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:

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 of simple_reflection() and product(). See CoxeterGroups().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:

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 of simple_reflection() and product(). See CoxeterGroups().example().