Generalized Coxeter Groups¶
- class sage.categories.generalized_coxeter_groups.GeneralizedCoxeterGroups[source]¶
- 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: from sage.categories.generalized_coxeter_groups import GeneralizedCoxeterGroups sage: C = GeneralizedCoxeterGroups(); C Category of generalized Coxeter groups - >>> from sage.all import * >>> from sage.categories.generalized_coxeter_groups import GeneralizedCoxeterGroups >>> C = GeneralizedCoxeterGroups(); C Category of generalized Coxeter groups - class Finite(base_category)[source]¶
- Bases: - CategoryWithAxiom_singleton- The category of finite generalized Coxeter groups. - extra_super_categories()[source]¶
- Implement that a finite generalized Coxeter group is a well-generated complex reflection group. - EXAMPLES: - 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 - >>> 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()[source]¶
- 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 - EXAMPLES: - sage: from sage.categories.generalized_coxeter_groups import GeneralizedCoxeterGroups sage: GeneralizedCoxeterGroups().additional_structure() - >>> from sage.all import * >>> from sage.categories.generalized_coxeter_groups import GeneralizedCoxeterGroups >>> GeneralizedCoxeterGroups().additional_structure() 
 - super_categories()[source]¶
- EXAMPLES: - sage: from sage.categories.generalized_coxeter_groups import GeneralizedCoxeterGroups sage: GeneralizedCoxeterGroups().super_categories() [Category of complex reflection or generalized Coxeter groups] - >>> from sage.all import * >>> from sage.categories.generalized_coxeter_groups import GeneralizedCoxeterGroups >>> GeneralizedCoxeterGroups().super_categories() [Category of complex reflection or generalized Coxeter groups]