# Examples of finite Coxeter groups¶

class sage.categories.examples.finite_coxeter_groups.DihedralGroup(n=5)

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)

class Element
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)]

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

coxeter_matrix()

Return the Coxeter matrix of self.

EXAMPLES:

sage: FiniteCoxeterGroups().example(6).coxeter_matrix()
[1 6]
[6 1]

degrees()

Return the degrees of self.

EXAMPLES:

sage: FiniteCoxeterGroups().example(6).degrees()
(2, 6)

index_set()

EXAMPLES:

sage: D4 = FiniteCoxeterGroups().example(4)
sage: D4.index_set()
(1, 2)

one()

Implements Monoids.ParentMethods.one().

EXAMPLES:

sage: D6 = FiniteCoxeterGroups().example(6)
sage: D6.one()
()

sage.categories.examples.finite_coxeter_groups.Example