Examples of finite Coxeter groups¶
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class
sage.categories.examples.finite_coxeter_groups.
DihedralGroup
(n=5)¶ Bases:
sage.structure.unique_representation.UniqueRepresentation
,sage.structure.parent.Parent
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)
TESTS:
sage: TestSuite(G).run() sage: c = FiniteCoxeterGroups().example(3).cayley_graph() sage: sorted(c.edges()) [((), (1,), 1), ((), (2,), 2), ((1,), (), 1), ((1,), (1, 2), 2), ((1, 2), (1,), 2), ((1, 2), (1, 2, 1), 1), ((1, 2, 1), (1, 2), 1), ((1, 2, 1), (2, 1), 2), ((2,), (), 2), ((2,), (2, 1), 1), ((2, 1), (1, 2, 1), 2), ((2, 1), (2,), 1)]
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class
Element
¶ Bases:
sage.structure.element_wrapper.ElementWrapper
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apply_simple_reflection_right
(i)¶ Implements
CoxeterGroups.ElementMethods.apply_simple_reflection()
.EXAMPLES:
sage: D5 = FiniteCoxeterGroups().example(5) sage: [i^2 for i in D5] # indirect doctest [(), (), (), (1, 2, 1, 2), (2, 1, 2, 1), (), (), (2, 1), (1, 2), ()] sage: [i^5 for i in D5] # indirect doctest [(), (1,), (2,), (), (), (1, 2, 1), (2, 1, 2), (), (), (1, 2, 1, 2, 1)]
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has_right_descent
(i, positive=False, side='right')¶ Implements
SemiGroups.ElementMethods.has_right_descent()
.EXAMPLES:
sage: D6 = FiniteCoxeterGroups().example(6) sage: s = D6.simple_reflections() sage: s[1].has_descent(1) True sage: s[1].has_descent(1) True sage: s[1].has_descent(2) False sage: D6.one().has_descent(1) False sage: D6.one().has_descent(2) False sage: D6.long_element().has_descent(1) True sage: D6.long_element().has_descent(2) True
TESTS:
sage: D6._test_has_descent()
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DihedralGroup.
degrees
()¶ Return the degrees of
self
.EXAMPLES:
sage: FiniteCoxeterGroups().example(6).degrees() (2, 6)
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DihedralGroup.
index_set
()¶ Implements
CoxeterGroups.ParentMethods.index_set()
.EXAMPLES:
sage: D4 = FiniteCoxeterGroups().example(4) sage: D4.index_set() (1, 2)
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DihedralGroup.
one
()¶ Implements
Monoids.ParentMethods.one()
.EXAMPLES:
sage: D6 = FiniteCoxeterGroups().example(6) sage: D6.one() ()
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class
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sage.categories.examples.finite_coxeter_groups.
Example
¶ alias of
DihedralGroup