Examples of finite Coxeter groups

sage.categories.examples.finite_coxeter_groups.DihedralGroup

An example of finite Coxeter group: the \(n\)-th dihedral group of order \(2n\).

The purpose of this class is to provide a minimal template for implementing finite Coxeter groups. See DihedralGroup for a full featured and optimized implementation.

EXAMPLES:

sage: G = FiniteCoxeterGroups().example()

This group is generated by two simple reflections \(s_1\) and \(s_2\) subject to the relation \((s_1s_2)^n = 1\):

sage: G.simple_reflections()
Finite family {1: (1,), 2: (2,)}

sage: s1, s2 = G.simple_reflections()
sage: (s1*s2)^5 == G.one()
True

An element is represented by its reduced word (a tuple of elements of \(self.index_set()\)):

sage: G.an_element()
(1, 2)

sage: list(G)
[(),
 (1,),
 (2,),
 (1, 2),
 (2, 1),
 (1, 2, 1),
 (2, 1, 2),
 (1, 2, 1, 2),
 (2, 1, 2, 1),
 (1, 2, 1, 2, 1)]

This reduced word is unique, except for the longest element where the choosen reduced word is \((1,2,1,2\dots)\):

sage: G.long_element()
(1, 2, 1, 2, 1)
sage.categories.examples.finite_coxeter_groups.Example

An example of finite Coxeter group: the \(n\)-th dihedral group of order \(2n\).

The purpose of this class is to provide a minimal template for implementing finite Coxeter groups. See DihedralGroup for a full featured and optimized implementation.

EXAMPLES:

sage: G = FiniteCoxeterGroups().example()

This group is generated by two simple reflections \(s_1\) and \(s_2\) subject to the relation \((s_1s_2)^n = 1\):

sage: G.simple_reflections()
Finite family {1: (1,), 2: (2,)}

sage: s1, s2 = G.simple_reflections()
sage: (s1*s2)^5 == G.one()
True

An element is represented by its reduced word (a tuple of elements of \(self.index_set()\)):

sage: G.an_element()
(1, 2)

sage: list(G)
[(),
 (1,),
 (2,),
 (1, 2),
 (2, 1),
 (1, 2, 1),
 (2, 1, 2),
 (1, 2, 1, 2),
 (2, 1, 2, 1),
 (1, 2, 1, 2, 1)]

This reduced word is unique, except for the longest element where the choosen reduced word is \((1,2,1,2\dots)\):

sage: G.long_element()
(1, 2, 1, 2, 1)