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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. | ||
| REFERENCES: | ||
| .. [LT2009] G.I. Lehrer and D.E. Taylor. *Unitary reflection groups*. | ||
| Australian Mathematical Society Lecture Series, 2009. | ||
| 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] | ||
| """ | ||
| return [ComplexReflectionOrGeneralizedCoxeterGroups()] | ||
|
|
||
| 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() | ||
| """ | ||
| return None | ||
|
|
||
| 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 | ||
| """ | ||
| from sage.combinat.colored_permutations import ColoredPermutations | ||
| return ColoredPermutations(5, 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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