Main entry points:
AUTHORS:
Bases: sage.combinat.crystals.tensor_product.CrystalOfWords
A class for crystals of tableaux with integer valued shapes
INPUT:
This constructs a classical crystal with the given Cartan type and highest weight(s) corresponding to the given shape(s).
If the type is \(D_r\), the shape is permitted to have a negative value in the \(r\)-th position. Thus if the shape equals \([s_1,\ldots,s_r]\), then \(s_r\) may be negative but in any case \(s_1 \geq \cdots \geq s_{r-1} \geq |s_r|\). This crystal is related to that of shape \([s_1,\ldots,|s_r|]\) by the outer automorphism of \(SO(2r)\).
If the type is \(D_r\) or \(B_r\), the shape is permitted to be of length \(r\) with all parts of half integer value. This corresponds to having one spin column at the beginning of the tableau. If several shapes are provided, they currently should all or none have this property.
Crystals of tableaux are constructed using an embedding into tensor products following Kashiwara and Nakashima [KN94]. Sage’s tensor product rule for crystals differs from that of Kashiwara and Nakashima by reversing the order of the tensor factors. Sage produces the same crystals of tableaux as Kashiwara and Nakashima. With Sage’s convention, the tensor product of crystals is the same as the monoid operation on tableaux and hence the plactic monoid.
See also
sage.combinat.crystals.crystals for general help on crystals, and in particular plotting and \(\LaTeX\) output.
EXAMPLES:
We create the crystal of tableaux for type \(A_2\), with highest weight given by the partition \([2,1,1]\):
sage: T = crystals.Tableaux(['A',3], shape = [2,1,1])
Here is the list of its elements:
sage: T.list()
[[[1, 1], [2], [3]], [[1, 2], [2], [3]], [[1, 3], [2], [3]],
[[1, 4], [2], [3]], [[1, 4], [2], [4]], [[1, 4], [3], [4]],
[[2, 4], [3], [4]], [[1, 1], [2], [4]], [[1, 2], [2], [4]],
[[1, 3], [2], [4]], [[1, 3], [3], [4]], [[2, 3], [3], [4]],
[[1, 1], [3], [4]], [[1, 2], [3], [4]], [[2, 2], [3], [4]]]
Internally, a tableau of a given Cartan type is represented as a tensor product of letters of the same type. The order in which the tensor factors appear is by reading the columns of the tableaux left to right, top to bottom (in French notation). As an example:
sage: T = crystals.Tableaux(['A',2], shape = [3,2])
sage: T.module_generators[0]
[[1, 1, 1], [2, 2]]
sage: T.module_generators[0]._list
[2, 1, 2, 1, 1]
To create a tableau, one can use:
sage: Tab = crystals.Tableaux(['A',3], shape = [2,2])
sage: Tab(rows=[[1,2],[3,4]])
[[1, 2], [3, 4]]
sage: Tab(columns=[[3,1],[4,2]])
[[1, 2], [3, 4]]
Todo
FIXME:
We illustrate the use of a shape with a negative last entry in type \(D\):
sage: T = crystals.Tableaux(['D',4],shape=[1,1,1,-1])
sage: T.cardinality()
35
sage: TestSuite(T).run()
We illustrate the construction of crystals of spin tableaux when the partitions have half integer values in type \(B\) and \(D\):
sage: T = crystals.Tableaux(['B',3],shape=[3/2,1/2,1/2]); T
The crystal of tableaux of type ['B', 3] and shape(s) [[3/2, 1/2, 1/2]]
sage: T.cardinality()
48
sage: T.module_generators
[[+++, [[1]]]]
sage: TestSuite(T).run()
sage: T = crystals.Tableaux(['D',3],shape=[3/2,1/2,-1/2]); T
The crystal of tableaux of type ['D', 3] and shape(s) [[3/2, 1/2, -1/2]]
sage: T.cardinality()
20
sage: T.module_generators
[[++-, [[1]]]]
sage: TestSuite(T).run()
TESTS:
Base cases:
sage: T = crystals.Tableaux(['A',2], shape = [])
sage: T.list()
[[]]
sage: TestSuite(T).run()
sage: T = crystals.Tableaux(['C',2], shape = [1])
sage: T.list()
[[[1]], [[2]], [[-2]], [[-1]]]
sage: TestSuite(T).run()
sage: T = crystals.Tableaux(['A',2], shapes = [[],[1],[2]])
sage: T.list()
[[], [[1]], [[2]], [[3]], [[1, 1]], [[1, 2]], [[2, 2]], [[1, 3]], [[2, 3]], [[3, 3]]]
sage: T.module_generators
([], [[1]], [[1, 1]])
sage: T = crystals.Tableaux(['B',2], shape=[3])
sage: T(rows=[[1,1,0]])
[[1, 1, 0]]
Input tests:
sage: T = crystals.Tableaux(['A',3], shape = [2,2])
sage: C = T.letters
sage: Tab(rows = [[1,2],[3,4]])._list == [C(3),C(1),C(4),C(2)]
True
sage: Tab(columns = [[3,1],[4,2]])._list == [C(3),C(1),C(4),C(2)]
True
For compatibility with TensorProductOfCrystals() we need to accept as input the internal list or sequence of elements:
sage: Tab(list = [3,1,4,2])._list == [C(3),C(1),C(4),C(2)]
True
sage: Tab(3,1,4,2)._list == [C(3),C(1),C(4),C(2)]
True
The next example checks whether a given tableau is in fact a valid type \(C\) tableau or not:
sage: T = crystals.Tableaux(['C',3], shape = [2,2,2])
sage: Tab = T(rows=[[1,3],[2,-3],[3,-1]])
sage: Tab in T.list()
True
sage: Tab = T(rows=[[2,3],[3,-3],[-3,-2]])
sage: Tab in T.list()
False
alias of CrystalOfTableauxElement
Returns the Cartan type of the associated crystal
EXAMPLES:
sage: T = crystals.Tableaux(['A',3], shape = [2,2])
sage: T.cartan_type()
['A', 3]
This yields the module generator (or highest weight element) of a classical crystal of given shape. The module generator is the unique tableau with equal shape and content.
EXAMPLE:
sage: T = crystals.Tableaux(['D',3], shape = [1,1])
sage: T.module_generator([1,1])
[[1], [2]]
sage: T = crystals.Tableaux(['D',4],shape=[2,2,2,-2])
sage: T.module_generator(tuple([2,2,2,-2]))
[[1, 1], [2, 2], [3, 3], [-4, -4]]
sage: T.cardinality()
294
sage: T = crystals.Tableaux(['D',4],shape=[2,2,2,2])
sage: T.module_generator(tuple([2,2,2,2]))
[[1, 1], [2, 2], [3, 3], [4, 4]]
sage: T.cardinality()
294
Bases: sage.combinat.crystals.tensor_product.TensorProductOfRegularCrystalsElement
Element in a crystal of tableaux.
EXAMPLES:
sage: T = crystals.Tableaux(['A',3], shape = [2,2])
sage: t = T(rows=[[1,2],[3,4]])
sage: t.pp()
1 2
3 4
Promotion for type A crystals of tableaux of rectangular shape
Returns the result of applying promotion on this tableau.
This method only makes sense in type A with rectangular shapes.
EXAMPLES:
sage: C = crystals.Tableaux(["A",3], shape = [3,3,3])
sage: t = C(Tableau([[1,1,1],[2,2,3],[3,4,4]]))
sage: t
[[1, 1, 1], [2, 2, 3], [3, 4, 4]]
sage: t.promotion()
[[1, 1, 2], [2, 2, 3], [3, 4, 4]]
sage: t.promotion().parent()
The crystal of tableaux of type ['A', 3] and shape(s) [[3, 3, 3]]
Inverse promotion for type A crystals of tableaux of rectangular shape
Returns the result of applying inverse promotion on this tableau.
This method only makes sense in type A with rectangular shapes.
EXAMPLES:
sage: C = crystals.Tableaux(["A",3], shape = [3,3,3])
sage: t = C(Tableau([[1,1,1],[2,2,3],[3,4,4]]))
sage: t
[[1, 1, 1], [2, 2, 3], [3, 4, 4]]
sage: t.promotion_inverse()
[[1, 1, 2], [2, 3, 3], [4, 4, 4]]
sage: t.promotion_inverse().parent()
The crystal of tableaux of type ['A', 3] and shape(s) [[3, 3, 3]]
Returns the Tableau object corresponding to self.
EXAMPLES:
sage: T = crystals.Tableaux(['A',3], shape = [2,2])
sage: t = T(rows=[[1,2],[3,4]]).to_tableau(); t
[[1, 2], [3, 4]]
sage: type(t)
<class 'sage.combinat.tableau.Tableaux_all_with_category.element_class'>
sage: type(t[0][0])
<type 'int'>
sage: T = crystals.Tableaux(['D',3], shape = [1,1])
sage: t=T(rows=[[-3],[3]]).to_tableau(); t
[[-3], [3]]
sage: t=T(rows=[[3],[-3]]).to_tableau(); t
[[3], [-3]]
sage: T = crystals.Tableaux(['B',2], shape = [1,1])
sage: t = T(rows=[[0],[0]]).to_tableau(); t
[[0], [0]]
Bases: sage.structure.unique_representation.UniqueRepresentation, sage.structure.parent.Parent
Auxiliary class to provide a call method to create tensor product elements. This class is shared with several tensor product classes and is also used in CrystalOfTableaux to allow tableaux of different tensor product structures in column-reading (and hence different shapes) to be considered elements in the same crystal.
alias of TensorProductOfCrystalsElement
Computes the one-dimensional configuration sum.
INPUT:
The one-dimensional configuration sum is the sum of the weights of all elements in the crystal weighted by the energy function.
EXAMPLES:
sage: K = crystals.KirillovReshetikhin(['A',2,1],1,1)
sage: T = crystals.TensorProduct(K,K)
sage: T.one_dimensional_configuration_sum()
B[-2*Lambda[1] + 2*Lambda[2]] + (q+1)*B[-Lambda[1]] + (q+1)*B[Lambda[1] - Lambda[2]]
+ B[2*Lambda[1]] + B[-2*Lambda[2]] + (q+1)*B[Lambda[2]]
sage: R.<t> = ZZ[]
sage: T.one_dimensional_configuration_sum(t, False)
B[-2*Lambda[1] + 2*Lambda[2]] + (t+1)*B[-Lambda[1]] + (t+1)*B[Lambda[1] - Lambda[2]]
+ B[2*Lambda[1]] + B[-2*Lambda[2]] + (t+1)*B[Lambda[2]]
sage: R = RootSystem(['A',2,1])
sage: La = R.weight_space().basis()
sage: LS = crystals.ProjectedLevelZeroLSPaths(2*La[1])
sage: LS.one_dimensional_configuration_sum() == T.one_dimensional_configuration_sum() # long time
True
TESTS:
sage: K1 = crystals.KirillovReshetikhin(['A',2,1],1,1)
sage: K2 = crystals.KirillovReshetikhin(['A',2,1],2,1)
sage: T = crystals.TensorProduct(K1,K2)
sage: T.one_dimensional_configuration_sum() == T.one_dimensional_configuration_sum(group_components=False)
True
Bases: sage.combinat.crystals.tensor_product.TensorProductOfCrystals
Full tensor product of crystals.
Return the cardinality of self.
EXAMPLES:
sage: C = crystals.Letters(['A',2])
sage: T = crystals.TensorProduct(C,C)
sage: T.cardinality()
9
Bases: sage.combinat.crystals.tensor_product.FullTensorProductOfCrystals
Full tensor product of regular crystals.
Bases: sage.combinat.combinat.CombinatorialObject, sage.structure.element.Element
A class for lists having a parent
Specification: any subclass C should implement __init__ which accepts the following form C(parent, list = list)
EXAMPLES:
We create an immutable list whose parent is the class list:
sage: from sage.combinat.crystals.tensor_product import ImmutableListWithParent, TestParent
sage: l = ImmutableListWithParent(TestParent(), [1,2,3])
sage: l._list
[1, 2, 3]
sage: l.parent()
A parent for tests
sage: l.sibling([2,1]) == ImmutableListWithParent(TestParent(), [2,1])
True
sage: l.reversed()
[3, 2, 1]
sage: l.set_index(1,4)
[1, 4, 3]
Returns the sibling of self which is obtained by reversing the elements of`` self``.
EXAMPLES:
sage: from sage.combinat.crystals.tensor_product import ImmutableListWithParent, TestParent
sage: l = ImmutableListWithParent(TestParent(), [1,2,3])
sage: l.reversed()
[3, 2, 1]
Returns the sibling of self obtained by setting the \(k^{th}\) entry of self to value.
EXAMPLES:
sage: from sage.combinat.crystals.tensor_product import ImmutableListWithParent, TestParent
sage: l = ImmutableListWithParent(TestParent(), [1,2,3])
sage: l.set_index(0,2)
[2, 2, 3]
sage: l.set_index(1,4)
[1, 4, 3]
sage: _.parent()
A parent for tests
Returns an ImmutableListWithParent object whose list is l and whose parent is the same as the parent of self.
Note that the implementation of this function makes an assumption about the constructor for subclasses.
EXAMPLES:
sage: from sage.combinat.crystals.tensor_product import ImmutableListWithParent, TestParent
sage: l = ImmutableListWithParent(TestParent(), [1,2,3])
sage: m = l.sibling([2,3,4]); m
[2, 3, 4]
sage: m.parent()
A parent for tests
Bases: sage.combinat.crystals.tensor_product.CrystalOfWords
Tensor product of crystals.
Given two crystals \(B\) and \(B'\) of the same Cartan type, one can form the tensor product \(B \otimes B^{\prime}\). As a set \(B \otimes B^{\prime}\) is the Cartesian product \(B \times B^{\prime}\). The crystal operators \(f_i\) and \(e_i\) act on \(b \otimes b^{\prime} \in B \otimes B^{\prime}\) as follows:
and
We also define:
Note
This is the opposite of Kashiwara’s convention for tensor products of crystals.
Since tensor products are associative \((\mathcal{B} \otimes \mathcal{C}) \otimes \mathcal{D} \cong \mathcal{B} \otimes (\mathcal{C} \otimes \mathcal{D})\) via the natural isomorphism \((b \otimes c) \otimes d \mapsto b \otimes (c \otimes d)\), we can generalizing this to arbitrary tensor products. Thus consider \(B_N \otimes \cdots \otimes B_1\), where each \(B_k\) is an abstract crystal. The underlying set of the tensor product is \(B_N \times \cdots \times B_1\), while the crystal structure is given as follows. Let \(I\) be the index set, and fix some \(i \in I\) and \(b_N \otimes \cdots \otimes b_1 \in B_N \otimes \cdots \otimes B_1\). Define
Then
where \(\lambda_i = \langle \alpha_i^{\vee}, \mathrm{wt}(b_N \otimes \cdots \otimes b_1) \rangle\). Then for \(k = 1, \ldots, N\) the action of the Kashiwara operators is determined as follows.
If \(a_i(k) > a_i(j)\) for \(1 \leq j < k\) and \(a_i(k) \geq a_i(j)\) for \(k < j \leq N\):
If \(a_i(k) \geq a_i(j)\) for \(1 \leq j < k\) and \(a_i(k) > a_i(j)\) for \(k < j \leq N\):
Note that this is just recursively applying the definition of the tensor product on two crystals. Recall that \(\langle \alpha_i^{\vee}, \mathrm{wt}(b_j) \rangle = \varphi_i(b_j) - \varepsilon_i(b_j)\) by the definition of a crystal.
Regular crystals
Now if all crystals \(B_k\) are regular crystals, all \(\varepsilon_i\) and \(\varphi_i\) are non-negative and we can define tensor product by the signature rule. We start by writing a word in \(+\) and \(-\) as follows:
and then canceling ordered pairs of \(+-\) until the word is in the reduced form:
Here \(e_i\) acts on the factor corresponding to the leftmost \(+\) and \(f_i\) on the factor corresponding to the rightmost \(-\). If there is no \(+\) or \(-\) respectively, then the result is \(0\) (None).
EXAMPLES:
We construct the type \(A_2\)-crystal generated by \(2 \otimes 1 \otimes 1\):
sage: C = crystals.Letters(['A',2])
sage: T = crystals.TensorProduct(C,C,C,generators=[[C(2),C(1),C(1)]])
It has \(8\) elements:
sage: T.list()
[[2, 1, 1], [2, 1, 2], [2, 1, 3], [3, 1, 3], [3, 2, 3], [3, 1, 1], [3, 1, 2], [3, 2, 2]]
One can also check the Cartan type of the crystal:
sage: T.cartan_type()
['A', 2]
Other examples include crystals of tableaux (which internally are represented as tensor products obtained by reading the tableaux columnwise):
sage: C = crystals.Tableaux(['A',3], shape=[1,1,0])
sage: D = crystals.Tableaux(['A',3], shape=[1,0,0])
sage: T = crystals.TensorProduct(C,D, generators=[[C(rows=[[1], [2]]), D(rows=[[1]])], [C(rows=[[2], [3]]), D(rows=[[1]])]])
sage: T.cardinality()
24
sage: TestSuite(T).run()
sage: T.module_generators
[[[[1], [2]], [[1]]], [[[2], [3]], [[1]]]]
sage: [x.weight() for x in T.module_generators]
[(2, 1, 0, 0), (1, 1, 1, 0)]
If no module generators are specified, we obtain the full tensor product:
sage: C = crystals.Letters(['A',2])
sage: T = crystals.TensorProduct(C,C)
sage: T.list()
[[1, 1], [1, 2], [1, 3], [2, 1], [2, 2], [2, 3], [3, 1], [3, 2], [3, 3]]
sage: T.cardinality()
9
For a tensor product of crystals without module generators, the default implementation of module_generators contains all elements in the tensor product of the crystals. If there is a subset of elements in the tensor product that still generates the crystal, this needs to be implemented for the specific crystal separately:
sage: T.module_generators.list()
[[1, 1], [1, 2], [1, 3], [2, 1], [2, 2], [2, 3], [3, 1], [3, 2], [3, 3]]
For classical highest weight crystals, it is also possible to list all highest weight elements:
sage: C = crystals.Letters(['A',2])
sage: T = crystals.TensorProduct(C,C,C,generators=[[C(2),C(1),C(1)],[C(1),C(2),C(1)]])
sage: T.highest_weight_vectors()
([2, 1, 1], [1, 2, 1])
Examples with non-regular and infinite crystals (these did not work before trac ticket #14402):
sage: B = crystals.infinity.Tableaux(['D',10])
sage: T = crystals.TensorProduct(B,B)
sage: T
Full tensor product of the crystals
[The infinity crystal of tableaux of type ['D', 10],
The infinity crystal of tableaux of type ['D', 10]]
sage: B = crystals.infinity.GeneralizedYoungWalls(15)
sage: T = crystals.TensorProduct(B,B,B)
sage: T
Full tensor product of the crystals
[Crystal of generalized Young walls of type ['A', 15, 1],
Crystal of generalized Young walls of type ['A', 15, 1],
Crystal of generalized Young walls of type ['A', 15, 1]]
sage: La = RootSystem(['A',2,1]).weight_lattice().fundamental_weights()
sage: B = crystals.GeneralizedYoungWalls(2,La[0]+La[1])
sage: C = crystals.GeneralizedYoungWalls(2,2*La[2])
sage: D = crystals.GeneralizedYoungWalls(2,3*La[0]+La[2])
sage: T = crystals.TensorProduct(B,C,D)
sage: T
Full tensor product of the crystals
[Highest weight crystal of generalized Young walls of Cartan type ['A', 2, 1] and highest weight Lambda[0] + Lambda[1],
Highest weight crystal of generalized Young walls of Cartan type ['A', 2, 1] and highest weight 2*Lambda[2],
Highest weight crystal of generalized Young walls of Cartan type ['A', 2, 1] and highest weight 3*Lambda[0] + Lambda[2]]
There is also a global option for setting the convention (by default Sage uses anti-Kashiwara):
sage: C = crystals.Letters(['A',2])
sage: T = crystals.TensorProduct(C,C)
sage: elt = T(C(1), C(2)); elt
[1, 2]
sage: crystals.TensorProduct.global_options['convention'] = "Kashiwara"
sage: elt
[2, 1]
sage: crystals.TensorProduct.global_options.reset()
Sets the global options for tensor products of crystals. The default is to use the anti-Kashiwara convention.
There are two conventions for how \(e_i\) and \(f_i\) act on tensor products, and the difference between the two is the order of the tensor factors are reversed. This affects both the input and output. See the example below.
OPTIONS:
Note
Changing the convention also changes how the input is handled.
Warning
Internally, the crystals are always stored using the anti-Kashiwara convention.
If no parameters are set, then the function returns a copy of the options dictionary.
EXAMPLES:
sage: C = crystals.Letters(['A',2])
sage: T = crystals.TensorProduct(C,C)
sage: elt = T(C(1), C(2)); elt
[1, 2]
sage: crystals.TensorProduct.global_options['convention'] = "Kashiwara"
sage: elt
[2, 1]
sage: T(C(1), C(2)) == elt
False
sage: T(C(2), C(1)) == elt
True
sage: crystals.TensorProduct.global_options.reset()
See GlobalOptions for more features of these options.
Bases: sage.combinat.crystals.tensor_product.ImmutableListWithParent
A class for elements of tensor products of crystals.
Return the action of \(e_i\) on self.
INPUT:
EXAMPLES:
sage: B = crystals.infinity.Tableaux("D4")
sage: T = crystals.TensorProduct(B,B)
sage: b1 = B.highest_weight_vector().f_string([1,4,3])
sage: b2 = B.highest_weight_vector().f_string([2,2,3,1,4])
sage: t = T(b2, b1)
sage: t.e(1)
[[[1, 1, 1, 1, 1], [2, 2, 3, -3], [3]], [[1, 1, 1, 1, 2], [2, 2, 2], [3, -3]]]
sage: t.e(2)
sage: t.e(3)
[[[1, 1, 1, 1, 1, 2], [2, 2, 3, -4], [3]], [[1, 1, 1, 1, 2], [2, 2, 2], [3, -3]]]
sage: t.e(4)
[[[1, 1, 1, 1, 1, 2], [2, 2, 3, 4], [3]], [[1, 1, 1, 1, 2], [2, 2, 2], [3, -3]]]
Return \(\varepsilon_i\) of self.
INPUT:
EXAMPLES:
sage: B = crystals.infinity.Tableaux("G2")
sage: T = crystals.TensorProduct(B,B)
sage: b1 = B.highest_weight_vector().f(2)
sage: b2 = B.highest_weight_vector().f_string([2,2,1])
sage: t = T(b2, b1)
sage: [t.epsilon(i) for i in B.index_set()]
[0, 3]
Return the action of \(f_i\) on self.
INPUT:
EXAMPLES:
sage: La = RootSystem(['A',3,1]).weight_lattice().fundamental_weights()
sage: B = crystals.GeneralizedYoungWalls(3,La[0])
sage: T = crystals.TensorProduct(B,B,B)
sage: b1 = B.highest_weight_vector().f_string([0,3])
sage: b2 = B.highest_weight_vector().f_string([0])
sage: b3 = B.highest_weight_vector()
sage: t = T(b3, b2, b1)
sage: t.f(0)
[[[0]], [[0]], [[0, 3]]]
sage: t.f(1)
[[], [[0]], [[0, 3], [1]]]
sage: t.f(2)
[[], [[0]], [[0, 3, 2]]]
sage: t.f(3)
[[], [[0, 3]], [[0, 3]]]
Return \(\varphi_i\) of self.
INPUT:
EXAMPLES:
sage: La = RootSystem(['A',2,1]).weight_lattice().fundamental_weights()
sage: B = crystals.GeneralizedYoungWalls(2,La[0]+La[1])
sage: T = crystals.TensorProduct(B,B)
sage: b1 = B.highest_weight_vector().f_string([1,0])
sage: b2 = B.highest_weight_vector().f_string([0,1])
sage: t = T(b2, b1)
sage: [t.phi(i) for i in B.index_set()]
[1, 1, 4]
TESTS:
Check that trac ticket #15462 is fixed:
sage: B = crystals.Tableaux(['A',2], shape=[2,1])
sage: La = RootSystem(['A',2]).ambient_space().fundamental_weights()
sage: T = crystals.TensorProduct(crystals.elementary.T(['A',2], La[1]+La[2]), B)
sage: t = T.an_element()
sage: t.phi(1)
2
sage: t.phi(2)
2
Pretty print self.
EXAMPLES:
sage: C = crystals.Tableaux(['A',3], shape=[3,1])
sage: D = crystals.Tableaux(['A',3], shape=[1])
sage: E = crystals.Tableaux(['A',3], shape=[2,2,2])
sage: T = crystals.TensorProduct(C,D,E)
sage: T.module_generators[0].pp()
1 1 1 (X) 1 (X) 1 1
2 2 2
3 3
Return the weight of self.
EXAMPLES:
sage: B = crystals.infinity.Tableaux("A3")
sage: T = crystals.TensorProduct(B,B)
sage: b1 = B.highest_weight_vector().f_string([2,1,3])
sage: b2 = B.highest_weight_vector().f(1)
sage: t = T(b2, b1)
sage: t
[[[1, 1, 1, 2], [2, 2], [3]], [[1, 1, 1, 1, 2], [2, 2, 4], [3]]]
sage: t.weight()
(-2, 1, 0, 1)
Bases: sage.combinat.crystals.tensor_product.TensorProductOfCrystals
Tensor product of crystals with a generating set.
Bases: sage.combinat.crystals.tensor_product.TensorProductOfCrystalsElement
Element class for a tensor product of regular crystals.
TESTS:
sage: C = crystals.Letters(['A',2])
sage: T = crystals.TensorProduct(C, C)
sage: elt = T(C(1), C(2))
sage: from sage.combinat.crystals.tensor_product import TensorProductOfRegularCrystalsElement
sage: isinstance(elt, TensorProductOfRegularCrystalsElement)
True
Returns the affine grading of \(self\).
The affine grading is only defined when self is an element of a tensor product of affine Kirillov-Reshetikhin crystals. It is calculated by finding a path from self to a ground state path using the helper method e_string_to_ground_state() and counting the number of affine Kashiwara operators \(e_0\) applied on the way.
INPUT:
OUTPUT: an integer
EXAMPLES:
sage: K = crystals.KirillovReshetikhin(['A',2,1],1,1)
sage: T = crystals.TensorProduct(K,K)
sage: t = T.module_generators[0]
sage: t.affine_grading()
1
sage: K = crystals.KirillovReshetikhin(['A',2,1],1,1)
sage: T = crystals.TensorProduct(K,K,K)
sage: hw = [b for b in T if all(b.epsilon(i)==0 for i in [1,2])]
sage: for b in hw:
....: print b, b.affine_grading()
[[[1]], [[1]], [[1]]] 3
[[[1]], [[2]], [[1]]] 1
[[[2]], [[1]], [[1]]] 2
[[[3]], [[2]], [[1]]] 0
sage: K = crystals.KirillovReshetikhin(['C',2,1],1,1)
sage: T = crystals.TensorProduct(K,K,K)
sage: hw = [b for b in T if all(b.epsilon(i)==0 for i in [1,2])]
sage: for b in hw:
....: print b, b.affine_grading()
[[[1]], [[1]], [[1]]] 2
[[[1]], [[2]], [[1]]] 1
[[[1]], [[-1]], [[1]]] 0
[[[2]], [[1]], [[1]]] 1
[[[-2]], [[2]], [[1]]] 0
[[[-1]], [[1]], [[1]]] 1
Return the action of \(e_i\) on self.
EXAMPLES:
sage: C = crystals.Letters(['A',5])
sage: T = crystals.TensorProduct(C,C)
sage: T(C(1),C(2)).e(1) == T(C(1),C(1))
True
sage: T(C(2),C(1)).e(1) is None
True
sage: T(C(2),C(2)).e(1) == T(C(1),C(2))
True
Returns a string of integers in the index set \((i_1,\ldots,i_k)\) such that \(e_{i_k} \cdots e_{i_1}\) of self is the ground state.
This method is only defined when self is an element of a tensor product of affine Kirillov-Reshetikhin crystals. It calculates a path from self to a ground state path using Demazure arrows as defined in Lemma 7.3 in [SchillingTingley2011].
INPUT:
OUTPUT: a tuple of integers \((i_1,\ldots,i_k)\)
EXAMPLES:
sage: K = crystals.KirillovReshetikhin(['A',2,1],1,1)
sage: T = crystals.TensorProduct(K,K)
sage: t = T.module_generators[0]
sage: t.e_string_to_ground_state()
(0, 2)
sage: K = crystals.KirillovReshetikhin(['C',2,1],1,1)
sage: T = crystals.TensorProduct(K,K)
sage: t = T.module_generators[0]; t
[[[1]], [[1]]]
sage: t.e_string_to_ground_state()
(0,)
sage: x=t.e(0)
sage: x.e_string_to_ground_state()
()
sage: y=t.f_string([1,2,1,1,0]); y
[[[2]], [[1]]]
sage: y.e_string_to_ground_state()
()
Return the energy function of self.
The energy is only defined when self is an element of a tensor product of affine Kirillov-Reshetikhin crystals. In this implementation, it is assumed that self is an element of a tensor product of perfect crystals of the same level, see Theorem 7.5 in [SchillingTingley2011].
INPUT:
OUTPUT: an integer
REFERENCES:
| [SchillingTingley2011] | (1, 2) A. Schilling, P. Tingley. Demazure crystals, Kirillov-Reshetikhin crystals, and the energy function. Electronic Journal of Combinatorics. 19(2). 2012. Arxiv 1104.2359 |
EXAMPLES:
sage: K = crystals.KirillovReshetikhin(['A',2,1],1,1)
sage: T = crystals.TensorProduct(K,K,K)
sage: hw = [b for b in T if all(b.epsilon(i)==0 for i in [1,2])]
sage: for b in hw:
....: print b, b.energy_function()
[[[1]], [[1]], [[1]]] 0
[[[1]], [[2]], [[1]]] 2
[[[2]], [[1]], [[1]]] 1
[[[3]], [[2]], [[1]]] 3
sage: K = crystals.KirillovReshetikhin(['C',2,1],1,2)
sage: T = crystals.TensorProduct(K,K)
sage: hw = [b for b in T if all(b.epsilon(i)==0 for i in [1,2])]
sage: for b in hw: # long time (5s on sage.math, 2011)
....: print b, b.energy_function()
[[], []] 4
[[], [[1, 1]]] 1
[[[1, 1]], []] 3
[[[1, 1]], [[1, 1]]] 0
[[[1, 2]], [[1, 1]]] 1
[[[2, 2]], [[1, 1]]] 2
[[[-1, -1]], [[1, 1]]] 2
[[[1, -1]], [[1, 1]]] 2
[[[2, -1]], [[1, 1]]] 2
sage: K = crystals.KirillovReshetikhin(['C',2,1],1,1)
sage: T = crystals.TensorProduct(K)
sage: t = T.module_generators[0]
sage: t.energy_function()
Traceback (most recent call last):
...
ValueError: All crystals in the tensor product need to be perfect of the same level
Return \(\varepsilon_i\) of self.
EXAMPLES:
sage: C = crystals.Letters(['A',5])
sage: T = crystals.TensorProduct(C,C)
sage: T(C(1),C(1)).epsilon(1)
0
sage: T(C(1),C(2)).epsilon(1)
1
sage: T(C(2),C(1)).epsilon(1)
0
Return the action of \(f_i\) on self.
EXAMPLES:
sage: C = crystals.Letters(['A',5])
sage: T = crystals.TensorProduct(C,C)
sage: T(C(1),C(1)).f(1)
[1, 2]
sage: T(C(1),C(2)).f(1)
[2, 2]
sage: T(C(2),C(1)).f(1) is None
True
Return \(\varphi_i\) of self.
EXAMPLES:
sage: C = crystals.Letters(['A',5])
sage: T = crystals.TensorProduct(C,C)
sage: T(C(1),C(1)).phi(1)
2
sage: T(C(1),C(2)).phi(1)
1
sage: T(C(2),C(1)).phi(1)
0
EXAMPLES:
sage: C = crystals.Letters(['A',5])
sage: T = crystals.TensorProduct(C,C)
sage: T(C(2),C(1)).positions_of_unmatched_minus(1)
[]
sage: T(C(1),C(2)).positions_of_unmatched_minus(1)
[0]
EXAMPLES:
sage: C = crystals.Letters(['A',5])
sage: T = crystals.TensorProduct(C,C)
sage: T(C(2),C(1)).positions_of_unmatched_plus(1)
[]
sage: T(C(1),C(2)).positions_of_unmatched_plus(1)
[1]
Return the weight of self.
EXAMPLES:
sage: C = crystals.Letters(['A',3])
sage: T = crystals.TensorProduct(C,C)
sage: T(C(1),C(2)).weight()
(1, 1, 0, 0)
sage: T = crystals.Tableaux(['D',4],shape=[])
sage: T.list()[0].weight()
(0, 0, 0, 0)
Bases: sage.combinat.crystals.tensor_product.TensorProductOfCrystalsWithGenerators
Tensor product of regular crystals with a generating set.
Bases: sage.structure.unique_representation.UniqueRepresentation, sage.structure.parent.Parent
A parent for tests.
Truncates to the integer closer to zero
EXAMPLES:
sage: from sage.combinat.crystals.tensor_product import trunc
sage: trunc(-3/2), trunc(-1), trunc(-1/2), trunc(0), trunc(1/2), trunc(1), trunc(3/2)
(-1, -1, 0, 0, 0, 1, 1)
sage: isinstance(trunc(3/2), Integer)
True