Highest Weight Crystals

class sage.categories.highest_weight_crystals.HighestWeightCrystals(s=None)

Bases: sage.categories.category_singleton.Category_singleton

The category of highest weight crystals.

A crystal is highest weight if it is acyclic; in particular, every connected component has a unique highest weight element, and that element generate the component.

EXAMPLES:

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

TESTS:

sage: TestSuite(C).run()
sage: B = HighestWeightCrystals().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
  running ._test_stembridge_local_axioms() . . . pass
  pass
running ._test_elements_eq_reflexive() . . . pass
running ._test_elements_eq_symmetric() . . . pass
running ._test_elements_eq_transitive() . . . pass
running ._test_elements_neq() . . . 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
running ._test_stembridge_local_axioms() . . . pass
class ElementMethods
class HighestWeightCrystals.ParentMethods
highest_weight_vector()

Returns the highest weight vector if there is a single one; otherwise, raises an error.

Caveat: this assumes that highest_weight_vectors() returns a list or tuple.

EXAMPLES:

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

Returns the highest weight vectors of self

This default implementation selects among the module generators those that are highest weight, and caches the result. A crystal element \(b\) is highest weight if \(e_i(b)=0\) for all \(i\) in the index set.

EXAMPLES:

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

Returns the lowest weight vectors of self

This default implementation selects among all elements of the crystal those that are lowest weight, and cache the result. A crystal element \(b\) is lowest weight if \(f_i(b)=0\) for all \(i\) in the index set.

EXAMPLES:

sage: C = CrystalOfLetters(['A',5])
sage: C.lowest_weight_vectors()
[6]
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.lowest_weight_vectors()
[[3, 2, 3], [3, 3, 2]]
HighestWeightCrystals.example()

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

EXAMPLES:

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

EXAMPLES:

sage: HighestWeightCrystals().super_categories()
[Category of crystals]

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