crystals.KirillovReshetikhin(['D',5,1], 3,1)
sage: b = K(rows=[[1]])
sage: b.e(0)
[[3], [-3], [-2]]

sage: K = crystals.KirillovReshetikhin(['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 = crystals.KirillovReshetikhin(['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 \(B^{r,s} \cong \bigoplus_\Lambda B(\Lambda)\), where \(\Lambda\) are weights obtained from a rectangle of width \(s\) and height \(r\) by removing vertical dominoes. Here we identify the fundamental weight \(\Lambda_i\) with a column of height \(i\).

EXAMPLES:

sage: K = crystals.KirillovReshetikhin(['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 = crystals.KirillovReshetikhin(['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 \({2, 3, \ldots, n}\)-highest weight and \(\pm\) diagrams.

EXAMPLES:

sage: K = crystals.KirillovReshetikhin(['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.pp()
+
-
sage: b = T(rows=[])
sage: pm=K.from_highest_weight_vector_to_pm_diagram(b); pm
[[0, 2], [0, 0], [0]]
sage: pm.pp()

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 \({2, 3, \ldots, n}\)-highest weight.

EXAMPLES:

sage: K = crystals.KirillovReshetikhin(['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 = crystals.KirillovReshetikhin(['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()

Return inverse of promotion.

In this case promotion is an involution, so promotion inverse equals promotion.

EXAMPLES:

sage: K = crystals.KirillovReshetikhin(['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

promotion_on_highest_weight_vector(b)

Calculates promotion on a \({2,3,...,n}\) highest weight vector b.

EXAMPLES:

sage: K = crystals.KirillovReshetikhin(['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_vector(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.KashiwaraNakashimaTableaux(cartan_type, r, s)

Return the Kashiwara-Nakashima model for the Kirillov-Reshetikhin crystal \(B^{r,s}\) in the given type.

The Kashiwara-Nakashima (KN) model constructs the KR crystal from the KN tableaux model for the corresponding classical crystals. This model is named for the underlying KN tableaux.

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 [Shi2002], [Sch2008], and [JS2010].

Other Kirillov-Reshetikhin crystals are constructed using similarity methods. See Section 4 of [FOS2009].

For more information on Kirillov-Reshetikhin crystals, see KirillovReshetikhinCrystal().

EXAMPLES:

sage: K = crystals.KirillovReshetikhin(['A',3,1], 2, 1)
sage: K2 = crystals.kirillov_reshetikhin.KashiwaraNakashimaTableaux(['A',3,1], 2, 1)
sage: K is K2
True

sage.combinat.crystals.kirillov_reshetikhin.KirillovReshetikhinCrystal(cartan_type, r, s, model='KN')

Return the Kirillov-Reshetikhin crystal \(B^{r,s}\) of the given type in the given model.

For more information about general crystals see sage.combinat.crystals.crystals.

There are a variety of models for Kirillov-Reshetikhin crystals. There is one using the classical crystal with Kashiwara-Nakashima tableaux. There is one using rigged configurations. Another tableaux model comes from the bijection between rigged configurations and tensor products of tableaux called Kirillov-Reshetikhin tableaux Lastly there is a model of Kirillov-Reshetikhin crystals for \(s = 1\) from crystals of LS paths.

INPUT:

• cartan_type – an affine Cartan type

• r – a label of finite Dynkin diagram

• s – a positive integer

• model – (default: 'KN') can be one of the following:

  • 'KN' or 'KashiwaraNakashimaTableaux' - use the Kashiwara-Nakashima tableaux model

  • 'KR' or 'KirillovReshetkihinTableaux' - use the Kirillov-Reshetkihin tableaux model

  • 'RC' or 'RiggedConfiguration' - use the rigged configuration model

  • 'LSPaths' - use the LS path model

EXAMPLES:

sage: K = crystals.KirillovReshetikhin(['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 = crystals.KirillovReshetikhin(['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 = crystals.KirillovReshetikhin(['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 [LOS2012].

REFERENCES:

• [Shi2002]

• [Sch2008]

• [JS2010]

• [FOS2009]

• [LOS2012]

sage.combinat.crystals.kirillov_reshetikhin.KirillovReshetikhinCrystalFromLSPaths(cartan_type, r, s=1)

Single column Kirillov-Reshetikhin crystals.

This yields the single column Kirillov-Reshetikhin crystals from the projected level zero LS paths, see CrystalOfLSPaths. This works for all types (even exceptional types). The weight of the canonical element in this crystal is \(\Lambda_r\). For other implementation see KirillovReshetikhinCrystal().

EXAMPLES:

sage: K = crystals.kirillov_reshetikhin.LSPaths(['A',2,1],2) # indirect doctest
sage: KR = crystals.KirillovReshetikhin(['A',2,1],2,1)
sage: G = K.digraph()
sage: GR = KR.digraph()
sage: G.is_isomorphic(GR, edge_labels = True)
True

sage: K = crystals.kirillov_reshetikhin.LSPaths(['C',3,1],2)
sage: KR = crystals.KirillovReshetikhin(['C',3,1],2,1)
sage: G = K.digraph()
sage: GR = KR.digraph()
sage: G.is_isomorphic(GR, edge_labels = True)
True

sage: K = crystals.kirillov_reshetikhin.LSPaths(['E',6,1],1)
sage: KR = crystals.KirillovReshetikhin(['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 = crystals.kirillov_reshetikhin.LSPaths(['G',2,1],1)
sage: K.cardinality()
7

sage: K = crystals.kirillov_reshetikhin.LSPaths(['B',3,1],2)
sage: KR = crystals.KirillovReshetikhin(['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

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

Bases: KirillovReshetikhinGenericCrystal, AffineCrystalFromClassicalAndPromotion

This generic class assumes that the Kirillov-Reshetikhin crystal is constructed from a classical crystal using the 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

Element

alias of KirillovReshetikhinCrystalFromPromotionElement

class sage.combinat.crystals.kirillov_reshetikhin.KirillovReshetikhinCrystalFromPromotionElement

Bases: AffineCrystalFromClassicalAndPromotionElement, KirillovReshetikhinGenericCrystalElement

Element for a Kirillov-Reshetikhin crystal from promotion.

class sage.combinat.crystals.kirillov_reshetikhin.KirillovReshetikhinGenericCrystal(cartan_type, r, s, dual=None)

Bases: AffineCrystalFromClassical

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.

Element

alias of KirillovReshetikhinGenericCrystalElement

classically_highest_weight_vectors()

Return the classically highest weight vectors of self.

EXAMPLES:

sage: K = crystals.KirillovReshetikhin(['D', 4, 1], 2, 2)
sage: K.classically_highest_weight_vectors()
([], [[1], [2]], [[1, 1], [2, 2]])

kirillov_reshetikhin_tableaux()

Return the corresponding set of KirillovReshetikhinTableaux.

EXAMPLES:

sage: KRC = crystals.KirillovReshetikhin(['D', 4, 1], 2, 2)
sage: KRC.kirillov_reshetikhin_tableaux()
Kirillov-Reshetikhin tableaux of type ['D', 4, 1] and shape (2, 2)

module_generator()

Return the unique module generator of classical weight \(s \Lambda_r\) of a Kirillov-Reshetikhin crystal \(B^{r,s}\)

EXAMPLES:

sage: K = crystals.KirillovReshetikhin(['C',2,1],1,2)
sage: K.module_generator()
[[1, 1]]
sage: K = crystals.KirillovReshetikhin(['E',6,1],1,1)
sage: K.module_generator()
[(1,)]

sage: K = crystals.KirillovReshetikhin(['D',4,1],2,1)
sage: K.module_generator()
[[1], [2]]

r()

Return \(r\) of the underlying Kirillov-Reshetikhin crystal \(B^{r,s}\).

EXAMPLES:

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

s()

Return \(s\) of the underlying Kirillov-Reshetikhin crystal \(B^{r,s}\).

EXAMPLES:

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

class sage.combinat.crystals.kirillov_reshetikhin.KirillovReshetikhinGenericCrystalElement

Bases: AffineCrystalFromClassicalElement

Abstract class for all Kirillov-Reshetikhin crystal elements.

lusztig_involution()

Return the classical Lusztig involution on self.

EXAMPLES:

sage: KRC = crystals.KirillovReshetikhin(['D',4,1], 2,2)
sage: elt = KRC(-1,2); elt
[[2], [-1]]
sage: elt.lusztig_involution()
[[1], [-2]]

pp()

Pretty print self.

EXAMPLES:

sage: C = crystals.KirillovReshetikhin(['D',4,1], 2,1)
sage: C(2,1).pp()
  1
  2
sage: C = crystals.KirillovReshetikhin(['B',3,1], 3,3)
sage: C.module_generators[0].pp()
+ (X)   1
+
+

to_kirillov_reshetikhin_tableau()

Construct the corresponding KirillovReshetikhinTableauxElement from self.

We construct the Kirillov-Reshetikhin tableau element as follows:

1. Let \(\lambda\) be the shape of self.

2. Determine a path \(e_{i_1} e_{i_2} \cdots e_{i_k}\) to the highest weight.

3. Apply \(f_{i_k} \cdots f_{i_2} f_{i_1}\) to a highest weight KR tableau from filling the shape \(\lambda\).

EXAMPLES:

sage: KRC = crystals.KirillovReshetikhin(['A', 4, 1], 2, 1)
sage: KRC(columns=[[2,1]]).to_kirillov_reshetikhin_tableau()
[[1], [2]]
sage: KRC = crystals.KirillovReshetikhin(['D', 4, 1], 2, 1)
sage: KRC(rows=[]).to_kirillov_reshetikhin_tableau()
[[1], [-1]]

to_tableau()

Return the Tableau corresponding to self.

EXAMPLES:

sage: C = crystals.KirillovReshetikhin(['D',4,1], 2,1)
sage: t = C(2,1).to_tableau(); t
[[1], [2]]
sage: type(t)
<class 'sage.combinat.tableau.Tableaux_all_with_category.element_class'>

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

Bases: 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 [Sch2008].

The input is a list \(pm = [[a_0,b_0], [a_1,b_1], ..., [a_{n-1},b_{n-1}], [b_n]]\) of pairs and a last 1-tuple (or list of length 1). The pair \([a_i,b_i]\) 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 \(a_i\) \(\pm\) 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 list [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 shapes, respectively.

EXAMPLES:

sage: from sage.combinat.crystals.kirillov_reshetikhin import PMDiagram
sage: pm = 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.pp()
.  .  .  .
.  +  -  -
sage: PMDiagram([2,5,[4,4],[4,2],[4,1]], from_shapes=True)
[[0, 1], [1, 2], [1]]

heights_of_addable_plus()

Return a list with the heights of all addable plus in the \(\pm\) diagram.

EXAMPLES:

sage: from sage.combinat.crystals.kirillov_reshetikhin import PMDiagram
sage: pm = 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 = PMDiagram([[1,2],[1,1],[1,1],[1,1],[1]])
sage: pm.heights_of_addable_plus()
[1, 2, 3, 4]

heights_of_minus()

Return a list with the heights of all minus in the \(\pm\) diagram.

EXAMPLES:

sage: from sage.combinat.crystals.kirillov_reshetikhin import PMDiagram
sage: pm = 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 = PMDiagram([[1,2],[1,1],[1,1],[1,1],[1]])
sage: pm.heights_of_minus()
[4, 4, 2, 2]

inner_shape()

Return the inner shape of the pm diagram

EXAMPLES:

sage: from sage.combinat.crystals.kirillov_reshetikhin import PMDiagram
sage: pm = PMDiagram([[0,1],[1,2],[1]])
sage: pm.inner_shape()
[4, 1]
sage: pm = PMDiagram([[1,2],[1,1],[1,1],[1,1],[1]])
sage: pm.inner_shape()
[7, 5, 3, 1]
sage: pm = PMDiagram([[1,2],[1,2],[1,1],[1,1],[1,1],[1]])
sage: pm.inner_shape()
[10, 7, 5, 3, 1]

intermediate_shape()

Return the intermediate shape of the pm diagram (inner shape plus positions of plusses)

EXAMPLES:

sage: from sage.combinat.crystals.kirillov_reshetikhin import PMDiagram
sage: pm = PMDiagram([[0,1],[1,2],[1]])
sage: pm.intermediate_shape()
[4, 2]
sage: pm = PMDiagram([[1,2],[1,1],[1,1],[1,1],[1]])
sage: pm.intermediate_shape()
[8, 6, 4, 2]
sage: pm = PMDiagram([[1,2],[1,2],[1,1],[1,1],[1,1],[1]])
sage: pm.intermediate_shape()
[11, 8, 6, 4, 2]
sage: pm = PMDiagram([[1,0],[0,1],[2,0],[0,0],[0]])
sage: pm.intermediate_shape()
[4, 2, 2]
sage: pm = PMDiagram([[1, 0], [0, 0], [0, 0], [0, 0], [0]])
sage: pm.intermediate_shape()
[1]

outer_shape()

Return the outer shape of the \(\pm\) diagram

EXAMPLES:

sage: from sage.combinat.crystals.kirillov_reshetikhin import PMDiagram
sage: pm = PMDiagram([[0,1],[1,2],[1]])
sage: pm.outer_shape()
[4, 4]
sage: pm = PMDiagram([[1,2],[1,1],[1,1],[1,1],[1]])
sage: pm.outer_shape()
[8, 8, 4, 4]
sage: pm = PMDiagram([[1,2],[1,2],[1,1],[1,1],[1,1],[1]])
sage: pm.outer_shape()
[13, 8, 8, 4, 4]

pp()

Pretty print self.

EXAMPLES:

sage: from sage.combinat.crystals.kirillov_reshetikhin import PMDiagram
sage: pm = PMDiagram([[1,0],[0,1],[2,0],[0,0],[0]])
sage: pm.pp()
.  .  .  +
.  .  -  -
+  +
-  -
sage: pm = PMDiagram([[0,2], [0,0], [0]])
sage: pm.pp()

sigma()

Return 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()
[[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]]