Complex reflection groups

class sage.categories.complex_reflection_groups.ComplexReflectionGroups

Bases: Category_singleton

The category of complex reflection groups.

Let \(V\) be a complex vector space. A complex reflection is an element of \(\operatorname{GL}(V)\) fixing a 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

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]

Python

>>> from sage.all import *
>>> from sage.categories.complex_reflection_groups import ComplexReflectionGroups
>>> ComplexReflectionGroups()
Category of complex reflection groups
>>> ComplexReflectionGroups().super_categories()
[Category of complex reflection or generalized Coxeter groups]
>>> 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

sage: W = ComplexReflectionGroups().example(); W                                # needs sage.combinat
5-colored permutations of size 3

Python

>>> from sage.all import *
>>> W = ComplexReflectionGroups().example(); W                                # needs sage.combinat
5-colored permutations of size 3

W is in the category of complex reflection groups:

Sage

sage: W in ComplexReflectionGroups()                                            # needs sage.combinat
True

Python

>>> from sage.all import *
>>> W in ComplexReflectionGroups()                                            # needs sage.combinat
True

Finite

alias of FiniteComplexReflectionGroups

class ParentMethods

Bases: object

rank()

Return the rank of self.

The rank of self is the dimension of the smallest faithfull reflection representation of self.

EXAMPLES:

Sage

sage: W = CoxeterGroups().example(); W
The symmetric group on {0, ..., 3}
sage: W.rank()
3

Python

>>> from sage.all import *
>>> W = CoxeterGroups().example(); W
The symmetric group on {0, ..., 3}
>>> 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

Category.additional_structure()

EXAMPLES:

Sage

sage: from sage.categories.complex_reflection_groups import ComplexReflectionGroups
sage: ComplexReflectionGroups().additional_structure()

Python

>>> from sage.all import *
>>> from sage.categories.complex_reflection_groups import ComplexReflectionGroups
>>> ComplexReflectionGroups().additional_structure()

example()

Return an example of a complex reflection group.

EXAMPLES:

Sage

sage: from sage.categories.complex_reflection_groups import ComplexReflectionGroups
sage: ComplexReflectionGroups().example()                                   # needs sage.combinat
5-colored permutations of size 3

Python

>>> from sage.all import *
>>> from sage.categories.complex_reflection_groups import ComplexReflectionGroups
>>> ComplexReflectionGroups().example()                                   # needs sage.combinat
5-colored permutations of size 3

super_categories()

Return the super categories of self.

EXAMPLES:

Sage

sage: from sage.categories.complex_reflection_groups import ComplexReflectionGroups
sage: ComplexReflectionGroups().super_categories()
[Category of complex reflection or generalized Coxeter groups]

Python

>>> from sage.all import *
>>> from sage.categories.complex_reflection_groups import ComplexReflectionGroups
>>> ComplexReflectionGroups().super_categories()
[Category of complex reflection or generalized Coxeter groups]