Examples of finite Coxeter groups¶
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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)
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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)