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 , the shape is permitted to have a negative
value in the
-th position. Thus if the shape equals
,
then
may be negative but in any case
. This crystal is related to that of shape
by the outer automorphism of
.
If the type is or
, the shape is permitted to be of
length
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
output.EXAMPLES:
We create the crystal of tableaux for type , with
highest weight given by the partition
:
sage: T = CrystalOfTableaux(['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 = CrystalOfTableaux(['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 = CrystalOfTableaux(['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 :
sage: T = CrystalOfTableaux(['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 and
:
sage: T = CrystalOfTableaux(['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 = CrystalOfTableaux(['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 = CrystalOfTableaux(['A',2], shape = [])
sage: T.list()
[[]]
sage: TestSuite(T).run()
sage: T = CrystalOfTableaux(['C',2], shape = [1])
sage: T.list()
[[[1]], [[2]], [[-2]], [[-1]]]
sage: TestSuite(T).run()
sage: T = CrystalOfTableaux(['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 = CrystalOfTableaux(['B',2], shape=[3])
sage: T(rows=[[1,1,0]])
[[1, 1, 0]]
Input tests:
sage: T = CrystalOfTableaux(['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 tableau or not:
sage: T = CrystalOfTableaux(['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 = CrystalOfTableaux(['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 = CrystalOfTableaux(['D',3], shape = [1,1])
sage: T.module_generator([1,1])
[[1], [2]]
sage: T = CrystalOfTableaux(['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 = CrystalOfTableaux(['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 = CrystalOfTableaux(['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 = CrystalOfTableaux(["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 = CrystalOfTableaux(["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 = CrystalOfTableaux(['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 = CrystalOfTableaux(['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 = CrystalOfTableaux(['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 = KirillovReshetikhinCrystal(['A',2,1],1,1)
sage: T = TensorProductOfCrystals(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 = CrystalOfProjectedLevelZeroLSPaths(2*La[1])
sage: LS.one_dimensional_configuration_sum() == T.one_dimensional_configuration_sum()
True
TESTS:
sage: K1 = KirillovReshetikhinCrystal(['A',2,1],1,1)
sage: K2 = KirillovReshetikhinCrystal(['A',2,1],2,1)
sage: T = TensorProductOfCrystals(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 = CrystalOfLetters(['A',2])
sage: T = TensorProductOfCrystals(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
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 self’s parent.
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 and
of the same Cartan type,
one can form the tensor product
. As a set
is the Cartesian product
. The crystal operators
and
act on
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 via the natural isomorphism
, we can generalizing this to arbitrary tensor
products. Thus consider
, where each
is an abstract crystal. The underlying set of the tensor product is
, while the crystal structure is given
as follows. Let
be the index set, and fix some
and
. Define
Then
where . Then for
the action of the
Kashiwara operators is determined as follows.
If for
and
for
:
If for
and
for
:
Note that this is just recursively applying the definition of the tensor
product on two crystals. Recall that by the
definition of a crystal.
Regular crystals
Now if all crystals are regular crystals, all
and
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 acts on the factor corresponding to the leftmost
and
on the factor corresponding to the rightmost
. If there is no
or
respectively, then the result is
(None).
EXAMPLES:
We construct the type -crystal generated by
:
sage: C = CrystalOfLetters(['A',2])
sage: T = TensorProductOfCrystals(C,C,C,generators=[[C(2),C(1),C(1)]])
It has 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 = CrystalOfTableaux(['A',3], shape=[1,1,0])
sage: D = CrystalOfTableaux(['A',3], shape=[1,0,0])
sage: T = TensorProductOfCrystals(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 = CrystalOfLetters(['A',2])
sage: T = TensorProductOfCrystals(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 = CrystalOfLetters(['A',2])
sage: T = TensorProductOfCrystals(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 = InfinityCrystalOfTableaux(['D',10])
sage: T = TensorProductOfCrystals(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 = InfinityCrystalOfGeneralizedYoungWalls(15)
sage: T = TensorProductOfCrystals(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 = CrystalOfGeneralizedYoungWalls(2,La[0]+La[1])
sage: C = CrystalOfGeneralizedYoungWalls(2,2*La[2])
sage: D = CrystalOfGeneralizedYoungWalls(2,3*La[0]+La[2])
sage: T = TensorProductOfCrystals(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 = CrystalOfLetters(['A',2])
sage: T = TensorProductOfCrystals(C,C)
sage: elt = T(C(1), C(2)); elt
[1, 2]
sage: TensorProductOfCrystals.global_options['convention'] = "Kashiwara"
sage: elt
[2, 1]
sage: TensorProductOfCrystals.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 and
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 = CrystalOfLetters(['A',2])
sage: T = TensorProductOfCrystals(C,C)
sage: elt = T(C(1), C(2)); elt
[1, 2]
sage: TensorProductOfCrystals.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: TensorProductOfCrystals.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 on self.
INPUT:
EXAMPLES:
sage: B = InfinityCrystalOfTableaux("D4")
sage: T = TensorProductOfCrystals(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 of self.
INPUT:
EXAMPLES:
sage: B = InfinityCrystalOfTableaux("G2")
sage: T = TensorProductOfCrystals(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 on self.
INPUT:
EXAMPLES:
sage: La = RootSystem(['A',3,1]).weight_lattice().fundamental_weights()
sage: B = CrystalOfGeneralizedYoungWalls(3,La[0])
sage: T = TensorProductOfCrystals(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 of self.
INPUT:
EXAMPLES:
sage: La = RootSystem(['A',2,1]).weight_lattice().fundamental_weights()
sage: B = CrystalOfGeneralizedYoungWalls(2,La[0]+La[1])
sage: T = TensorProductOfCrystals(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 = CrystalOfTableaux(['A',2], shape=[2,1])
sage: La = RootSystem(['A',2]).ambient_space().fundamental_weights()
sage: T = TensorProductOfCrystals(TCrystal(['A',2], La[1]+La[2]), B)
sage: t = T.an_element()
sage: t.phi(1)
2
sage: t.phi(2)
2
Return the weight of self.
EXAMPLES:
sage: B = InfinityCrystalOfTableaux("A3")
sage: T = TensorProductOfCrystals(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 = CrystalOfLetters(['A',2])
sage: T = TensorProductOfCrystals(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 .
INPUT:
OUTPUT: an integer
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 applied on the way.
EXAMPLES:
sage: K = KirillovReshetikhinCrystal(['A',2,1],1,1)
sage: T = TensorProductOfCrystals(K,K)
sage: t = T.module_generators[0]
sage: t.affine_grading()
1
sage: K = KirillovReshetikhinCrystal(['A',2,1],1,1)
sage: T = TensorProductOfCrystals(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 = KirillovReshetikhinCrystal(['C',2,1],1,1)
sage: T = TensorProductOfCrystals(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 on self.
EXAMPLES:
sage: C = CrystalOfLetters(['A',5])
sage: T = TensorProductOfCrystals(C,C)
sage: T(C(1),C(2)).e(1) == T(C(1),C(1))
True
sage: T(C(2),C(1)).e(1) == 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 such that
is
the ground state.
INPUT:
OUTPUT: a tuple of integers
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].
EXAMPLES:
sage: K = KirillovReshetikhinCrystal(['A',2,1],1,1)
sage: T = TensorProductOfCrystals(K,K)
sage: t = T.module_generators[0]
sage: t.e_string_to_ground_state()
(0, 2)
sage: K = KirillovReshetikhinCrystal(['C',2,1],1,1)
sage: T = TensorProductOfCrystals(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.
INPUT:
OUTPUT: an integer
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].
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 = KirillovReshetikhinCrystal(['A',2,1],1,1)
sage: T = TensorProductOfCrystals(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 = KirillovReshetikhinCrystal(['C',2,1],1,2)
sage: T = TensorProductOfCrystals(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 = KirillovReshetikhinCrystal(['C',2,1],1,1)
sage: T = TensorProductOfCrystals(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 of self.
EXAMPLES:
sage: C = CrystalOfLetters(['A',5])
sage: T = TensorProductOfCrystals(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 on self.
EXAMPLES:
sage: C = CrystalOfLetters(['A',5])
sage: T = TensorProductOfCrystals(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 of self.
EXAMPLES:
sage: C = CrystalOfLetters(['A',5])
sage: T = TensorProductOfCrystals(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 = CrystalOfLetters(['A',5])
sage: T = TensorProductOfCrystals(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 = CrystalOfLetters(['A',5])
sage: T = TensorProductOfCrystals(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 = CrystalOfLetters(['A',3])
sage: T = TensorProductOfCrystals(C,C)
sage: T(C(1),C(2)).weight()
(1, 1, 0, 0)
sage: T=CrystalOfTableaux(['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
Base class for all parents.
Parents are the Sage/mathematical analogues of container objects in computer science.
INPUT:
If facade is specified, then Sets().Facades() is added to the categories of the parent. Furthermore, if facade is not True, the internal attribute _facade_for is set accordingly for use by Sets.Facades.ParentMethods.facade_for().
Internal invariants:
Todo
Eventually, category should be Sets by default.
This is the generic call method for all parents.
When called, it will find a map based on the Parent (or type) of x. If a coercion exists, it will always be chosen. This map will then be called (with the arguments and keywords if any).
By default this will dispatch as quickly as possible to _element_constructor_() though faster pathways are possible if so desired.
TESTS:
We check that the invariant:
self._element_init_pass_parent == guess_pass_parent(self, self._element_constructor)
is preserved (see trac ticket #5979):
sage: class MyParent(Parent):
....: def _element_constructor_(self, x):
....: print self, x
....: return sage.structure.element.Element(parent = self)
....: def _repr_(self):
....: return "my_parent"
....:
sage: my_parent = MyParent()
sage: x = my_parent("bla")
my_parent bla
sage: x.parent() # indirect doctest
my_parent
sage: x = my_parent() # shouldn't this one raise an error?
my_parent 0
sage: x = my_parent(3) # todo: not implemented why does this one fail???
my_parent 3
This function allows one to specify coercions, actions, conversions and embeddings involving this parent.
IT SHOULD ONLY BE CALLED DURING THE __INIT__ method, often at the end.
INPUT:
coerce_list – a list of coercion Morphisms to self and parents with canonical coercions to self
action_list – a list of actions on and by self
parents with conversions to self
embedding – a single Morphism from self
convert_method_name – a name to look for that other elements can implement to create elements of self (e.g. _integer_)
element_constructor – A callable object used by the __call__ method to construct new elements. Typically the element class or a bound method (defaults to self._element_constructor_).
init_no_parent – if True omit passing self in as the first argument of element_constructor for conversion. This is useful if parents are unique, or element_constructor is a bound method (this latter case can be detected automatically).
This is a multiplication method that more or less directly calls another attribute _mul_ (single underscore). This is because __mul__ can not be implemented via inheritance from the parent methods of the category, but _mul_ can be inherited. This is, e.g., used when creating twosided ideals of matrix algebras. See trac ticket #7797.
EXAMPLE:
sage: MS = MatrixSpace(QQ,2,2)
This matrix space is in fact an algebra, and in particular it is a ring, from the point of view of categories:
sage: MS.category()
Category of algebras over Rational Field
sage: MS in Rings()
True
However, its class does not inherit from the base class Ring:
sage: isinstance(MS,Ring)
False
Its _mul_ method is inherited from the category, and can be used to create a left or right ideal:
sage: MS._mul_.__module__
'sage.categories.rings'
sage: MS*MS.1 # indirect doctest
Left Ideal
(
[0 1]
[0 0]
)
of Full MatrixSpace of 2 by 2 dense matrices over Rational Field
sage: MS*[MS.1,2]
Left Ideal
(
[0 1]
[0 0],
[2 0]
[0 2]
)
of Full MatrixSpace of 2 by 2 dense matrices over Rational Field
sage: MS.1*MS
Right Ideal
(
[0 1]
[0 0]
)
of Full MatrixSpace of 2 by 2 dense matrices over Rational Field
sage: [MS.1,2]*MS
Right Ideal
(
[0 1]
[0 0],
[2 0]
[0 2]
)
of Full MatrixSpace of 2 by 2 dense matrices over Rational Field
True if there is an element of self that is equal to x under ==, or if x is already an element of self. Also, True in other cases involving the Symbolic Ring, which is handled specially.
For many structures we test this by using __call__() and then testing equality between x and the result.
The Symbolic Ring is treated differently because it is ultra-permissive about letting other rings coerce in, but ultra-strict about doing comparisons.
EXAMPLES:
sage: 2 in Integers(7)
True
sage: 2 in ZZ
True
sage: Integers(7)(3) in ZZ
True
sage: 3/1 in ZZ
True
sage: 5 in QQ
True
sage: I in RR
False
sage: SR(2) in ZZ
True
sage: RIF(1, 2) in RIF
True
sage: pi in RIF # there is no element of RIF equal to pi
False
sage: sqrt(2) in CC
True
sage: pi in RR
True
sage: pi in CC
True
sage: pi in RDF
True
sage: pi in CDF
True
TESTS:
Check that trac ticket #13824 is fixed:
sage: 4/3 in GF(3)
False
sage: 15/50 in GF(25, 'a')
False
sage: 7/4 in Integers(4)
False
sage: 15/36 in Integers(6)
False
Override this method to specify coercions beyond those specified in coerce_list.
If no such coercion exists, return None or False. Otherwise, it may return either an actual Map to use for the coercion, a callable (in which case it will be wrapped in a Map), or True (in which case a generic map will be provided).
Override this method to provide additional conversions beyond those given in convert_list.
This function is called after coercions are attempted. If there is a coercion morphism in the opposite direction, one should consider adding a section method to that.
This MUST return a Map from S to self, or None. If None is returned then a generic map will be provided.
Override this method to provide an action of self on S or S on self beyond what was specified in action_list.
This must return an action which accepts an element of self and an element of S (in the order specified by self_on_left).
Returns an element of self. Want it in sufficient generality that poorly-written functions won’t work when they’re not supposed to. This is cached so doesn’t have to be super fast.
EXAMPLES:
sage: QQ._an_element_()
1/2
sage: ZZ['x,y,z']._an_element_()
x
TESTS:
Since Parent comes before the parent classes provided by categories in the hierarchy of classes, we make sure that this default implementation of _an_element_() does not override some provided by the categories. Eventually, this default implementation should be moved into the categories to avoid this workaround:
sage: S = FiniteEnumeratedSet([1,2,3])
sage: S.category()
Category of facade finite enumerated sets
sage: super(Parent, S)._an_element_
Cached version of <function _an_element_from_iterator at ...>
sage: S._an_element_()
1
sage: S = FiniteEnumeratedSet([])
sage: S._an_element_()
Traceback (most recent call last):
...
EmptySetError
Metadata about the _repr_() output.
INPUT:
Valid key arguments are:
OUTPUT:
Boolean.
EXAMPLES:
sage: ZZ._repr_option('ascii_art')
False
sage: MatrixSpace(ZZ, 2)._repr_option('element_ascii_art')
True
Initialize the category framework
Most parents initialize their category upon construction, and this is the recommended behavior. For example, this happens when the constructor calls Parent.__init__() directly or indirectly. However, some parents defer this for performance reasons. For example, sage.matrix.matrix_space.MatrixSpace does not.
EXAMPLES:
sage: P = Parent()
sage: P.category()
Category of sets
sage: class MyParent(Parent):
....: def __init__(self):
....: self._init_category_(Groups())
sage: MyParent().category()
Category of groups
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