Coverage for local/lib/python2.7/site-packages/sage/categories/generalized_coxeter_groups.py : 31%
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r""" Generalized Coxeter Groups """ #***************************************************************************** # Copyright (C) 2016 Travis Scrimshaw <tscrim at ucdavis.edu> # # This program is free software: you can redistribute it and/or modify # it under the terms of the GNU General Public License as published by # the Free Software Foundation, either version 2 of the License, or # (at your option) any later version. # http://www.gnu.org/licenses/ #*****************************************************************************
from sage.misc.cachefunc import cached_method from sage.categories.category_singleton import Category_singleton from sage.categories.category_with_axiom import CategoryWithAxiom from sage.categories.complex_reflection_or_generalized_coxeter_groups import ComplexReflectionOrGeneralizedCoxeterGroups
class GeneralizedCoxeterGroups(Category_singleton): r""" The category of generalized Coxeter groups.
A generalized Coxeter group is a group with a presentation of the following form:
.. MATH::
\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
TESTS::
sage: TestSuite(C).run() """ @cached_method def super_categories(self): """ EXAMPLES::
sage: from sage.categories.generalized_coxeter_groups import GeneralizedCoxeterGroups sage: GeneralizedCoxeterGroups().super_categories() [Category of complex reflection or generalized coxeter groups] """
def additional_structure(self): r""" Return ``None``.
Indeed, all the structure generalized Coxeter groups have in addition to groups (simple reflections, ...) is already defined in the super category.
.. SEEALSO:: :meth:`Category.additional_structure`
EXAMPLES::
sage: from sage.categories.generalized_coxeter_groups import GeneralizedCoxeterGroups sage: GeneralizedCoxeterGroups().additional_structure() """
class Finite(CategoryWithAxiom): """ The category of finite generalized Coxeter groups. """ def extra_super_categories(self): """ 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 """
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