Examples of finite Weyl groups¶
-
sage.categories.examples.finite_weyl_groups.
Example
¶ An example of finite Weyl group: the symmetric group, with elements in list notation.
The purpose of this class is to provide a minimal template for implementing finite Weyl groups. See
SymmetricGroup
for a full featured and optimized implementation.EXAMPLES:
sage: S = FiniteWeylGroups().example() sage: S The symmetric group on {0, ..., 3} sage: S.category() Category of finite irreducible weyl groups
The elements of this group are permutations of the set \(\{0,\ldots,3\}\):
sage: S.one() (0, 1, 2, 3) sage: S.an_element() (1, 2, 3, 0)
The group itself is generated by the elementary transpositions:
sage: S.simple_reflections() Finite family {0: (1, 0, 2, 3), 1: (0, 2, 1, 3), 2: (0, 1, 3, 2)}
Only the following basic operations are implemented:
one()
product()
simple_reflection()
cartan_type()
Element.has_right_descent()
.
All the other usual Weyl group operations are inherited from the categories:
sage: S.cardinality() 24 sage: S.long_element() (3, 2, 1, 0) sage: S.cayley_graph(side = "left").plot() Graphics object consisting of 120 graphics primitives
Alternatively, one could have implemented
sage.categories.coxeter_groups.CoxeterGroups.ElementMethods.apply_simple_reflection()
instead ofsimple_reflection()
andproduct()
. SeeCoxeterGroups().example()
.
-
sage.categories.examples.finite_weyl_groups.
SymmetricGroup
¶ An example of finite Weyl group: the symmetric group, with elements in list notation.
The purpose of this class is to provide a minimal template for implementing finite Weyl groups. See
SymmetricGroup
for a full featured and optimized implementation.EXAMPLES:
sage: S = FiniteWeylGroups().example() sage: S The symmetric group on {0, ..., 3} sage: S.category() Category of finite irreducible weyl groups
The elements of this group are permutations of the set \(\{0,\ldots,3\}\):
sage: S.one() (0, 1, 2, 3) sage: S.an_element() (1, 2, 3, 0)
The group itself is generated by the elementary transpositions:
sage: S.simple_reflections() Finite family {0: (1, 0, 2, 3), 1: (0, 2, 1, 3), 2: (0, 1, 3, 2)}
Only the following basic operations are implemented:
one()
product()
simple_reflection()
cartan_type()
Element.has_right_descent()
.
All the other usual Weyl group operations are inherited from the categories:
sage: S.cardinality() 24 sage: S.long_element() (3, 2, 1, 0) sage: S.cayley_graph(side = "left").plot() Graphics object consisting of 120 graphics primitives
Alternatively, one could have implemented
sage.categories.coxeter_groups.CoxeterGroups.ElementMethods.apply_simple_reflection()
instead ofsimple_reflection()
andproduct()
. SeeCoxeterGroups().example()
.