Tensor Products of Crystals

Main entry points:

• TensorProductOfCrystals

• CrystalOfTableaux

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

• Travis Scrimshaw (2020): Added queer crystal

class sage.combinat.crystals.tensor_product.CrystalOfQueerTableaux(cartan_type, shape)

Bases: CrystalOfWords, QueerSuperCrystalsMixin

A queer crystal of the semistandard decomposition tableaux of a given shape.

INPUT:

• cartan_type – a Cartan type

• shape – a shape

class Element

Bases: TensorProductOfQueerSuperCrystalsElement

rows()

Return the list of rows of self.

EXAMPLES:

sage: B = crystals.Tableaux(['Q',3], shape=[3,2,1])
sage: t = B.an_element()
sage: t.rows()
[[3, 3, 3], [2, 2], [1]]

class sage.combinat.crystals.tensor_product.CrystalOfTableaux(cartan_type, shapes)

Bases: 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 [KN1994]. 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: list(T.module_generators[0])
[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()

We can also construct the tableaux for \(\mathfrak{gl}(m|n)\) as given by [BKK2000]:

sage: T = crystals.Tableaux(['A', [1,2]], shape=[4,2,1,1,1])
sage: T.cardinality()
1392

We can also construct the tableaux for \(\mathfrak{q}(n)\) as given by [GJK+2014]:

sage: T = crystals.Tableaux(['Q', 3], shape=[3,1])
sage: T.cardinality()
24

class Element

Bases: 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.

EXAMPLES:

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.CrystalOfWords

Bases: UniqueRepresentation, 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.

class Element

Bases: TensorProductOfCrystalsElement

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

Bases: TensorProductOfCrystals

Full tensor product of crystals.

Todo

Merge this into TensorProductOfCrystals.

cardinality()

Return the cardinality of self.

EXAMPLES:

sage: C = crystals.Letters(['A',2])
sage: T = crystals.TensorProduct(C,C)
sage: T.cardinality()
9

weight_lattice_realization()

Return the weight lattice realization used to express weights.

The weight lattice realization is the common parent which all weight lattice realizations of the crystals of self coerce into.

EXAMPLES:

sage: B = crystals.elementary.B(['A',4], 2)
sage: B.weight_lattice_realization()
Root lattice of the Root system of type ['A', 4]
sage: T = crystals.infinity.Tableaux(['A',4])
sage: T.weight_lattice_realization()
Ambient space of the Root system of type ['A', 4]
sage: TP = crystals.TensorProduct(B, T)
sage: TP.weight_lattice_realization()
Ambient space of the Root system of type ['A', 4]

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

Bases: FullTensorProductOfCrystals, QueerSuperCrystalsMixin

Tensor product of queer super crystals.

class Element

Bases: TensorProductOfQueerSuperCrystalsElement

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

Bases: FullTensorProductOfCrystals

Full tensor product of regular crystals.

class Element

Bases: TensorProductOfRegularCrystalsElement

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

Bases: FullTensorProductOfCrystals

Tensor product of super crystals.

EXAMPLES:

sage: L = crystals.Letters(['A', [1,1]])
sage: T = tensor([L,L,L])
sage: T.cardinality()
64

class Element

Bases: TensorProductOfSuperCrystalsElement

class sage.combinat.crystals.tensor_product.QueerSuperCrystalsMixin

Bases: object

Mixin class with methods for a finite queer supercrystal.

index_set()

Return the enlarged index set.

EXAMPLES:

sage: Q = crystals.Letters(['Q',3])
sage: T = tensor([Q,Q])
sage: T.index_set()
(-4, -3, -2, -1, 1, 2)

class sage.combinat.crystals.tensor_product.TensorProductOfCrystals

Bases: 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

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

We also define:

\[\begin{split}\begin{aligned} \varphi_i(b \otimes b') & = \max\left( \varphi_i(b), \varphi_i(b') + \langle \alpha_i^{\vee}, \mathrm{wt}(b) \rangle \right), \\ \varepsilon_i(b \otimes b') & = \max\left( \varepsilon_i(b'), \varepsilon_i(b) - \langle \alpha_i^{\vee}, \mathrm{wt}(b') \rangle \right). \end{aligned}\end{split}\]

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 github issue #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(extended=True).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.options.convention = "Kashiwara"
sage: elt
[2, 1]
sage: crystals.TensorProduct.options._reset()

options = Current options for TensorProductOfCrystals   - convention: antiKashiwara

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

Bases: TensorProductOfCrystals

Tensor product of crystals with a generating set.

Todo

Deprecate this class in favor of using subcrystal().

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

Bases: TensorProductOfCrystalsWithGenerators

Tensor product of regular crystals with a generating set.

class Element

Bases: TensorProductOfRegularCrystalsElement