# Generalized Coxeter Groups¶

class sage.categories.generalized_coxeter_groups.GeneralizedCoxeterGroups(s=None)

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

class Finite(base_category)

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: 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

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.

EXAMPLES:

sage: from sage.categories.generalized_coxeter_groups import GeneralizedCoxeterGroups

super_categories()
sage: from sage.categories.generalized_coxeter_groups import GeneralizedCoxeterGroups