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

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 --+
                                                              |
                                                             AmbientSpace
Known Subclasses:
type_F.ambient_space, type_reducible.ambient_space, type_E.ambient_space, type_G.ambient_space, type_A.ambient_space, type_C.ambient_space, type_B.ambient_space, type_D.ambient_space 

Abstract class for ambient spaces

Any implementation of this class should implement a class method
smallest_base_ring as described below, and a method dimension
working on a partially initialized instance with just root_system
as attribute. There is no safe default implementation for the later,
so none is provided.


Instance Methods [hide private]
 
__init__(self, root_system, base_ring)
EXAMPLES:...		 source code
 
dimension(self)
Returns the dimension of this ambient space.		 source code
 
__repr__(self)
EXAMPLES:...		 source code
 
__call__(self, v)
TESTS:...		 source code
 
__getitem__(self, i)
Note that indexing starts at 1.		 source code
 
coroot_lattice(self)
EXAMPLES:...		 source code
 
simple_coroot(self, i)
Returns the i-th simple coroot, as an element of this space...		 source code
 
reflection(self, root, coroot=['4ti2-20061025', 'R-2.6.0', 'atlas-3.7.37', 'atlas-3.8.1', 'a...)
EXAMPLES:...		 source code
 
_term(self, i)
Note that indexing starts at 0.		 source code
 
__cmp__(self, other)
EXAMPLES:...		 source code

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, fundamental_weights, 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, simple_roots, weyl_group

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

Class Methods [hide private]
 
smallest_base_ring(cls)
Returns the smallest ground ring over which the ambient space can be realized.		 source code
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]

__init__(self, root_system, base_ring)
(Constructor)

source code  

EXAMPLES:
    sage: e = RootSystem(['A',3]).ambient_lattice()
    sage: e == loads(dumps(e))
    True
Overrides: free_module.CombinatorialFreeModule.__init__

dimension(self)

source code  

Returns the dimension of this ambient space.

EXAMPLES:
    sage: from sage.combinat.root_system.ambient_space import AmbientSpace
    sage: e = RootSystem(['F',4]).ambient_space()
    sage: AmbientSpace.dimension(e)
    Traceback (most recent call last):
    ...
    NotImplementedError
Overrides: free_module.CombinatorialFreeModuleInterface.dimension

smallest_base_ring(cls)
Class Method

source code  

Returns the smallest ground ring over which the ambient space can be realized.

EXAMPLES:
    sage: e = RootSystem(['F',4]).ambient_space()
    sage: e.smallest_base_ring()
    Rational Field

__repr__(self)
(Representation operator)

source code  

EXAMPLES:
    sage: RootSystem(['A',4]).ambient_lattice()
    Ambient lattice for the Root system of type ['A', 4]
    sage: RootSystem(['B',4]).ambient_space()
    Ambient space for the Root system of type ['B', 4]
Overrides: free_module.CombinatorialFreeModuleInterface.__repr__

__call__(self, v)
(Call operator)

source code  

TESTS:
    sage: R = RootSystem(['A',4]).ambient_lattice()
    sage: R([1,2,3,4,5])
    (1, 2, 3, 4, 5)
Overrides: free_module.CombinatorialFreeModuleInterface.__call__

__getitem__(self, i)
(Indexing operator)

source code  

Note that indexing starts at 1.

EXAMPLES:
    sage: e = RootSystem(['A',2]).ambient_lattice()
    sage: e[1]
    (1, 0, 0)
Overrides: structure.parent_gens.ParentWithGens.__getitem__

coroot_lattice(self)

source code  

EXAMPLES:
    sage: e = RootSystem(["A", 3]).ambient_lattice()
    sage: e.coroot_lattice()
    Ambient lattice for the Root system of type ['A', 3]
Overrides: root_lattice_realization.RootLatticeRealization.coroot_lattice

simple_coroot(self, i)

source code  

Returns the i-th simple coroot, as an element of this space

EXAMPLES:
    sage: R = RootSystem(["A",3])
    sage: L = R.ambient_lattice()
    sage: L.simple_coroot(1)
    (1, -1, 0, 0)
    sage: L.simple_coroot(2)
    (0, 1, -1, 0)
    sage: L.simple_coroot(3)
    (0, 0, 1, -1)
Overrides: root_lattice_realization.RootLatticeRealization.simple_coroot

reflection(self, root, coroot=['4ti2-20061025', 'R-2.6.0', 'atlas-3.7.37', 'atlas-3.8.1', 'a...)

source code  

EXAMPLES:
    sage: e = RootSystem(["A", 3]).ambient_lattice()
    sage: a = e.simple_root(0); a
    (-1, 0, 0, 0)
    sage: b = e.simple_root(1); b
    (1, -1, 0, 0)
    sage: s_a = e.reflection(a)
    sage: s_a(b)
    (0, -1, 0, 0)
Overrides: root_lattice_realization.RootLatticeRealization.reflection

_term(self, i)

source code  

Note that indexing starts at 0.

EXAMPLES:
    sage: e = RootSystem(['A',2]).ambient_lattice()
    sage: e._term(0)
    (1, 0, 0)

__cmp__(self, other)
(Comparison operator)

source code  

EXAMPLES:
    sage: e1 = RootSystem(['A',3]).ambient_lattice()
    sage: e2 = RootSystem(['B',3]).ambient_lattice()
    sage: e1 == e1
    True
    sage: e1 == e2
    False
Overrides: free_module.CombinatorialFreeModuleInterface.__cmp__

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