Regular Supercrystals¶
- class sage.categories.regular_supercrystals.RegularSuperCrystals[source]¶
- Bases: - Category_singleton- The category of crystals for super Lie algebras. - EXAMPLES: - sage: from sage.categories.regular_supercrystals import RegularSuperCrystals sage: C = RegularSuperCrystals() sage: C Category of regular super crystals sage: C.super_categories() [Category of finite super crystals] - >>> from sage.all import * >>> from sage.categories.regular_supercrystals import RegularSuperCrystals >>> C = RegularSuperCrystals() >>> C Category of regular super crystals >>> C.super_categories() [Category of finite super crystals] - Parents in this category should implement the following methods: - either an attribute - _cartan_typeor a method- cartan_type
- module_generators– a list (or container) of distinct elements that generate the crystal using \(f_i\) and \(e_i\)
 - Furthermore, their elements - xshould implement the following methods:- x.e(i)(returning \(e_i(x)\))
- x.f(i)(returning \(f_i(x)\))
- x.weight()(returning \(\operatorname{wt}(x)\))
 - EXAMPLES: - sage: from sage.misc.abstract_method import abstract_methods_of_class sage: from sage.categories.regular_supercrystals import RegularSuperCrystals sage: abstract_methods_of_class(RegularSuperCrystals().element_class) {'optional': [], 'required': ['e', 'f', 'weight']} - >>> from sage.all import * >>> from sage.misc.abstract_method import abstract_methods_of_class >>> from sage.categories.regular_supercrystals import RegularSuperCrystals >>> abstract_methods_of_class(RegularSuperCrystals().element_class) {'optional': [], 'required': ['e', 'f', 'weight']} - class ElementMethods[source]¶
- Bases: - object- epsilon(i)[source]¶
- Return \(\varepsilon_i\) of - self.- EXAMPLES: - sage: C = crystals.Tableaux(['A',[1,2]], shape=[2,1]) sage: c = C.an_element(); c [[-2, -2], [-1]] sage: c.epsilon(2) 0 sage: c.epsilon(0) 0 sage: c.epsilon(-1) 0 - >>> from sage.all import * >>> C = crystals.Tableaux(['A',[Integer(1),Integer(2)]], shape=[Integer(2),Integer(1)]) >>> c = C.an_element(); c [[-2, -2], [-1]] >>> c.epsilon(Integer(2)) 0 >>> c.epsilon(Integer(0)) 0 >>> c.epsilon(-Integer(1)) 0 
 - phi(i)[source]¶
- Return \(\varphi_i\) of - self.- EXAMPLES: - sage: C = crystals.Tableaux(['A',[1,2]], shape=[2,1]) sage: c = C.an_element(); c [[-2, -2], [-1]] sage: c.phi(1) 0 sage: c.phi(2) 0 sage: c.phi(0) 1 - >>> from sage.all import * >>> C = crystals.Tableaux(['A',[Integer(1),Integer(2)]], shape=[Integer(2),Integer(1)]) >>> c = C.an_element(); c [[-2, -2], [-1]] >>> c.phi(Integer(1)) 0 >>> c.phi(Integer(2)) 0 >>> c.phi(Integer(0)) 1 
 
 - class TensorProducts(category, *args)[source]¶
- Bases: - TensorProductsCategory- The category of regular crystals constructed by tensor product of regular crystals. - extra_super_categories()[source]¶
- EXAMPLES: - sage: from sage.categories.regular_supercrystals import RegularSuperCrystals sage: RegularSuperCrystals().TensorProducts().extra_super_categories() [Category of regular super crystals] - >>> from sage.all import * >>> from sage.categories.regular_supercrystals import RegularSuperCrystals >>> RegularSuperCrystals().TensorProducts().extra_super_categories() [Category of regular super crystals] 
 
 - super_categories()[source]¶
- EXAMPLES: - sage: from sage.categories.regular_supercrystals import RegularSuperCrystals sage: C = RegularSuperCrystals() sage: C.super_categories() [Category of finite super crystals] - >>> from sage.all import * >>> from sage.categories.regular_supercrystals import RegularSuperCrystals >>> C = RegularSuperCrystals() >>> C.super_categories() [Category of finite super crystals]