Coverage for local/lib/python2.7/site-packages/sage/categories/complex_reflection_groups.py : 27%
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r""" Complex reflection groups """ #***************************************************************************** # Copyright (C) 2011-2015 Christian Stump <christian.stump at gmail.com> # # 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.misc.lazy_import import LazyImport from sage.categories.category_singleton import Category_singleton from sage.categories.complex_reflection_or_generalized_coxeter_groups import ComplexReflectionOrGeneralizedCoxeterGroups
class ComplexReflectionGroups(Category_singleton): r""" The category of complex reflection groups.
Let `V` be a complex vector space. A *complex reflection* is an element of `\operatorname{GL}(V)` fixing an 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:`Reflection_group`.
.. SEEALSO::
:func:`ReflectionGroup` for usage examples of this category.
EXAMPLES::
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]
An example of a reflection group::
sage: W = ComplexReflectionGroups().example(); W 5-colored permutations of size 3
``W`` is in the category of complex reflection groups::
sage: W in ComplexReflectionGroups() True
TESTS::
sage: TestSuite(W).run() sage: TestSuite(ComplexReflectionGroups()).run() """
@cached_method def super_categories(self): r""" Return the super categories of ``self``.
EXAMPLES::
sage: from sage.categories.complex_reflection_groups import ComplexReflectionGroups sage: ComplexReflectionGroups().super_categories() [Category of complex reflection or generalized coxeter groups] """
def additional_structure(self): r""" Return ``None``.
Indeed, all the structure complex reflection 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.complex_reflection_groups import ComplexReflectionGroups sage: ComplexReflectionGroups().additional_structure() """
def example(self): r""" Return an example of a complex reflection group.
EXAMPLES::
sage: from sage.categories.complex_reflection_groups import ComplexReflectionGroups sage: ComplexReflectionGroups().example() 5-colored permutations of size 3 """
class ParentMethods:
@cached_method def rank(self): r""" Return the rank of ``self``.
The rank of ``self`` is the dimension of the smallest faithfull reflection representation of ``self``.
EXAMPLES::
sage: W = CoxeterGroups().example(); W The symmetric group on {0, ..., 3} sage: W.rank() 3 """
Finite = LazyImport('sage.categories.finite_complex_reflection_groups', 'FiniteComplexReflectionGroups', as_name='Finite')
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