Complex reflection groups#
- class sage.categories.complex_reflection_groups.ComplexReflectionGroups(s=None)#
Bases:
Category_singletonThe category of complex reflection groups.
Let \(V\) be a complex vector space. A complex reflection is an element of \(\operatorname{GL}(V)\) fixing an hyperplane pointwise and acting by multiplication by a root of unity on a complementary line.
A complex reflection group is a group \(W\) that is (isomorphic to) a subgroup of some general linear group \(\operatorname{GL}(V)\) generated by a distinguished set of complex reflections.
The dimension of \(V\) is the rank of \(W\).
For a comprehensive treatment of complex reflection groups and many definitions and theorems used here, we refer to [LT2009]. See also Wikipedia article Reflection_group.
See also
ReflectionGroup()for usage examples of this category.EXAMPLES:
sage: from sage.categories.complex_reflection_groups import ComplexReflectionGroups sage: ComplexReflectionGroups() Category of complex reflection groups sage: ComplexReflectionGroups().super_categories() [Category of complex reflection or generalized coxeter groups] sage: ComplexReflectionGroups().all_super_categories() [Category of complex reflection groups, Category of complex reflection or generalized coxeter groups, Category of groups, Category of monoids, Category of finitely generated semigroups, Category of semigroups, Category of finitely generated magmas, Category of inverse unital magmas, Category of unital magmas, Category of magmas, Category of enumerated sets, Category of sets, Category of sets with partial maps, Category of objects]
An example of a reflection group:
sage: W = ComplexReflectionGroups().example(); W 5-colored permutations of size 3
Wis in the category of complex reflection groups:sage: W in ComplexReflectionGroups() True
- Finite#
alias of
FiniteComplexReflectionGroups
- class ParentMethods#
Bases:
object- rank()#
Return the rank of
self.The rank of
selfis the dimension of the smallest faithfull reflection representation ofself.EXAMPLES:
sage: W = CoxeterGroups().example(); W The symmetric group on {0, ..., 3} sage: W.rank() 3
- additional_structure()#
Return
None.Indeed, all the structure complex reflection groups have in addition to groups (simple reflections, …) is already defined in the super category.
See also
EXAMPLES:
sage: from sage.categories.complex_reflection_groups import ComplexReflectionGroups sage: ComplexReflectionGroups().additional_structure()
- example()#
Return an example of a complex reflection group.
EXAMPLES:
sage: from sage.categories.complex_reflection_groups import ComplexReflectionGroups sage: ComplexReflectionGroups().example() 5-colored permutations of size 3
- super_categories()#
Return the super categories of
self.EXAMPLES:
sage: from sage.categories.complex_reflection_groups import ComplexReflectionGroups sage: ComplexReflectionGroups().super_categories() [Category of complex reflection or generalized coxeter groups]