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Package sage :: Package combinat :: Package root_system :: Module type_reducible :: Class ambient_space
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Class ambient_space

source code

                                 object --+                        
                                          |                        
           structure.sage_object.SageObject --+                    
                                              |                    
                        structure.parent.Parent --+                
                                                  |                
               structure.parent_base.ParentWithBase --+            
                                                      |            
           free_module.CombinatorialFreeModuleInterface --+        
                                                          |        
                         object --+                       |        
                                  |                       |        
   structure.sage_object.SageObject --+                   |        
                                      |                   |        
                structure.parent.Parent --+               |        
                                          |               |        
       structure.parent_base.ParentWithBase --+           |        
                                              |           |        
           structure.parent_gens.ParentWithGens --+       |        
                                                  |       |        
structure.parent_gens.ParentWithAdditiveAbelianGens --+   |        
                                                      |   |        
                                  modules.module.Module --+        
                                                          |        
                        free_module.CombinatorialFreeModule --+    
                                                              |    
                                             object --+       |    
                                                      |       |    
        root_lattice_realization.RootLatticeRealization --+   |    
                                                          |   |    
        weight_lattice_realization.WeightLatticeRealization --+    
                                                              |    
                                     ambient_space.AmbientSpace --+
                                                                  |
                                                                 ambient_space


EXAMPLES:
    sage: RootSystem("A2xB2").ambient_space()
    Ambient space for the Root system of type A2xB2


Instance Methods [hide private]
 
cartan_type(self)
EXAMPLES:...		 source code
 
component_types(self)
EXAMPLES:...		 source code
 
dimension(self)
EXAMPLES:...		 source code
 
ambient_spaces(self)
Returns a list of the irreducible Cartan types of which the given reducible Cartan type is a product.		 source code
 
inject_weights(self, i, v)
Produces the corresponding element of the lattice.		 source code
 
simple_roots(self)
EXAMPLES:...		 source code
 
positive_roots(self)
EXAMPLES:...		 source code
 
negative_roots(self)
EXAMPLES:...		 source code
 
fundamental_weights(self)
EXAMPLES:...		 source code

Inherited from ambient_space.AmbientSpace: __call__, __cmp__, __getitem__, __init__, __repr__, coroot_lattice, reflection, simple_coroot

Inherited from ambient_space.AmbientSpace (private): _term

Inherited from free_module.CombinatorialFreeModuleInterface: basis, combinatorial_class, get_order, prefix, set_order, sum, term, zero

Inherited from free_module.CombinatorialFreeModuleInterface (private): _an_element_impl, _apply_module_endomorphism, _apply_module_morphism, _from_dict

Inherited from modules.module.Module: __hash__, __new__, category, endomorphism_ring, is_atomic_repr

Inherited from structure.parent_gens.ParentWithAdditiveAbelianGens: __iter__, generator_orders

Inherited from structure.parent_gens.ParentWithGens: __getslice__, __getstate__, __len__, __setstate__, _is_valid_homomorphism_, gen, gens, gens_dict, hom, inject_variables, injvar, latex_name, latex_variable_names, list, ngens, objgen, objgens, variable_name, variable_names

Inherited from structure.parent_gens.ParentWithGens (private): _assign_names, _first_ngens

Inherited from structure.parent_base.ParentWithBase: Hom, base, base_extend, base_extend_canonical, base_extend_canonical_sym, base_extend_recursive, base_ring

Inherited from structure.parent.Parent: __contains__, _coerce_, coerce_map_from, coerce_map_from_impl, construction, get_action, get_action_impl, has_coerce_map_from, has_coerce_map_from_impl, init_coerce

Inherited from structure.parent.Parent (private): _an_element, _coerce_impl, _coerce_self, _coerce_try

Inherited from structure.sage_object.SageObject: _axiom_, _axiom_init_, _gap_, _gap_init_, _gp_, _gp_init_, _interface_, _interface_init_, _interface_is_cached_, _kash_, _kash_init_, _macaulay2_, _macaulay2_init_, _magma_, _magma_init_, _maple_, _maple_init_, _mathematica_, _mathematica_init_, _maxima_, _maxima_init_, _octave_, _octave_init_, _pari_, _pari_init_, _r_init_, _sage_, _singular_, _singular_init_, db, dump, dumps, plot, rename, reset_name, save, version

Inherited from weight_lattice_realization.WeightLatticeRealization: check, rho, weyl_dimension

Inherited from root_lattice_realization.RootLatticeRealization: a_long_simple_root, alpha, alphacheck, associated_coroot, cohighest_root, dynkin_diagram, highest_root, index_set, pi, projection, roots, s, simple_coroots, simple_projection, simple_projections, simple_reflection, simple_reflections, simple_root, weyl_group

Inherited from object: __delattr__, __getattribute__, __reduce__, __reduce_ex__, __setattr__, __str__

Class Methods [hide private]

Inherited from ambient_space.AmbientSpace: smallest_base_ring

Class Variables [hide private]

Inherited from free_module.CombinatorialFreeModuleInterface (private): _name, _prefix

Properties [hide private]

Inherited from structure.parent_gens.ParentWithAdditiveAbelianGens (private): _generator_orders

Inherited from structure.parent_gens.ParentWithGens (private): _gens, _gens_dict, _latex_names, _list, _names

Inherited from structure.parent.Parent (private): _has_coerce_map_from

Inherited from object: __class__

Method Details [hide private]

cartan_type(self)

source code  

EXAMPLES:
    sage: RootSystem("A2xB2").ambient_space().cartan_type()
    A2xB2

component_types(self)

source code  

EXAMPLES:
    sage: RootSystem("A2xB2").ambient_space().component_types()
    [['A', 2], ['B', 2]]

dimension(self)

source code  

EXAMPLES:
    sage: RootSystem("A2xB2").ambient_space().dimension()
    5
Overrides: ambient_space.AmbientSpace.dimension

ambient_spaces(self)

source code  

Returns a list of the irreducible Cartan types of which the
given reducible Cartan type is a product.

EXAMPLES:
    sage: RootSystem("A2xB2").ambient_space().ambient_spaces()
    [Ambient space for the Root system of type ['A', 2],
     Ambient space for the Root system of type ['B', 2]]

inject_weights(self, i, v)

source code  

Produces the corresponding element of the lattice.

INPUT:
  i -- an integer in range(self.components)
  v -- a vector in the i-th component weight lattice

EXAMPLES:
    sage: V = RootSystem("A2xB2").ambient_space()
    sage: [V.inject_weights(i,V.ambient_spaces()[i].fundamental_weights()[1]) for i in range(2)]
    [(1, 0, 0, 0, 0), (0, 0, 0, 1, 0)]
    sage: [V.inject_weights(i,V.ambient_spaces()[i].fundamental_weights()[2]) for i in range(2)]
    [(1, 1, 0, 0, 0), (0, 0, 0, 1/2, 1/2)]

simple_roots(self)

source code  

EXAMPLES:
    sage: RootSystem("A1xA2").ambient_space().simple_roots()
    [(1, -1, 0, 0, 0), (0, 0, 1, -1, 0), (0, 0, 0, 1, -1)]
Overrides: root_lattice_realization.RootLatticeRealization.simple_roots

positive_roots(self)

source code  

EXAMPLES:
    sage: RootSystem("A1xA2").ambient_space().positive_roots()
    [(1, -1, 0, 0, 0), (0, 0, 1, -1, 0), (0, 0, 1, 0, -1), (0, 0, 0, 1, -1)]

negative_roots(self)

source code  

EXAMPLES:
    sage: RootSystem("A1xA2").ambient_space().negative_roots()
    [(-1, 1, 0, 0, 0), (0, 0, -1, 1, 0), (0, 0, -1, 0, 1), (0, 0, 0, -1, 1)]

fundamental_weights(self)

source code  

EXAMPLES:
    sage: RootSystem("A2xB2").ambient_space().fundamental_weights()
    [(1, 0, 0, 0, 0), (1, 1, 0, 0, 0), (0, 0, 0, 1, 0), (0, 0, 0, 1/2, 1/2)]
Overrides: weight_lattice_realization.WeightLatticeRealization.fundamental_weights

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