# Example of a crystal¶

class sage.categories.examples.crystals.HighestWeightCrystalOfTypeA(n=3)

An example of a crystal: the highest weight crystal of type $$A_n$$ of highest weight $$\omega_1$$.

The purpose of this class is to provide a minimal template for implementing crystals. See CrystalOfLetters for a full featured and optimized implementation.

EXAMPLES:

sage: C = Crystals().example()
sage: C
Highest weight crystal of type A_3 of highest weight omega_1
sage: C.category()
Category of classical crystals


The elements of this crystal are in the set $$\{1,\ldots,n+1\}$$:

sage: C.list()
[1, 2, 3,  4]
sage: C.module_generators[0]
1


The crystal operators themselves correspond to the elementary transpositions:

sage: b = C.module_generators[0]
sage: b.f(1)
2
sage: b.f(1).e(1) == b
True


TESTS:

sage: C = Crystals().example()
sage: TestSuite(C).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() . . . 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

Only the following basic operations are implemented:

All the other usual crystal operations are inherited from the categories; for example:

sage: C.cardinality()
4

class Element(value, parent)

EXAMPLES:

sage: from sage.structure.element_wrapper import DummyParent
sage: a = ElementWrapper(1, parent = DummyParent("A parent"))


TESTS:

sage: TestSuite(a).run(skip = "_test_category")


Note: ElementWrapper is not intended to be used directly, hence the failing category test.

e(i)

Returns the action of $$e_i$$ on self.

EXAMPLES:

sage: C = Crystals().example(4)
sage: [[c,i,c.e(i)] for i in C.index_set() for c in C if c.e(i) is not None]
[[2, 1, 1], [3, 2, 2], [4, 3, 3], [5, 4, 4]]

f(i)

Returns the action of $$f_i$$ on self.

EXAMPLES:

sage: C = Crystals().example(4)
sage: [[c,i,c.f(i)] for i in C.index_set() for c in C if c.f(i) is not None]
[[1, 1, 2], [2, 2, 3], [3, 3, 4], [4, 4, 5]]

class sage.categories.examples.crystals.NaiveCrystal

This is an example of a “crystal” which does not come from any kind of representation, designed primarily to test the Stembridge local rules with. The crystal has vertices labeled 0 through 5, with 0 the highest weight.

The code here could also possibly be generalized to create a class that automatically builds a crystal from an edge-colored digraph, if someone feels adventurous.

Currently, only the methods highest_weight_vector(), e(), and f() are guaranteed to work.

EXAMPLES:

sage: C = Crystals().example(choice='naive')
sage: C.highest_weight_vector()
0

class Element(value, parent)

EXAMPLES:

sage: from sage.structure.element_wrapper import DummyParent
sage: a = ElementWrapper(1, parent = DummyParent("A parent"))


TESTS:

sage: TestSuite(a).run(skip = "_test_category")


Note: ElementWrapper is not intended to be used directly, hence the failing category test.

e(i)

Returns the action of $$e_i$$ on self.

EXAMPLES:

sage: C = Crystals().example(choice='naive')
sage: [[c,i,c.e(i)] for i in C.index_set() for c in [C(j) for j in [0..5]] if c.e(i) is not None]
[[1, 1, 0], [2, 1, 1], [3, 1, 2], [5, 1, 3], [4, 2, 0], [5, 2, 4]]

f(i)

Returns the action of $$f_i$$ on self.

EXAMPLES:

sage: C = Crystals().example(choice='naive')
sage: [[c,i,c.f(i)] for i in C.index_set() for c in [C(j) for j in [0..5]] if c.f(i) is not None]
[[0, 1, 1], [1, 1, 2], [2, 1, 3], [3, 1, 5], [0, 2, 4], [4, 2, 5]]


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