Example of a crystal#

class sage.categories.examples.crystals.HighestWeightCrystalOfTypeA(n=3)[source]#

Bases: UniqueRepresentation, Parent

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

>>> from sage.all import *
>>> C = Crystals().example()
>>> C
Highest weight crystal of type A_3 of highest weight omega_1
>>> 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

>>> from sage.all import *
>>> C.list()
[1, 2, 3,  4]
>>> C.module_generators[Integer(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

>>> from sage.all import *
>>> b = C.module_generators[Integer(0)]
>>> b.f(Integer(1))
2
>>> b.f(Integer(1)).e(Integer(1)) == b
True

Only the following basic operations are implemented:

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

sage: C.cardinality()
4

>>> from sage.all import *
>>> C.cardinality()
4
class Element[source]#

Bases: ElementWrapper

e(i)[source]#

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]]

>>> from sage.all import *
>>> C = Crystals().example(Integer(4))
>>> [[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)[source]#

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]]

>>> from sage.all import *
>>> C = Crystals().example(Integer(4))
>>> [[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[source]#

Bases: UniqueRepresentation, Parent

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

>>> from sage.all import *
>>> C = Crystals().example(choice='naive')
>>> C.highest_weight_vector()
0
class Element[source]#

Bases: ElementWrapper

e(i)[source]#

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]]

>>> from sage.all import *
>>> C = Crystals().example(choice='naive')
>>> [[c,i,c.e(i)] for i in C.index_set() for c in [C(j) for j in (ellipsis_range(Integer(0),Ellipsis,Integer(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)[source]#

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]]

>>> from sage.all import *
>>> C = Crystals().example(choice='naive')
>>> [[c,i,c.f(i)] for i in C.index_set() for c in [C(j) for j in (ellipsis_range(Integer(0),Ellipsis,Integer(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]]