Examples of finite Weyl groups#
- sage.categories.examples.finite_weyl_groups.Example[source]#
alias of
SymmetricGroup
- class sage.categories.examples.finite_weyl_groups.SymmetricGroup(n=4)[source]#
Bases:
UniqueRepresentation
,Parent
An example of finite Weyl group: the symmetric group, with elements in list notation.
The purpose of this class is to provide a minimal template for implementing finite Weyl groups. See
SymmetricGroup
for a full featured and optimized implementation.EXAMPLES:
sage: S = FiniteWeylGroups().example() sage: S The symmetric group on {0, ..., 3} sage: S.category() Category of finite irreducible Weyl groups
>>> from sage.all import * >>> S = FiniteWeylGroups().example() >>> S The symmetric group on {0, ..., 3} >>> S.category() Category of finite irreducible Weyl groups
The elements of this group are permutations of the set \(\{0,\ldots,3\}\):
sage: S.one() (0, 1, 2, 3) sage: S.an_element() (1, 2, 3, 0)
>>> from sage.all import * >>> S.one() (0, 1, 2, 3) >>> S.an_element() (1, 2, 3, 0)
The group itself is generated by the elementary transpositions:
sage: S.simple_reflections() Finite family {0: (1, 0, 2, 3), 1: (0, 2, 1, 3), 2: (0, 1, 3, 2)}
>>> from sage.all import * >>> S.simple_reflections() Finite family {0: (1, 0, 2, 3), 1: (0, 2, 1, 3), 2: (0, 1, 3, 2)}
Only the following basic operations are implemented:
All the other usual Weyl group operations are inherited from the categories:
sage: S.cardinality() 24 sage: S.long_element() (3, 2, 1, 0) sage: S.cayley_graph(side="left").plot() # needs sage.graphs sage.plot Graphics object consisting of 120 graphics primitives
>>> from sage.all import * >>> S.cardinality() 24 >>> S.long_element() (3, 2, 1, 0) >>> S.cayley_graph(side="left").plot() # needs sage.graphs sage.plot Graphics object consisting of 120 graphics primitives
Alternatively, one could have implemented
sage.categories.coxeter_groups.CoxeterGroups.ElementMethods.apply_simple_reflection()
instead ofsimple_reflection()
andproduct()
. SeeCoxeterGroups().example()
.- class Element[source]#
Bases:
ElementWrapper
- has_right_descent(i)[source]#
Implements
CoxeterGroups.ElementMethods.has_right_descent()
.EXAMPLES:
sage: S = FiniteWeylGroups().example() sage: s = S.simple_reflections() sage: (s[1] * s[2]).has_descent(2) True sage: S._test_has_descent()
>>> from sage.all import * >>> S = FiniteWeylGroups().example() >>> s = S.simple_reflections() >>> (s[Integer(1)] * s[Integer(2)]).has_descent(Integer(2)) True >>> S._test_has_descent()
- cartan_type()[source]#
Return the Cartan type of
self
.EXAMPLES:
sage: FiniteWeylGroups().example().cartan_type() # needs sage.modules ['A', 3] relabelled by {1: 0, 2: 1, 3: 2}
>>> from sage.all import * >>> FiniteWeylGroups().example().cartan_type() # needs sage.modules ['A', 3] relabelled by {1: 0, 2: 1, 3: 2}
- degrees()[source]#
Return the degrees of
self
.EXAMPLES:
sage: W = FiniteWeylGroups().example() sage: W.degrees() (2, 3, 4)
>>> from sage.all import * >>> W = FiniteWeylGroups().example() >>> W.degrees() (2, 3, 4)
- index_set()[source]#
Implements
CoxeterGroups.ParentMethods.index_set()
.EXAMPLES:
sage: FiniteWeylGroups().example().index_set() [0, 1, 2]
>>> from sage.all import * >>> FiniteWeylGroups().example().index_set() [0, 1, 2]
- one()[source]#
Implements
Monoids.ParentMethods.one()
.EXAMPLES:
sage: FiniteWeylGroups().example().one() (0, 1, 2, 3)
>>> from sage.all import * >>> FiniteWeylGroups().example().one() (0, 1, 2, 3)
- product(x, y)[source]#
Implements
Semigroups.ParentMethods.product()
.EXAMPLES:
sage: s = FiniteWeylGroups().example().simple_reflections() sage: s[1] * s[2] (0, 2, 3, 1)
>>> from sage.all import * >>> s = FiniteWeylGroups().example().simple_reflections() >>> s[Integer(1)] * s[Integer(2)] (0, 2, 3, 1)
- simple_reflection(i)[source]#
Implement
CoxeterGroups.ParentMethods.simple_reflection()
by returning the transposition \((i, i+1)\).EXAMPLES:
sage: FiniteWeylGroups().example().simple_reflection(2) (0, 1, 3, 2)
>>> from sage.all import * >>> FiniteWeylGroups().example().simple_reflection(Integer(2)) (0, 1, 3, 2)