Examples of finite Weyl groups¶
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sage.categories.examples.finite_weyl_groups.Example¶ alias of
sage.categories.examples.finite_weyl_groups.SymmetricGroup
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class
sage.categories.examples.finite_weyl_groups.SymmetricGroup(n=4)¶ Bases:
sage.structure.unique_representation.UniqueRepresentation,sage.structure.parent.ParentAn 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
SymmetricGroupfor 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
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)
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)}
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() 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¶ Bases:
sage.structure.element_wrapper.ElementWrapper-
has_right_descent(i)¶ 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()
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cartan_type()¶ Return the Cartan type of
self.EXAMPLES:
sage: FiniteWeylGroups().example().cartan_type() ['A', 3] relabelled by {1: 0, 2: 1, 3: 2}
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degrees()¶ Return the degrees of
self.EXAMPLES:
sage: W = FiniteWeylGroups().example() sage: W.degrees() (2, 3, 4)
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index_set()¶ Implements
CoxeterGroups.ParentMethods.index_set().EXAMPLES:
sage: FiniteWeylGroups().example().index_set() [0, 1, 2]
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one()¶ Implements
Monoids.ParentMethods.one().EXAMPLES:
sage: FiniteWeylGroups().example().one() (0, 1, 2, 3)
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product(x, y)¶ Implements
Semigroups.ParentMethods.product().EXAMPLES:
sage: s = FiniteWeylGroups().example().simple_reflections() sage: s[1] * s[2] (0, 2, 3, 1)
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simple_reflection(i)¶ Implement
CoxeterGroups.ParentMethods.simple_reflection()by returning the transposition \((i, i+1)\).EXAMPLES:
sage: FiniteWeylGroups().example().simple_reflection(2) (0, 1, 3, 2)
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