# Examples of finite Coxeter groups#

class sage.categories.examples.finite_coxeter_groups.DihedralGroup(n=5)[source]#

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()

>>> from sage.all import *
>>> 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

>>> from sage.all import *
>>> G.simple_reflections()
Finite family {1: (1,), 2: (2,)}

>>> s1, s2 = G.simple_reflections()
>>> (s1*s2)**Integer(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)]

>>> from sage.all import *
>>> G.an_element()
(1, 2)

>>> 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 chosen reduced word is $$(1,2,1,2\dots)$$:

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

>>> from sage.all import *
>>> G.long_element()
(1, 2, 1, 2, 1)

class Element[source]#

Bases: ElementWrapper

apply_simple_reflection_right(i)[source]#

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)]

>>> from sage.all import *
>>> D5 = FiniteCoxeterGroups().example(Integer(5))
>>> [i**Integer(2) for i in D5]  # indirect doctest
[(), (), (), (1, 2, 1, 2), (2, 1, 2, 1), (), (), (2, 1), (1, 2), ()]
>>> [i**Integer(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')[source]#

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

>>> from sage.all import *
>>> D6 = FiniteCoxeterGroups().example(Integer(6))
>>> s = D6.simple_reflections()
>>> s[Integer(1)].has_descent(Integer(1))
True
>>> s[Integer(1)].has_descent(Integer(1))
True
>>> s[Integer(1)].has_descent(Integer(2))
False
>>> D6.one().has_descent(Integer(1))
False
>>> D6.one().has_descent(Integer(2))
False
>>> D6.long_element().has_descent(Integer(1))
True
>>> D6.long_element().has_descent(Integer(2))
True

wrapped_class#

alias of tuple

coxeter_matrix()[source]#

Return the Coxeter matrix of self.

EXAMPLES:

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

>>> from sage.all import *
>>> FiniteCoxeterGroups().example(Integer(6)).coxeter_matrix()
[1 6]
[6 1]

degrees()[source]#

Return the degrees of self.

EXAMPLES:

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

>>> from sage.all import *
>>> FiniteCoxeterGroups().example(Integer(6)).degrees()
(2, 6)

index_set()[source]#

EXAMPLES:

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

>>> from sage.all import *
>>> D4 = FiniteCoxeterGroups().example(Integer(4))
>>> D4.index_set()
(1, 2)

one()[source]#

Implements Monoids.ParentMethods.one().

EXAMPLES:

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

>>> from sage.all import *
>>> D6 = FiniteCoxeterGroups().example(Integer(6))
>>> D6.one()
()

sage.categories.examples.finite_coxeter_groups.Example[source]#

alias of DihedralGroup