Generalized Coxeter Groups

class sage.categories.generalized_coxeter_groups.GeneralizedCoxeterGroups

Bases: Category_singleton

The category of generalized Coxeter groups.

A generalized Coxeter group is a group with a presentation of the following form:

\[\langle s_i \mid s_i^{p_i}, s_i s_j \cdots = s_j s_i \cdots \rangle,\]

where \(p_i > 1\), \(i \in I\), and the factors in the braid relation occur \(m_{ij} = m_{ji}\) times for all \(i \neq j \in I\).

EXAMPLES:

Sage

sage: from sage.categories.generalized_coxeter_groups import GeneralizedCoxeterGroups
sage: C = GeneralizedCoxeterGroups(); C
Category of generalized Coxeter groups

Python

>>> from sage.all import *
>>> from sage.categories.generalized_coxeter_groups import GeneralizedCoxeterGroups
>>> C = GeneralizedCoxeterGroups(); C
Category of generalized Coxeter groups

class Finite(base_category)

Bases: CategoryWithAxiom_singleton

The category of finite generalized Coxeter groups.

extra_super_categories()

Implement that a finite generalized Coxeter group is a well-generated complex reflection group.

EXAMPLES:

Sage

sage: from sage.categories.generalized_coxeter_groups import GeneralizedCoxeterGroups
sage: from sage.categories.complex_reflection_groups import ComplexReflectionGroups

sage: Cat = GeneralizedCoxeterGroups().Finite()
sage: Cat.extra_super_categories()
[Category of well generated finite complex reflection groups]
sage: Cat.is_subcategory(ComplexReflectionGroups().Finite().WellGenerated())
True

Python

>>> from sage.all import *
>>> from sage.categories.generalized_coxeter_groups import GeneralizedCoxeterGroups
>>> from sage.categories.complex_reflection_groups import ComplexReflectionGroups

>>> Cat = GeneralizedCoxeterGroups().Finite()
>>> Cat.extra_super_categories()
[Category of well generated finite complex reflection groups]
>>> Cat.is_subcategory(ComplexReflectionGroups().Finite().WellGenerated())
True

additional_structure()

Return None.

Indeed, all the structure generalized Coxeter 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.generalized_coxeter_groups import GeneralizedCoxeterGroups
sage: GeneralizedCoxeterGroups().additional_structure()

Python

>>> from sage.all import *
>>> from sage.categories.generalized_coxeter_groups import GeneralizedCoxeterGroups
>>> GeneralizedCoxeterGroups().additional_structure()

super_categories()

EXAMPLES:

Sage

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

Python

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