Crystals¶

class sage.categories.crystals.Crystals(s=None)¶

Bases: sage.categories.category.Category

The category of crystals

See sage.combinat.crystals for an introduction to crystals.

EXAMPLES:

sage: C = Crystals()
sage: C
Category of crystals
sage: C.super_categories()
[Category of... enumerated sets]
sage: C.example()
Highest weight crystal of type A_3 of highest weight omega_1

Parents in this category should implement the following methods:

either a method cartan_type or an attribute _cartan_type

module_generators: a list (or container) of distinct elements which generate the crystal using $f_i$

Furthermore, their elements should implement the following methods:

x.e(i) (returning $e_i(x)$ )

x.f(i) (returning $f_i(x)$ )

EXAMPLES:

sage: from sage.misc.abstract_method import abstract_methods_of_class
sage: abstract_methods_of_class(Crystals().element_class)
{'required': ['e', 'f'], 'optional': []}

TESTS:

sage: TestSuite(C).run()
sage: B = Crystals().example()
sage: TestSuite(B).run(verbose = True)
running ._test_an_element() . . . pass
running ._test_category() . . . pass
running ._test_elements() . . .
  Running the test suite of self.an_element()
  running ._test_category() . . . pass
  running ._test_eq() . . . pass
  running ._test_not_implemented_methods() . . . pass
  running ._test_pickling() . . . pass
  pass
running ._test_elements_eq() . . . pass
running ._test_enumerated_set_contains() . . . pass
running ._test_enumerated_set_iter_cardinality() . . . pass
running ._test_enumerated_set_iter_list() . . . pass
running ._test_eq() . . . pass
running ._test_fast_iter() . . . pass
running ._test_not_implemented_methods() . . . pass
running ._test_pickling() . . . pass
running ._test_some_elements() . . . pass

class ElementMethods¶

Epsilon()¶

EXAMPLES:

sage: C = CrystalOfLetters(['A',5])
sage: C(0).Epsilon()
(0, 0, 0, 0, 0, 0)
sage: C(1).Epsilon()
(0, 0, 0, 0, 0, 0)
sage: C(2).Epsilon()
(1, 0, 0, 0, 0, 0)

Phi()¶

EXAMPLES:

sage: C = CrystalOfLetters(['A',5])
sage: C(0).Phi()
(0, 0, 0, 0, 0, 0)
sage: C(1).Phi()
(1, 0, 0, 0, 0, 0)
sage: C(2).Phi()
(1, 1, 0, 0, 0, 0)

cartan_type()¶

Returns the cartan type associated to self

EXAMPLES:

sage: C = CrystalOfLetters(['A', 5])
sage: C(1).cartan_type()
['A', 5]

demazure_operator(i, truncated=False)¶

Returns the list of the elements one can obtain from self by application of $f_i$ . If the option “truncated” is set to True, then self is not included in the list.

REFERENCES:

[L1995] Peter Littelmann, Crystal graphs and Young tableaux, J. Algebra 175 (1995), no. 1, 65–87.

[K1993] Masaki Kashiwara, The crystal base and Littelmann’s refined Demazure character formula, Duke Math. J. 71 (1993), no. 3, 839–858.

EXAMPLES:

sage: T = CrystalOfTableaux(['A',2], shape=[2,1])
sage: t=T(rows=[[1,2],[2]])
sage: t.demazure_operator(2)
[[[1, 2], [2]], [[1, 3], [2]], [[1, 3], [3]]]
sage: t.demazure_operator(2, truncated = True)
[[[1, 3], [2]], [[1, 3], [3]]]
sage: t.demazure_operator(1, truncated = True)
[]
sage: t.demazure_operator(1)
[[[1, 2], [2]]]

sage: K = KirillovReshetikhinCrystal(['A',2,1],2,1)
sage: t = K(rows=[[3],[2]])
sage: t.demazure_operator(0)
[[[2, 3]], [[1, 2]]]

e(i)¶

Returns $e_i(x)$ if it exists or None otherwise.

This method should be implemented by the element class of the crystal.

EXAMPLES:

sage: C = Crystals().example(5)
sage: x = C[2]; x
3
sage: x.e(1), x.e(2), x.e(3)
(None, 2, None)

e_string(list)¶

Applies $e_{i_r} ... e_{i_1}$ to self for $list = [i_1, ..., i_r]$

EXAMPLES:

sage: C = CrystalOfLetters(['A',3])
sage: b = C(3)
sage: b.e_string([2,1])
1
sage: b.e_string([1,2])

epsilon(i)¶

EXAMPLES:

sage: C = CrystalOfLetters(['A',5])
sage: C(1).epsilon(1)
0
sage: C(2).epsilon(1)
1

f(i)¶

Returns $f_i(x)$ if it exists or None otherwise.

This method should be implemented by the element class of the crystal.

EXAMPLES:

sage: C = Crystals().example(5)
sage: x = C[1]; x
2
sage: x.f(1), x.f(2), x.f(3)
(None, 3, None)

f_string(list)¶

Applies $f_{i_r} ... f_{i_1}$ to self for $list = [i_1, ..., i_r]$

EXAMPLES:

sage: C = CrystalOfLetters(['A',3])
sage: b = C(1)
sage: b.f_string([1,2])
3
sage: b.f_string([2,1])

index_set()¶

EXAMPLES:

sage: C = CrystalOfLetters(['A',5])
sage: C(1).index_set()
[1, 2, 3, 4, 5]

is_highest_weight(index_set=None)¶

Returns True if self is a highest weight. Specifying the option index_set to be a subset $I$ of the index set of the underlying crystal, finds all highest weight vectors for arrows in $I$ .

EXAMPLES:

sage: C = CrystalOfLetters(['A',5])
sage: C(1).is_highest_weight()
True
sage: C(2).is_highest_weight()
False
sage: C(2).is_highest_weight(index_set = [2,3,4,5])
True

phi(i)¶

EXAMPLES:

sage: C = CrystalOfLetters(['A',5])
sage: C(1).phi(1)
1
sage: C(2).phi(1)
0

phi_minus_epsilon(i)¶

Returns $\phi_i - \epsilon_i$ of self. There are sometimes better implementations using the weight for this. It is used for reflections along a string.

EXAMPLES:

sage: C = CrystalOfLetters(['A',5])
sage: C(1).phi_minus_epsilon(1)
1

s(i)¶

Returns the reflection of self along its $i$ -string

EXAMPLES:

sage: C = CrystalOfTableaux(['A',2], shape=[2,1])
sage: b=C(rows=[[1,1],[3]])
sage: b.s(1)
[[2, 2], [3]]
sage: b=C(rows=[[1,2],[3]])
sage: b.s(2)
[[1, 2], [3]]
sage: T=CrystalOfTableaux(['A',2],shape=[4])
sage: t=T(rows=[[1,2,2,2]])
sage: t.s(1)
[[1, 1, 1, 2]]

weight()¶

Returns the weight of this crystal element

EXAMPLES:

sage: C = CrystalOfLetters(['A',5])
sage: C(1).weight()
(1, 0, 0, 0, 0, 0)

class Crystals.ParentMethods¶

Lambda()¶

Returns the fundamentals weights in the weight lattice realization for the root system associated the crystal

EXAMPLES:

sage: C = CrystalOfLetters(['A', 5])
sage: C.Lambda()
Finite family {1: (1, 0, 0, 0, 0, 0), 2: (1, 1, 0, 0, 0, 0), 3: (1, 1, 1, 0, 0, 0), 4: (1, 1, 1, 1, 0, 0), 5: (1, 1, 1, 1, 1, 0)}

cartan_type()¶

Returns the Cartan type of the crystal

EXAMPLES::: sage: C = CrystalOfLetters([‘A’,2]) sage: C.cartan_type() [‘A’, 2]

crystal_morphism(g, index_set=None, automorphism=<function <lambda> at 0x228de60>, direction='down', direction_image='down', similarity_factor=None, similarity_factor_domain=None, cached=False, acyclic=True)¶

Constructs a morphism from the crystal self to another crystal. The input $g$ can either be a function of a (sub)set of elements of self to element in another crystal or a dictionary between certain elements. Usually one would map highest weight elements or crystal generators to each other using g. Specifying index_set gives the opportunity to define the morphism as $I$ -crystals where $I =$ index_set. If index_set is not specified, the index set of self is used. It is also possible to define twisted-morphisms by specifying an automorphism on the nodes in te Dynkin diagram (or the index_set). The option direction and direction_image indicate whether to use $f_i$ or $e_i$ in self or the image crystal to construct the morphism, depending on whether the direction is set to ‘down’ or ‘up’. It is also possible to set a similarity_factor. This should be a dictionary between the elements in the index set and positive integers. The crystal operator $f_i$ then gets mapped to $f_i^{m_i}$ where $m_i =$ similarity_factor[i]. Setting similarity_factor_domain to a dictionary between the index set and positive integers has the effect that $f_i^{m_i}$ gets mapped to $f_i$ where $m_i =$ similarity_factor_domain[i]. Finally, it is possible to set the option $acyclic = False$ . This calculates an isomorphism for cyclic crystals (for example finite affine crystals). In this case the input function $g$ is supposed to be given as a dictionary.

EXAMPLES:

sage: C2 = CrystalOfLetters(['A',2])
sage: C3 = CrystalOfLetters(['A',3])
sage: g = {C2.module_generators[0] : C3.module_generators[0]}
sage: g_full = C2.crystal_morphism(g)
sage: g_full(C2(1))
1
sage: g_full(C2(2))
2
sage: g = {C2(1) : C2(3)}
sage: g_full = C2.crystal_morphism(g, automorphism = lambda i : 3-i, direction_image = 'up')
sage: [g_full(b) for b in C2]
[3, 2, 1]
sage: T = CrystalOfTableaux(['A',2], shape = [2])
sage: g = {C2(1) : T(rows=[[1,1]])}
sage: g_full = C2.crystal_morphism(g, similarity_factor = {1:2, 2:2})
sage: [g_full(b) for b in C2]
[[[1, 1]], [[2, 2]], [[3, 3]]]
sage: g = {T(rows=[[1,1]]) : C2(1)}
sage: g_full = T.crystal_morphism(g, similarity_factor_domain = {1:2, 2:2})
sage: g_full(T(rows=[[2,2]]))
2

sage: B1=KirillovReshetikhinCrystal(['A',2,1],1,1)
sage: B2=KirillovReshetikhinCrystal(['A',2,1],1,2)
sage: T=TensorProductOfCrystals(B1,B2)
sage: T1=TensorProductOfCrystals(B2,B1)
sage: La = T.weight_lattice_realization().fundamental_weights()
sage: t = [b for b in T if b.weight() == -3*La[0] + 3*La[1]][0]
sage: t1 = [b for b in T1 if b.weight() == -3*La[0] + 3*La[1]][0]
sage: g={t:t1}
sage: f=T.crystal_morphism(g,acyclic = False)
sage: [[b,f(b)] for b in T]
[[[[[1]], [[1, 1]]], [[[1, 1]], [[1]]]],
[[[[1]], [[1, 2]]], [[[1, 1]], [[2]]]],
[[[[1]], [[2, 2]]], [[[1, 2]], [[2]]]],
[[[[1]], [[1, 3]]], [[[1, 1]], [[3]]]],
[[[[1]], [[2, 3]]], [[[1, 2]], [[3]]]],
[[[[1]], [[3, 3]]], [[[1, 3]], [[3]]]],
[[[[2]], [[1, 1]]], [[[1, 2]], [[1]]]],
[[[[2]], [[1, 2]]], [[[2, 2]], [[1]]]],
[[[[2]], [[2, 2]]], [[[2, 2]], [[2]]]],
[[[[2]], [[1, 3]]], [[[2, 3]], [[1]]]],
[[[[2]], [[2, 3]]], [[[2, 2]], [[3]]]],
[[[[2]], [[3, 3]]], [[[2, 3]], [[3]]]],
[[[[3]], [[1, 1]]], [[[1, 3]], [[1]]]],
[[[[3]], [[1, 2]]], [[[1, 3]], [[2]]]],
[[[[3]], [[2, 2]]], [[[2, 3]], [[2]]]],
[[[[3]], [[1, 3]]], [[[3, 3]], [[1]]]],
[[[[3]], [[2, 3]]], [[[3, 3]], [[2]]]],
[[[[3]], [[3, 3]]], [[[3, 3]], [[3]]]]]

digraph()¶

Returns the DiGraph associated to self.

EXAMPLES:

sage: C = Crystals().example(5)
sage: C.digraph()
Digraph on 6 vertices

TODO: add more tests

dot_tex()¶

Returns a dot_tex string representation of self.

EXAMPLES:

sage: C = CrystalOfLetters(['A',2])
sage: C.dot_tex()
'digraph G { \n  node [ shape=plaintext ];\n  N_0 [ label = " ", texlbl = "$1$" ];\n  N_1 [ label = " ", texlbl = "$2$" ];\n  N_2 [ label = " ", texlbl = "$3$" ];\n  N_0 -> N_1 [ label = " ", texlbl = "1" ];\n  N_1 -> N_2 [ label = " ", texlbl = "2" ];\n}'

index_set()¶

Returns the index set of the Dynkin diagram underlying the crystal

EXAMPLES:: sage: C = CrystalOfLetters([‘A’, 5]) sage: C.index_set() [1, 2, 3, 4, 5]

latex()¶

Returns the crystal graph as a latex string. This can be exported to a file with self.latex_file(‘filename’).

See Graph.layout_graphviz() for further documentation on which packages are required.

EXAMPLES:

sage: C = CrystalOfLetters(['A', 5])
sage: C.latex()         #optional requires dot2tex
...
sage: view(C, pdflatex = True, tightpage = True) # optional

latex_file(filename)¶

Exports a file, suitable for pdflatex, to ‘filename’. This requires a proper installation of dot2tex in sage-python. For more information see the documentation for self.latex().

EXAMPLES:

sage: C = CrystalOfLetters(['A', 5])
sage: C.latex_file('/tmp/test.tex') #optional requires dot2tex

metapost(filename, thicklines=False, labels=True, scaling_factor=1.0, tallness=1.0)¶

Use C.metapost(“filename.mp”,[options]), where options can be:

thicklines = True (for thicker edges) labels = False (to suppress labeling of the vertices) scaling_factor=value, where value is a floating point number, 1.0 by default. Increasing or decreasing the scaling factor changes the size of the image. tallness=1.0. Increasing makes the image taller without increasing the width.

Root operators e(1) or f(1) move along red lines, e(2) or f(2) along green. The highest weight is in the lower left. Vertices with the same weight are kept close together. The concise labels on the nodes are strings introduced by Berenstein and Zelevinsky and Littelmann; see Littelmann’s paper Cones, Crystals, Patterns, sections 5 and 6.

For Cartan types B2 or C2, the pattern has the form

a2 a3 a4 a1

where c*a2 = a3 = 2*a4 =0 and a1=0, with c=2 for B2, c=1 for C2. Applying e(2) a1 times, e(1) a2 times, e(2) a3 times, e(1) a4 times returns to the highest weight. (Observe that Littelmann writes the roots in opposite of the usual order, so our e(1) is his e(2) for these Cartan types.) For type A2, the pattern has the form

a3 a2 a1

where applying e(1) a1 times, e(2) a2 times then e(3) a1 times returns to the highest weight. These data determine the vertex and may be translated into a Gelfand-Tsetlin pattern or tableau.

EXAMPLES:

sage: C = CrystalOfLetters(['A', 2])
sage: C.metapost('/tmp/test.mp') #optional

sage: C = CrystalOfLetters(['A', 5])
sage: C.metapost('/tmp/test.mp')
...
NotImplementedError

plot(**options)¶

Returns the plot of self as a directed graph.

EXAMPLES:

sage: C = CrystalOfLetters(['A', 5])
sage: show_default(False) #do not show the plot by default
sage: C.plot()
Graphics object consisting of 17 graphics primitives

weight_lattice_realization()¶

Returns the weight lattice realization for the root system associated to self. This default implementation uses the ambient space of the root system.

EXAMPLES:

sage: C = CrystalOfLetters(['A', 5])
sage: C.weight_lattice_realization()
Ambient space of the Root system of type ['A', 5]
sage: K = KirillovReshetikhinCrystal(['A',2,1], 1, 1)
sage: K.weight_lattice_realization()
Weight lattice of the Root system of type ['A', 2, 1]

Crystals.super_categories(*args, **kwds)¶

EXAMPLES:

sage: Crystals().super_categories()
[Category of enumerated sets]

Crystals¶

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