Classical Crystals

class sage.categories.classical_crystals.ClassicalCrystals(s=None)

Bases: sage.categories.category.Category

The category of classical crystals, that is crystals of finite Cartan type.

EXAMPLES:

sage: C = ClassicalCrystals()
sage: C
Category of classical crystals
sage: C.super_categories()
[Category of finite crystals, Category of highest weight crystals]
sage: C.example()
Highest weight crystal of type A_3 of highest weight omega_1

TESTS:

sage: TestSuite(C).run()
sage: B = FiniteCrystals().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 ParentMethods

cardinality()

Returns the number of elements of the crystal, using Weyl’s dimension formula on each connected component.

EXAMPLES:

sage: C = ClassicalCrystals().example(5)
sage: C.cardinality()
6

character(R)

Returns the character of the crystal.

INPUT:

• R – a WeylCharacterRing

EXAMPLES:

sage: C = CrystalOfLetters(['A',2])
sage: T = TensorProductOfCrystals(C, C)
sage: A2 = WeylCharacterRing(C.cartan_type()); A2
The Weyl Character Ring of Type ['A', 2] with Integer Ring coefficients
sage: chi = T.character(A2); chi
A2(1,1,0) + A2(2,0,0)
sage: chi.check(verbose = true)
[9, 9]

TODO: add default value for R

demazure_character(weight, reduced_word=False)

Returns the Demazure character associated to the specified weight in the ambient weight lattice.

INPUT:

• weight – an element of the weight lattice realization of the crystal, or a reduced word
• reduced_word – a boolean (default: False) whether weight is given as a reduced word

This is currently only supported for crystals whose underlying weight space is the ambient space.

EXAMPLES:

sage: T = CrystalOfTableaux(['A',2], shape = [2,1])
sage: e = T.weight_lattice_realization().basis()
sage: weight = e[0] + 2*e[2]
sage: weight.reduced_word()
[2, 1]
sage: T.demazure_character(weight)
x1^2*x2 + x1^2*x3 + x1*x2^2 + x1*x2*x3 + x1*x3^2

sage: T = CrystalOfTableaux(['A',3],shape=[2,1])
sage: T.demazure_character([1,2,3], reduced_word = True)
x1^2*x2 + x1^2*x3 + x1*x2^2 + x1*x2*x3 + x2^2*x3

sage: T = CrystalOfTableaux(['B',2], shape = [2])
sage: e = T.weight_lattice_realization().basis()
sage: weight = -2*e[1]
sage: T.demazure_character(weight)
x1^2 + x1*x2 + x2^2 + x1 + x2 + x1/x2 + 1/x2 + 1/x2^2 + 1

TODO: detect automatically if weight is a reduced word, and remove the (untested!) reduced_word option.

REFERENCES:

.. [D1974] M. Demazure, Desingularisation des varietes de Schubert,
   Ann. E. N. S., Vol. 6, (1974), p. 163-172

.. [M2009] Sarah Mason, An Explicit Construction of Type A Demazure Atoms,
   Journal of Algebraic Combinatorics, Vol. 29, (2009), No. 3, p.295-313
   (arXiv:0707.4267)

list()

Returns the list of the elements of self, as per FiniteEnumeratedSets.ParentMethods.list()

EXAMPLES:

sage: C = CrystalOfLetters(['D',4])
sage: C.list()
[1, 2, 3, 4, -4, -3, -2, -1]

FIXME: this is just there to reinstate the default implementation of list() from __iter__() which is overriden in Crystals.

ClassicalCrystals.example(n=3)

Returns an example of highest weight crystals, as per Category.example().

EXAMPLES:

sage: B = ClassicalCrystals().example(); B
Highest weight crystal of type A_3 of highest weight omega_1

ClassicalCrystals.super_categories(*args, **kwds)

EXAMPLES:

sage: ClassicalCrystals().super_categories()
[Category of finite crystals, Category of highest weight crystals]