Tensor Products of Crystals

Main entry points:

AUTHORS:

  • Anne Schilling, Nicolas Thiery (2007): Initial version
  • Ben Salisbury, Travis Scrimshaw (2013): Refactored tensor products to handle non-regular crystals and created new subclass to take advantage of the regularity
class sage.combinat.crystals.tensor_product.CrystalOfTableaux(cartan_type, shapes)

Bases: sage.combinat.crystals.tensor_product.CrystalOfWords

A class for crystals of tableaux with integer valued shapes

INPUT:

  • cartan_type – a Cartan type
  • shape – a partition of length at most cartan_type.rank()
  • shapes – a list of such partitions

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:

  • Do we want to specify the columns increasingly or decreasingly? That is, should this be Tab(columns = [[1,3],[2,4]])?
  • Make this fully consistent with Tableau()!

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
Element

alias of CrystalOfTableauxElement

cartan_type()

Returns the Cartan type of the associated crystal

EXAMPLES:

sage: T = crystals.Tableaux(['A',3], shape = [2,2])
sage: T.cartan_type()
['A', 3]
module_generator(shape)

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
class sage.combinat.crystals.tensor_product.CrystalOfTableauxElement(parent, *args, **options)

Bases: sage.combinat.crystals.tensor_product.TensorProductOfRegularCrystalsElement

Element in a crystal of tableaux.

pp()

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()

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]]
promotion_inverse()

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]]
to_tableau()

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]]
class sage.combinat.crystals.tensor_product.CrystalOfWords

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.

Element

alias of TensorProductOfCrystalsElement

one_dimensional_configuration_sum(q=None, group_components=True)

Computes the one-dimensional configuration sum.

INPUT:

  • q – (default: None) a variable or None; if None, a variable \(q\) is set in the code
  • group_components – (default: True) boolean; if True, then the terms are grouped by classical component

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
class sage.combinat.crystals.tensor_product.FullTensorProductOfCrystals(crystals, **options)

Bases: sage.combinat.crystals.tensor_product.TensorProductOfCrystals

Full tensor product of crystals.

cardinality()

Return the cardinality of self.

EXAMPLES:

sage: C = crystals.Letters(['A',2])
sage: T = crystals.TensorProduct(C,C)
sage: T.cardinality()
9
class sage.combinat.crystals.tensor_product.FullTensorProductOfRegularCrystals(crystals, **options)

Bases: sage.combinat.crystals.tensor_product.FullTensorProductOfCrystals

Full tensor product of regular crystals.

Element

alias of TensorProductOfRegularCrystalsElement

class sage.combinat.crystals.tensor_product.ImmutableListWithParent(parent, list)

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]
reversed()

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]
set_index(k, value)

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
sibling(l)

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
class sage.combinat.crystals.tensor_product.TensorProductOfCrystals

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:

\[\begin{split}f_i(b \otimes b^{\prime}) = \begin{cases} f_i(b) \otimes b^{\prime} & \text{if } \varepsilon_i(b) \geq \varphi_i(b^{\prime}) \\ b \otimes f_i(b^{\prime}) & \text{otherwise} \end{cases}\end{split}\]

and

\[\begin{split}e_i(b \otimes b^{\prime}) = \begin{cases} e_i(b) \otimes b^{\prime} & \text{if } \varepsilon_i(b) > \varphi_i(b^{\prime}) \\ b \otimes e_i(b^{\prime}) & \text{otherwise.} \end{cases}\end{split}\]

We also define:

\[\begin{split}\begin{aligned} \varphi_i(b \otimes b^{\prime}) & = \max\left( \varphi_i(b), \varphi_i(b) + \varphi_i(b^{\prime}) - \varepsilon_i(b) \right) \\ \varepsilon_i(b \otimes b^{\prime}) & = \max\left( \varepsilon_i(b^{\prime}), \varepsilon_i(b^{\prime}) + \varepsilon_i(b) - \varphi_i(b^{\prime}) \right). \end{aligned}\end{split}\]

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

\[a_i(k) := \varepsilon_i(b_k) - \sum_{j=1}^{k-1} \langle \alpha_i^{\vee}, \mathrm{wt}(b_j) \rangle.\]

Then

\[\begin{split}\begin{aligned} \mathrm{wt}(b_N \otimes \cdots \otimes b_1) &= \mathrm{wt}(b_N) + \cdots + \mathrm{wt}(b_1), \\ \varepsilon_i(b_N \otimes \cdots \otimes b_1) &= \max_{1 \leq k \leq n}\left( \sum_{j=1}^k \varepsilon_i(b_j) - \sum_{j=1}^{k-1} \varphi_i(b_j) \right) \\ & = \max_{1 \leq k \leq N}\bigl( a_i(k) \bigr), \\ \varphi_i(b_N \otimes \cdots \otimes b_1) &= \max_{1 \leq k \leq N} \left( \varphi_i(b_N) + \sum_{j=k}^{N-1} \big( \varphi_i(b_j) - \varepsilon_i(b_{j+1}) \big) \right) \\ & = \max_{1 \leq k \leq N}\bigl( \lambda_i + a_i(k) \bigr) \end{aligned}\end{split}\]

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 \(k < j \leq N\):

    \[e_i(b_N \otimes \cdots \otimes b_1) = b_N \otimes \cdots \otimes e_i b_k \otimes \cdots \otimes b_1.\]
  • If \(a_i(k) \geq a_i(j)\) for \(1 \leq j < k\) and \(a_i(k) > a_i(j)\) for \(k < j \leq N\):

    \[f_i(b_N \otimes \cdots \otimes b_1) = b_N \otimes \cdots \otimes f_i b_k \otimes \cdots \otimes b_1.\]

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:

\[\underbrace{- \cdots -}_{\varphi_i(b_N) \text{ times}} \quad \underbrace{+ \cdots +}_{\varepsilon_i(b_N) \text{ times}} \quad \cdots \quad \underbrace{- \cdots -}_{\varphi_i(b_1) \text{ times}} \quad \underbrace{+ \cdots +}_{\varepsilon_i(b_1) \text{ times}},\]

and then canceling ordered pairs of \(+-\) until the word is in the reduced form:

\[\underbrace{- \cdots -}_{\varphi_i \text{ times}} \quad \underbrace{+ \cdots +}_{\varepsilon_i \text{ times}}.\]

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()
global_options(*get_value, **set_value)

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:

  • convention – (default: antiKashiwara) Sets the convention used for displaying/inputting tensor product of crystals
    • Kashiwara – use the Kashiwara convention
    • anti – alias for antiKashiwara
    • antiKashiwara – use the anti-Kashiwara convention
    • opposite – alias for antiKashiwara

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.

class sage.combinat.crystals.tensor_product.TensorProductOfCrystalsElement(parent, list)

Bases: sage.combinat.crystals.tensor_product.ImmutableListWithParent

A class for elements of tensor products of crystals.

e(i)

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

INPUT:

  • i – An element of the index set

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]]]
epsilon(i)

Return \(\varepsilon_i\) of self.

INPUT:

  • i – An element of the index set

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]
f(i)

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

INPUT:

  • i – An element of the index set

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]]]
phi(i)

Return \(\varphi_i\) of self.

INPUT:

  • i – An element of the index set

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
pp()

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
weight()

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)
class sage.combinat.crystals.tensor_product.TensorProductOfCrystalsWithGenerators(crystals, generators, cartan_type)

Bases: sage.combinat.crystals.tensor_product.TensorProductOfCrystals

Tensor product of crystals with a generating set.

class sage.combinat.crystals.tensor_product.TensorProductOfRegularCrystalsElement(parent, list)

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
affine_grading()

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:

  • self – an element of a tensor product of Kirillov-Reshetikhin crystals

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
e(i)

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
e_string_to_ground_state()

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:

  • self – an element of a tensor product of Kirillov-Reshetikhin crystals

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()
()
energy_function()

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:

  • self – an element of a tensor product of perfect Kirillov-Reshetkhin crystals of the same level

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
epsilon(i)

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
f(i)

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
phi(i)

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
positions_of_unmatched_minus(i, dual=False, reverse=False)

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]
positions_of_unmatched_plus(i)

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]
weight()

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)
class sage.combinat.crystals.tensor_product.TensorProductOfRegularCrystalsWithGenerators(crystals, generators, cartan_type)

Bases: sage.combinat.crystals.tensor_product.TensorProductOfCrystalsWithGenerators

Tensor product of regular crystals with a generating set.

Element

alias of TensorProductOfRegularCrystalsElement

class sage.combinat.crystals.tensor_product.TestParent

Bases: sage.structure.unique_representation.UniqueRepresentation, sage.structure.parent.Parent

A parent for tests.

sage.combinat.crystals.tensor_product.trunc(i)

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

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