Free modules¶
- class sage.combinat.free_module.CartesianProductWithFlattening(flatten)[source]¶
Bases:
object
A class for Cartesian product constructor, with partial flattening
- class sage.combinat.free_module.CombinatorialFreeModule(R, basis_keys=None, element_class=None, category=None, prefix=None, names=None, **kwds)[source]¶
Bases:
UniqueRepresentation
,Module
,IndexedGenerators
Class for free modules with a named basis.
INPUT:
R
– base ringbasis_keys
– list; tuple, family, set, etc. defining the indexing set for the basis of this moduleelement_class
– the class of which elements of this module should be instances (default: None, in which case the elements are instances ofIndexedFreeModuleElement
)category
– the category in which this module lies (optional, default None, in which case use the “category of modules with basis” over the base ringR
); this should be a subcategory ofModulesWithBasis
For the options controlling the printing of elements, see
IndexedGenerators
.Note
These print options may also be accessed and modified using the
print_options()
method, after the module has been defined.EXAMPLES:
We construct a free module whose basis is indexed by the letters a, b, c:
sage: F = CombinatorialFreeModule(QQ, ['a','b','c']) sage: F Free module generated by {'a', 'b', 'c'} over Rational Field
>>> from sage.all import * >>> F = CombinatorialFreeModule(QQ, ['a','b','c']) >>> F Free module generated by {'a', 'b', 'c'} over Rational Field
Its basis is a family, indexed by a, b, c:
sage: e = F.basis() sage: e Finite family {'a': B['a'], 'b': B['b'], 'c': B['c']}
>>> from sage.all import * >>> e = F.basis() >>> e Finite family {'a': B['a'], 'b': B['b'], 'c': B['c']}
sage: [x for x in e] [B['a'], B['b'], B['c']] sage: [k for k in e.keys()] ['a', 'b', 'c']
>>> from sage.all import * >>> [x for x in e] [B['a'], B['b'], B['c']] >>> [k for k in e.keys()] ['a', 'b', 'c']
Let us construct some elements, and compute with them:
sage: e['a'] B['a'] sage: 2*e['a'] 2*B['a'] sage: e['a'] + 3*e['b'] B['a'] + 3*B['b']
>>> from sage.all import * >>> e['a'] B['a'] >>> Integer(2)*e['a'] 2*B['a'] >>> e['a'] + Integer(3)*e['b'] B['a'] + 3*B['b']
Some uses of
sage.categories.commutative_additive_semigroups.CommutativeAdditiveSemigroups.ParentMethods.summation()
andsum()
:sage: F = CombinatorialFreeModule(QQ, [1,2,3,4]) sage: F.summation(F.monomial(1), F.monomial(3)) B[1] + B[3] sage: F = CombinatorialFreeModule(QQ, [1,2,3,4]) sage: F.sum(F.monomial(i) for i in [1,2,3]) B[1] + B[2] + B[3]
>>> from sage.all import * >>> F = CombinatorialFreeModule(QQ, [Integer(1),Integer(2),Integer(3),Integer(4)]) >>> F.summation(F.monomial(Integer(1)), F.monomial(Integer(3))) B[1] + B[3] >>> F = CombinatorialFreeModule(QQ, [Integer(1),Integer(2),Integer(3),Integer(4)]) >>> F.sum(F.monomial(i) for i in [Integer(1),Integer(2),Integer(3)]) B[1] + B[2] + B[3]
Note that free modules with a given basis and parameters are unique:
sage: F1 = CombinatorialFreeModule(QQ, (1,2,3,4)) sage: F1 is F True
>>> from sage.all import * >>> F1 = CombinatorialFreeModule(QQ, (Integer(1),Integer(2),Integer(3),Integer(4))) >>> F1 is F True
The identity of the constructed free module depends on the order of the basis and on the other parameters, like the prefix. Note that
CombinatorialFreeModule
is aUniqueRepresentation
. Hence, two combinatorial free modules evaluate equal if and only if they are identical:sage: F1 = CombinatorialFreeModule(QQ, (1,2,3,4)) sage: F1 is F True sage: F1 = CombinatorialFreeModule(QQ, [4,3,2,1]) sage: F1 == F False sage: F2 = CombinatorialFreeModule(QQ, [1,2,3,4], prefix='F') sage: F2 == F False
>>> from sage.all import * >>> F1 = CombinatorialFreeModule(QQ, (Integer(1),Integer(2),Integer(3),Integer(4))) >>> F1 is F True >>> F1 = CombinatorialFreeModule(QQ, [Integer(4),Integer(3),Integer(2),Integer(1)]) >>> F1 == F False >>> F2 = CombinatorialFreeModule(QQ, [Integer(1),Integer(2),Integer(3),Integer(4)], prefix='F') >>> F2 == F False
Because of this, if you create a free module with certain parameters and then modify its prefix or other print options, this affects all modules which were defined using the same parameters.
sage: F2.print_options(prefix='x') sage: F2.prefix() 'x' sage: F3 = CombinatorialFreeModule(QQ, [1,2,3,4], prefix='F') sage: F3 is F2 # F3 was defined just like F2 True sage: F3.prefix() 'x' sage: F4 = CombinatorialFreeModule(QQ, [1,2,3,4], prefix='F', bracket=True) sage: F4 == F2 # F4 was NOT defined just like F2 False sage: F4.prefix() 'F' sage: F2.print_options(prefix='F') #reset for following doctests
>>> from sage.all import * >>> F2.print_options(prefix='x') >>> F2.prefix() 'x' >>> F3 = CombinatorialFreeModule(QQ, [Integer(1),Integer(2),Integer(3),Integer(4)], prefix='F') >>> F3 is F2 # F3 was defined just like F2 True >>> F3.prefix() 'x' >>> F4 = CombinatorialFreeModule(QQ, [Integer(1),Integer(2),Integer(3),Integer(4)], prefix='F', bracket=True) >>> F4 == F2 # F4 was NOT defined just like F2 False >>> F4.prefix() 'F' >>> F2.print_options(prefix='F') #reset for following doctests
The constructed module is in the category of modules with basis over the base ring:
sage: CombinatorialFreeModule(QQ, Partitions()).category() # needs sage.combinat Category of vector spaces with basis over Rational Field
>>> from sage.all import * >>> CombinatorialFreeModule(QQ, Partitions()).category() # needs sage.combinat Category of vector spaces with basis over Rational Field
If furthermore the index set is finite (i.e. in the category
Sets().Finite()
), then the module is declared as being finite dimensional:sage: CombinatorialFreeModule(QQ, [1,2,3,4]).category() Category of finite dimensional vector spaces with basis over Rational Field sage: CombinatorialFreeModule(QQ, Partitions(3), # needs sage.combinat ....: category=Algebras(QQ).WithBasis()).category() Category of finite dimensional algebras with basis over Rational Field
>>> from sage.all import * >>> CombinatorialFreeModule(QQ, [Integer(1),Integer(2),Integer(3),Integer(4)]).category() Category of finite dimensional vector spaces with basis over Rational Field >>> CombinatorialFreeModule(QQ, Partitions(Integer(3)), # needs sage.combinat ... category=Algebras(QQ).WithBasis()).category() Category of finite dimensional algebras with basis over Rational Field
See
sage.categories.examples.algebras_with_basis
andsage.categories.examples.hopf_algebras_with_basis
for illustrations of the use of thecategory
keyword, and seesage.combinat.root_system.weight_space.WeightSpace
for an example of the use ofelement_class
.Customizing print and LaTeX representations of elements:
sage: F = CombinatorialFreeModule(QQ, ['a','b'], prefix='x') sage: original_print_options = F.print_options() sage: sorted(original_print_options.items()) [('bracket', None), ('iterate_key', False), ('latex_bracket', False), ('latex_names', None), ('latex_prefix', None), ('latex_scalar_mult', None), ('names', None), ('prefix', 'x'), ('scalar_mult', '*'), ('sorting_key', <function ...<lambda> at ...>), ('sorting_reverse', False), ('string_quotes', True), ('tensor_symbol', None)] sage: e = F.basis() sage: e['a'] - 3 * e['b'] x['a'] - 3*x['b'] sage: F.print_options(prefix='x', scalar_mult=' ', bracket='{') sage: e['a'] - 3 * e['b'] x{'a'} - 3 x{'b'} sage: latex(e['a'] - 3 * e['b']) x_{a} - 3 x_{b} sage: F.print_options(latex_prefix='y') sage: latex(e['a'] - 3 * e['b']) y_{a} - 3 y_{b} sage: F.print_options(sorting_reverse=True) sage: e['a'] - 3 * e['b'] -3 x{'b'} + x{'a'} sage: F.print_options(**original_print_options) # reset print options sage: F = CombinatorialFreeModule(QQ, [(1,2), (3,4)]) sage: e = F.basis() sage: e[(1,2)] - 3 * e[(3,4)] B[(1, 2)] - 3*B[(3, 4)] sage: F.print_options(bracket=['_{', '}']) sage: e[(1,2)] - 3 * e[(3,4)] B_{(1, 2)} - 3*B_{(3, 4)} sage: F.print_options(prefix='', bracket=False) sage: e[(1,2)] - 3 * e[(3,4)] (1, 2) - 3*(3, 4)
>>> from sage.all import * >>> F = CombinatorialFreeModule(QQ, ['a','b'], prefix='x') >>> original_print_options = F.print_options() >>> sorted(original_print_options.items()) [('bracket', None), ('iterate_key', False), ('latex_bracket', False), ('latex_names', None), ('latex_prefix', None), ('latex_scalar_mult', None), ('names', None), ('prefix', 'x'), ('scalar_mult', '*'), ('sorting_key', <function ...<lambda> at ...>), ('sorting_reverse', False), ('string_quotes', True), ('tensor_symbol', None)] >>> e = F.basis() >>> e['a'] - Integer(3) * e['b'] x['a'] - 3*x['b'] >>> F.print_options(prefix='x', scalar_mult=' ', bracket='{') >>> e['a'] - Integer(3) * e['b'] x{'a'} - 3 x{'b'} >>> latex(e['a'] - Integer(3) * e['b']) x_{a} - 3 x_{b} >>> F.print_options(latex_prefix='y') >>> latex(e['a'] - Integer(3) * e['b']) y_{a} - 3 y_{b} >>> F.print_options(sorting_reverse=True) >>> e['a'] - Integer(3) * e['b'] -3 x{'b'} + x{'a'} >>> F.print_options(**original_print_options) # reset print options >>> F = CombinatorialFreeModule(QQ, [(Integer(1),Integer(2)), (Integer(3),Integer(4))]) >>> e = F.basis() >>> e[(Integer(1),Integer(2))] - Integer(3) * e[(Integer(3),Integer(4))] B[(1, 2)] - 3*B[(3, 4)] >>> F.print_options(bracket=['_{', '}']) >>> e[(Integer(1),Integer(2))] - Integer(3) * e[(Integer(3),Integer(4))] B_{(1, 2)} - 3*B_{(3, 4)} >>> F.print_options(prefix='', bracket=False) >>> e[(Integer(1),Integer(2))] - Integer(3) * e[(Integer(3),Integer(4))] (1, 2) - 3*(3, 4)
- Element[source]¶
alias of
IndexedFreeModuleElement
- Tensor[source]¶
alias of
CombinatorialFreeModule_Tensor
- change_ring(R)[source]¶
Return the base change of
self
to \(R\).EXAMPLES:
sage: F = CombinatorialFreeModule(ZZ, ['a','b','c']); F Free module generated by {'a', 'b', 'c'} over Integer Ring sage: F_QQ = F.change_ring(QQ); F_QQ Free module generated by {'a', 'b', 'c'} over Rational Field sage: F_QQ.change_ring(ZZ) == F True sage: T = F.tensor(F); T Free module generated by {'a', 'b', 'c'} over Integer Ring # Free module generated by {'a', 'b', 'c'} over Integer Ring sage: T.change_ring(QQ) Free module generated by {'a', 'b', 'c'} over Rational Field # Free module generated by {'a', 'b', 'c'} over Rational Field sage: G = CombinatorialFreeModule(ZZ, ['x','y']); G Free module generated by {'x', 'y'} over Integer Ring sage: T = F.cartesian_product(G); T Free module generated by {'a', 'b', 'c'} over Integer Ring (+) Free module generated by {'x', 'y'} over Integer Ring sage: T.change_ring(QQ) Free module generated by {'a', 'b', 'c'} over Rational Field (+) Free module generated by {'x', 'y'} over Rational Field
>>> from sage.all import * >>> F = CombinatorialFreeModule(ZZ, ['a','b','c']); F Free module generated by {'a', 'b', 'c'} over Integer Ring >>> F_QQ = F.change_ring(QQ); F_QQ Free module generated by {'a', 'b', 'c'} over Rational Field >>> F_QQ.change_ring(ZZ) == F True >>> T = F.tensor(F); T Free module generated by {'a', 'b', 'c'} over Integer Ring # Free module generated by {'a', 'b', 'c'} over Integer Ring >>> T.change_ring(QQ) Free module generated by {'a', 'b', 'c'} over Rational Field # Free module generated by {'a', 'b', 'c'} over Rational Field >>> G = CombinatorialFreeModule(ZZ, ['x','y']); G Free module generated by {'x', 'y'} over Integer Ring >>> T = F.cartesian_product(G); T Free module generated by {'a', 'b', 'c'} over Integer Ring (+) Free module generated by {'x', 'y'} over Integer Ring >>> T.change_ring(QQ) Free module generated by {'a', 'b', 'c'} over Rational Field (+) Free module generated by {'x', 'y'} over Rational Field
- construction()[source]¶
The construction functor and base ring for
self
.EXAMPLES:
sage: F = CombinatorialFreeModule(QQ, ['a','b','c'], category=AlgebrasWithBasis(QQ)) sage: F.construction() (VectorFunctor, Rational Field)
>>> from sage.all import * >>> F = CombinatorialFreeModule(QQ, ['a','b','c'], category=AlgebrasWithBasis(QQ)) >>> F.construction() (VectorFunctor, Rational Field)
- dimension()[source]¶
Return the dimension of the free module (which is given by the number of elements in the basis).
EXAMPLES:
sage: F = CombinatorialFreeModule(QQ, ['a', 'b', 'c']) sage: F.dimension() 3 sage: F.basis().cardinality() 3 sage: F.basis().keys().cardinality() 3
>>> from sage.all import * >>> F = CombinatorialFreeModule(QQ, ['a', 'b', 'c']) >>> F.dimension() 3 >>> F.basis().cardinality() 3 >>> F.basis().keys().cardinality() 3
Rank is available as a synonym:
sage: F.rank() 3
>>> from sage.all import * >>> F.rank() 3
sage: s = SymmetricFunctions(QQ).schur() # needs sage.combinat sage: s.dimension() # needs sage.combinat +Infinity
>>> from sage.all import * >>> s = SymmetricFunctions(QQ).schur() # needs sage.combinat >>> s.dimension() # needs sage.combinat +Infinity
- element_class()[source]¶
The (default) class for the elements of this parent.
Overrides
Parent.element_class()
to force the construction of Python class. This is currently needed to inherit really all the features from categories, and in particular the initialization of_mul_
inMagmas.ParentMethods.__init_extra__()
.EXAMPLES:
sage: # needs sage.combinat sage: A = Algebras(QQ).WithBasis().example(); A An example of an algebra with basis: the free algebra on the generators ('a', 'b', 'c') over Rational Field sage: A.element_class.mro() [<class 'sage.categories.examples.algebras_with_basis.FreeAlgebra_with_category.element_class'>, <class 'sage.modules.with_basis.indexed_element.IndexedFreeModuleElement'>, ...] sage: a,b,c = A.algebra_generators() sage: a * b B[word: ab]
>>> from sage.all import * >>> # needs sage.combinat >>> A = Algebras(QQ).WithBasis().example(); A An example of an algebra with basis: the free algebra on the generators ('a', 'b', 'c') over Rational Field >>> A.element_class.mro() [<class 'sage.categories.examples.algebras_with_basis.FreeAlgebra_with_category.element_class'>, <class 'sage.modules.with_basis.indexed_element.IndexedFreeModuleElement'>, ...] >>> a,b,c = A.algebra_generators() >>> a * b B[word: ab]
- from_vector(vector, order=None, coerce=True)[source]¶
Build an element of
self
from a (sparse) vector.See also
get_order()
,CombinatorialFreeModule.Element._vector_()
EXAMPLES:
sage: # needs sage.combinat sage: QS3 = SymmetricGroupAlgebra(QQ, 3) sage: b = QS3.from_vector(vector((2, 0, 0, 0, 0, 4))); b 2*[1, 2, 3] + 4*[3, 2, 1] sage: a = 2*QS3([1,2,3]) + 4*QS3([3,2,1]) sage: a == b True
>>> from sage.all import * >>> # needs sage.combinat >>> QS3 = SymmetricGroupAlgebra(QQ, Integer(3)) >>> b = QS3.from_vector(vector((Integer(2), Integer(0), Integer(0), Integer(0), Integer(0), Integer(4)))); b 2*[1, 2, 3] + 4*[3, 2, 1] >>> a = Integer(2)*QS3([Integer(1),Integer(2),Integer(3)]) + Integer(4)*QS3([Integer(3),Integer(2),Integer(1)]) >>> a == b True
- get_order()[source]¶
Return the order of the elements in the basis.
EXAMPLES:
sage: QS2 = SymmetricGroupAlgebra(QQ,2) # needs sage.combinat sage: QS2.get_order() # note: order changed on 2009-03-13 # needs sage.combinat [[2, 1], [1, 2]]
>>> from sage.all import * >>> QS2 = SymmetricGroupAlgebra(QQ,Integer(2)) # needs sage.combinat >>> QS2.get_order() # note: order changed on 2009-03-13 # needs sage.combinat [[2, 1], [1, 2]]
- get_order_key()[source]¶
Return a comparison key on the basis indices that is compatible with the current term order.
EXAMPLES:
sage: A = FiniteDimensionalAlgebrasWithBasis(QQ).example() sage: A.set_order(['x', 'y', 'a', 'b']) sage: Akey = A.get_order_key() sage: sorted(A.basis().keys(), key=Akey) ['x', 'y', 'a', 'b'] sage: A.set_order(list(reversed(A.basis().keys()))) sage: Akey = A.get_order_key() sage: sorted(A.basis().keys(), key=Akey) ['b', 'a', 'y', 'x']
>>> from sage.all import * >>> A = FiniteDimensionalAlgebrasWithBasis(QQ).example() >>> A.set_order(['x', 'y', 'a', 'b']) >>> Akey = A.get_order_key() >>> sorted(A.basis().keys(), key=Akey) ['x', 'y', 'a', 'b'] >>> A.set_order(list(reversed(A.basis().keys()))) >>> Akey = A.get_order_key() >>> sorted(A.basis().keys(), key=Akey) ['b', 'a', 'y', 'x']
- is_exact()[source]¶
Return
True
if elements ofself
have exact representations, which is true ofself
if and only if it is true ofself.basis().keys()
andself.base_ring()
.EXAMPLES:
sage: GroupAlgebra(GL(3, GF(7))).is_exact() # needs sage.groups sage.rings.finite_rings True sage: GroupAlgebra(GL(3, GF(7)), RR).is_exact() # needs sage.groups sage.rings.finite_rings False sage: GroupAlgebra(GL(3, pAdicRing(7))).is_exact() # not implemented, needs sage.groups sage.rings.padics False
>>> from sage.all import * >>> GroupAlgebra(GL(Integer(3), GF(Integer(7)))).is_exact() # needs sage.groups sage.rings.finite_rings True >>> GroupAlgebra(GL(Integer(3), GF(Integer(7))), RR).is_exact() # needs sage.groups sage.rings.finite_rings False >>> GroupAlgebra(GL(Integer(3), pAdicRing(Integer(7)))).is_exact() # not implemented, needs sage.groups sage.rings.padics False
- linear_combination(iter_of_elements_coeff, factor_on_left=True)[source]¶
Return the linear combination \(\lambda_1 v_1 + \cdots + \lambda_k v_k\) (resp. the linear combination \(v_1 \lambda_1 + \cdots + v_k \lambda_k\)) where
iter_of_elements_coeff
iterates through the sequence \(((v_1, \lambda_1), ..., (v_k, \lambda_k))\).INPUT:
iter_of_elements_coeff
– iterator of pairs(element, coeff)
withelement
inself
andcoeff
inself.base_ring()
factor_on_left
– (optional) ifTrue
, the coefficients are multiplied from the left ifFalse
, the coefficients are multiplied from the right
EXAMPLES:
sage: F = CombinatorialFreeModule(QQ, [1,2]) sage: f = F.an_element(); f 2*B[1] + 2*B[2] sage: F.linear_combination( (f,i) for i in range(5) ) 20*B[1] + 20*B[2]
>>> from sage.all import * >>> F = CombinatorialFreeModule(QQ, [Integer(1),Integer(2)]) >>> f = F.an_element(); f 2*B[1] + 2*B[2] >>> F.linear_combination( (f,i) for i in range(Integer(5)) ) 20*B[1] + 20*B[2]
- monomial()[source]¶
Return the basis element indexed by
i
.INPUT:
i
– an element of the index set
EXAMPLES:
sage: F = CombinatorialFreeModule(QQ, ['a', 'b', 'c']) sage: F.monomial('a') B['a']
>>> from sage.all import * >>> F = CombinatorialFreeModule(QQ, ['a', 'b', 'c']) >>> F.monomial('a') B['a']
F.monomial
is in fact (almost) a map:sage: F.monomial Term map from {'a', 'b', 'c'} to Free module generated by {'a', 'b', 'c'} over Rational Field
>>> from sage.all import * >>> F.monomial Term map from {'a', 'b', 'c'} to Free module generated by {'a', 'b', 'c'} over Rational Field
- rank()[source]¶
Return the dimension of the free module (which is given by the number of elements in the basis).
EXAMPLES:
sage: F = CombinatorialFreeModule(QQ, ['a', 'b', 'c']) sage: F.dimension() 3 sage: F.basis().cardinality() 3 sage: F.basis().keys().cardinality() 3
>>> from sage.all import * >>> F = CombinatorialFreeModule(QQ, ['a', 'b', 'c']) >>> F.dimension() 3 >>> F.basis().cardinality() 3 >>> F.basis().keys().cardinality() 3
Rank is available as a synonym:
sage: F.rank() 3
>>> from sage.all import * >>> F.rank() 3
sage: s = SymmetricFunctions(QQ).schur() # needs sage.combinat sage: s.dimension() # needs sage.combinat +Infinity
>>> from sage.all import * >>> s = SymmetricFunctions(QQ).schur() # needs sage.combinat >>> s.dimension() # needs sage.combinat +Infinity
- set_order(order)[source]¶
Set the order of the elements of the basis.
If
set_order()
has not been called, then the ordering is the one used in the generation of the elements ofself
’s associated enumerated set.Warning
Many cached methods depend on this order, in particular for constructing subspaces and quotients. Changing the order after some computations have been cached does not invalidate the cache, and is likely to introduce inconsistencies.
EXAMPLES:
sage: # needs sage.combinat sage: QS2 = SymmetricGroupAlgebra(QQ,2) sage: b = list(QS2.basis().keys()) sage: b.reverse() sage: QS2.set_order(b) sage: QS2.get_order() [[2, 1], [1, 2]]
>>> from sage.all import * >>> # needs sage.combinat >>> QS2 = SymmetricGroupAlgebra(QQ,Integer(2)) >>> b = list(QS2.basis().keys()) >>> b.reverse() >>> QS2.set_order(b) >>> QS2.get_order() [[2, 1], [1, 2]]
- sum(iter_of_elements)[source]¶
Return the sum of all elements in
iter_of_elements
.Overrides method inherited from commutative additive monoid as it is much faster on dicts directly.
INPUT:
iter_of_elements
– iterator of elements ofself
EXAMPLES:
sage: F = CombinatorialFreeModule(QQ,[1,2]) sage: f = F.an_element(); f 2*B[1] + 2*B[2] sage: F.sum( f for _ in range(5) ) 10*B[1] + 10*B[2]
>>> from sage.all import * >>> F = CombinatorialFreeModule(QQ,[Integer(1),Integer(2)]) >>> f = F.an_element(); f 2*B[1] + 2*B[2] >>> F.sum( f for _ in range(Integer(5)) ) 10*B[1] + 10*B[2]
- sum_of_terms(terms, distinct=False)[source]¶
Construct a sum of terms of
self
.INPUT:
terms
– list (or iterable) of pairs(index, coeff)
distinct
– boolean (default:False
); whether the indices are guaranteed to be distinct
EXAMPLES:
sage: F = CombinatorialFreeModule(QQ, ['a', 'b', 'c']) sage: F.sum_of_terms([('a',2), ('c',3)]) 2*B['a'] + 3*B['c']
>>> from sage.all import * >>> F = CombinatorialFreeModule(QQ, ['a', 'b', 'c']) >>> F.sum_of_terms([('a',Integer(2)), ('c',Integer(3))]) 2*B['a'] + 3*B['c']
If
distinct
is True, then the construction is optimized:sage: F.sum_of_terms([('a',2), ('c',3)], distinct = True) 2*B['a'] + 3*B['c']
>>> from sage.all import * >>> F.sum_of_terms([('a',Integer(2)), ('c',Integer(3))], distinct = True) 2*B['a'] + 3*B['c']
Warning
Use
distinct=True
only if you are sure that the indices are indeed distinct:sage: F.sum_of_terms([('a',2), ('a',3)], distinct = True) 3*B['a']
>>> from sage.all import * >>> F.sum_of_terms([('a',Integer(2)), ('a',Integer(3))], distinct = True) 3*B['a']
Extreme case:
sage: F.sum_of_terms([]) 0
>>> from sage.all import * >>> F.sum_of_terms([]) 0
- term(index, coeff=None)[source]¶
Construct a term in
self
.INPUT:
index
– the index of a basis elementcoeff
– an element of the coefficient ring (default: one)
EXAMPLES:
sage: F = CombinatorialFreeModule(QQ, ['a', 'b', 'c']) sage: F.term('a',3) 3*B['a'] sage: F.term('a') B['a']
>>> from sage.all import * >>> F = CombinatorialFreeModule(QQ, ['a', 'b', 'c']) >>> F.term('a',Integer(3)) 3*B['a'] >>> F.term('a') B['a']
Design: should this do coercion on the coefficient ring?
- class sage.combinat.free_module.CombinatorialFreeModule_CartesianProduct(modules, **options)[source]¶
Bases:
CombinatorialFreeModule
An implementation of Cartesian products of modules with basis.
EXAMPLES:
We construct two free modules, assign them short names, and construct their Cartesian product:
sage: F = CombinatorialFreeModule(ZZ, [4,5]); F.rename('F') sage: G = CombinatorialFreeModule(ZZ, [4,6]); G.rename('G') sage: H = CombinatorialFreeModule(ZZ, [4,7]); H.rename('H') sage: S = cartesian_product([F, G]) sage: S F (+) G sage: S.basis() Lazy family (Term map from Disjoint union of Family ({4, 5}, {4, 6}) to F (+) G(i))_{i in Disjoint union of Family ({4, 5}, {4, 6})}
>>> from sage.all import * >>> F = CombinatorialFreeModule(ZZ, [Integer(4),Integer(5)]); F.rename('F') >>> G = CombinatorialFreeModule(ZZ, [Integer(4),Integer(6)]); G.rename('G') >>> H = CombinatorialFreeModule(ZZ, [Integer(4),Integer(7)]); H.rename('H') >>> S = cartesian_product([F, G]) >>> S F (+) G >>> S.basis() Lazy family (Term map from Disjoint union of Family ({4, 5}, {4, 6}) to F (+) G(i))_{i in Disjoint union of Family ({4, 5}, {4, 6})}
Note that the indices of the basis elements of F and G intersect non trivially. This is handled by forcing the union to be disjoint:
sage: list(S.basis()) [B[(0, 4)], B[(0, 5)], B[(1, 4)], B[(1, 6)]]
>>> from sage.all import * >>> list(S.basis()) [B[(0, 4)], B[(0, 5)], B[(1, 4)], B[(1, 6)]]
We now compute the Cartesian product of elements of free modules:
sage: f = F.monomial(4) + 2*F.monomial(5) sage: g = 2*G.monomial(4) + G.monomial(6) sage: h = H.monomial(4) + H.monomial(7) sage: cartesian_product([f, g]) B[(0, 4)] + 2*B[(0, 5)] + 2*B[(1, 4)] + B[(1, 6)] sage: cartesian_product([f, g, h]) B[(0, 4)] + 2*B[(0, 5)] + 2*B[(1, 4)] + B[(1, 6)] + B[(2, 4)] + B[(2, 7)] sage: cartesian_product([f, g, h]).parent() F (+) G (+) H
>>> from sage.all import * >>> f = F.monomial(Integer(4)) + Integer(2)*F.monomial(Integer(5)) >>> g = Integer(2)*G.monomial(Integer(4)) + G.monomial(Integer(6)) >>> h = H.monomial(Integer(4)) + H.monomial(Integer(7)) >>> cartesian_product([f, g]) B[(0, 4)] + 2*B[(0, 5)] + 2*B[(1, 4)] + B[(1, 6)] >>> cartesian_product([f, g, h]) B[(0, 4)] + 2*B[(0, 5)] + 2*B[(1, 4)] + B[(1, 6)] + B[(2, 4)] + B[(2, 7)] >>> cartesian_product([f, g, h]).parent() F (+) G (+) H
TODO: choose an appropriate semantic for Cartesian products of Cartesian products (associativity?):
sage: S = cartesian_product([cartesian_product([F, G]), H]) # todo: not implemented F (+) G (+) H
>>> from sage.all import * >>> S = cartesian_product([cartesian_product([F, G]), H]) # todo: not implemented F (+) G (+) H
- class Element[source]¶
Bases:
IndexedFreeModuleElement
- cartesian_embedding(i)[source]¶
Return the natural embedding morphism of the
i
-th Cartesian factor (summand) ofself
intoself
.INPUT:
i
– integer
EXAMPLES:
sage: F = CombinatorialFreeModule(ZZ, [4,5]); F.rename('F') sage: G = CombinatorialFreeModule(ZZ, [4,6]); G.rename('G') sage: S = cartesian_product([F, G]) sage: phi = S.cartesian_embedding(0) sage: phi(F.monomial(4) + 2 * F.monomial(5)) B[(0, 4)] + 2*B[(0, 5)] sage: phi(F.monomial(4) + 2 * F.monomial(6)).parent() == S True
>>> from sage.all import * >>> F = CombinatorialFreeModule(ZZ, [Integer(4),Integer(5)]); F.rename('F') >>> G = CombinatorialFreeModule(ZZ, [Integer(4),Integer(6)]); G.rename('G') >>> S = cartesian_product([F, G]) >>> phi = S.cartesian_embedding(Integer(0)) >>> phi(F.monomial(Integer(4)) + Integer(2) * F.monomial(Integer(5))) B[(0, 4)] + 2*B[(0, 5)] >>> phi(F.monomial(Integer(4)) + Integer(2) * F.monomial(Integer(6))).parent() == S True
- cartesian_factors()[source]¶
Return the factors of the Cartesian product.
EXAMPLES:
sage: F = CombinatorialFreeModule(ZZ, [4,5]); F.rename('F') sage: G = CombinatorialFreeModule(ZZ, [4,6]); G.rename('G') sage: S = cartesian_product([F, G]) sage: S.cartesian_factors() (F, G)
>>> from sage.all import * >>> F = CombinatorialFreeModule(ZZ, [Integer(4),Integer(5)]); F.rename('F') >>> G = CombinatorialFreeModule(ZZ, [Integer(4),Integer(6)]); G.rename('G') >>> S = cartesian_product([F, G]) >>> S.cartesian_factors() (F, G)
- cartesian_projection(i)[source]¶
Return the natural projection onto the \(i\)-th Cartesian factor (summand) of
self
.INPUT:
i
– integer
EXAMPLES:
sage: F = CombinatorialFreeModule(ZZ, [4,5]); F.rename('F') sage: G = CombinatorialFreeModule(ZZ, [4,6]); G.rename('G') sage: S = cartesian_product([F, G]) sage: x = S.monomial((0,4)) + 2 * S.monomial((0,5)) + 3 * S.monomial((1,6)) sage: S.cartesian_projection(0)(x) B[4] + 2*B[5] sage: S.cartesian_projection(1)(x) 3*B[6] sage: S.cartesian_projection(0)(x).parent() == F True sage: S.cartesian_projection(1)(x).parent() == G True
>>> from sage.all import * >>> F = CombinatorialFreeModule(ZZ, [Integer(4),Integer(5)]); F.rename('F') >>> G = CombinatorialFreeModule(ZZ, [Integer(4),Integer(6)]); G.rename('G') >>> S = cartesian_product([F, G]) >>> x = S.monomial((Integer(0),Integer(4))) + Integer(2) * S.monomial((Integer(0),Integer(5))) + Integer(3) * S.monomial((Integer(1),Integer(6))) >>> S.cartesian_projection(Integer(0))(x) B[4] + 2*B[5] >>> S.cartesian_projection(Integer(1))(x) 3*B[6] >>> S.cartesian_projection(Integer(0))(x).parent() == F True >>> S.cartesian_projection(Integer(1))(x).parent() == G True
- summand_embedding(i)[source]¶
Return the natural embedding morphism of the
i
-th Cartesian factor (summand) ofself
intoself
.INPUT:
i
– integer
EXAMPLES:
sage: F = CombinatorialFreeModule(ZZ, [4,5]); F.rename('F') sage: G = CombinatorialFreeModule(ZZ, [4,6]); G.rename('G') sage: S = cartesian_product([F, G]) sage: phi = S.cartesian_embedding(0) sage: phi(F.monomial(4) + 2 * F.monomial(5)) B[(0, 4)] + 2*B[(0, 5)] sage: phi(F.monomial(4) + 2 * F.monomial(6)).parent() == S True
>>> from sage.all import * >>> F = CombinatorialFreeModule(ZZ, [Integer(4),Integer(5)]); F.rename('F') >>> G = CombinatorialFreeModule(ZZ, [Integer(4),Integer(6)]); G.rename('G') >>> S = cartesian_product([F, G]) >>> phi = S.cartesian_embedding(Integer(0)) >>> phi(F.monomial(Integer(4)) + Integer(2) * F.monomial(Integer(5))) B[(0, 4)] + 2*B[(0, 5)] >>> phi(F.monomial(Integer(4)) + Integer(2) * F.monomial(Integer(6))).parent() == S True
- summand_projection(i)[source]¶
Return the natural projection onto the \(i\)-th Cartesian factor (summand) of
self
.INPUT:
i
– integer
EXAMPLES:
sage: F = CombinatorialFreeModule(ZZ, [4,5]); F.rename('F') sage: G = CombinatorialFreeModule(ZZ, [4,6]); G.rename('G') sage: S = cartesian_product([F, G]) sage: x = S.monomial((0,4)) + 2 * S.monomial((0,5)) + 3 * S.monomial((1,6)) sage: S.cartesian_projection(0)(x) B[4] + 2*B[5] sage: S.cartesian_projection(1)(x) 3*B[6] sage: S.cartesian_projection(0)(x).parent() == F True sage: S.cartesian_projection(1)(x).parent() == G True
>>> from sage.all import * >>> F = CombinatorialFreeModule(ZZ, [Integer(4),Integer(5)]); F.rename('F') >>> G = CombinatorialFreeModule(ZZ, [Integer(4),Integer(6)]); G.rename('G') >>> S = cartesian_product([F, G]) >>> x = S.monomial((Integer(0),Integer(4))) + Integer(2) * S.monomial((Integer(0),Integer(5))) + Integer(3) * S.monomial((Integer(1),Integer(6))) >>> S.cartesian_projection(Integer(0))(x) B[4] + 2*B[5] >>> S.cartesian_projection(Integer(1))(x) 3*B[6] >>> S.cartesian_projection(Integer(0))(x).parent() == F True >>> S.cartesian_projection(Integer(1))(x).parent() == G True
- class sage.combinat.free_module.CombinatorialFreeModule_Tensor(modules, **options)[source]¶
Bases:
CombinatorialFreeModule
Tensor Product of Free Modules.
EXAMPLES:
We construct two free modules, assign them short names, and construct their tensor product:
sage: F = CombinatorialFreeModule(ZZ, [1,2]); F.rename('F') sage: G = CombinatorialFreeModule(ZZ, [3,4]); G.rename('G') sage: T = tensor([F, G]); T F # G sage: T.category() Category of tensor products of finite dimensional modules with basis over Integer Ring sage: T.construction() # todo: not implemented [tensor, ]
>>> from sage.all import * >>> F = CombinatorialFreeModule(ZZ, [Integer(1),Integer(2)]); F.rename('F') >>> G = CombinatorialFreeModule(ZZ, [Integer(3),Integer(4)]); G.rename('G') >>> T = tensor([F, G]); T F # G >>> T.category() Category of tensor products of finite dimensional modules with basis over Integer Ring >>> T.construction() # todo: not implemented [tensor, ]
T is a free module, with same base ring as F and G:
sage: T.base_ring() Integer Ring
>>> from sage.all import * >>> T.base_ring() Integer Ring
The basis of T is indexed by tuples of basis indices of F and G:
sage: T.basis().keys() Image of Cartesian product of {1, 2}, {3, 4} by The map <class 'tuple'> from Cartesian product of {1, 2}, {3, 4} sage: T.basis().keys().list() [(1, 3), (1, 4), (2, 3), (2, 4)]
>>> from sage.all import * >>> T.basis().keys() Image of Cartesian product of {1, 2}, {3, 4} by The map <class 'tuple'> from Cartesian product of {1, 2}, {3, 4} >>> T.basis().keys().list() [(1, 3), (1, 4), (2, 3), (2, 4)]
FIXME: Should elements of a CartesianProduct be tuples (making them hashable)?
Here are the basis elements themselves:
sage: T.basis().cardinality() 4 sage: list(T.basis()) [B[1] # B[3], B[1] # B[4], B[2] # B[3], B[2] # B[4]]
>>> from sage.all import * >>> T.basis().cardinality() 4 >>> list(T.basis()) [B[1] # B[3], B[1] # B[4], B[2] # B[3], B[2] # B[4]]
The tensor product is associative and flattens sub tensor products:
sage: H = CombinatorialFreeModule(ZZ, [5,6]); H.rename('H') sage: tensor([F, tensor([G, H])]) F # G # H sage: tensor([tensor([F, G]), H]) F # G # H sage: tensor([F, G, H]) F # G # H
>>> from sage.all import * >>> H = CombinatorialFreeModule(ZZ, [Integer(5),Integer(6)]); H.rename('H') >>> tensor([F, tensor([G, H])]) F # G # H >>> tensor([tensor([F, G]), H]) F # G # H >>> tensor([F, G, H]) F # G # H
We now compute the tensor product of elements of free modules:
sage: f = F.monomial(1) + 2 * F.monomial(2) sage: g = 2*G.monomial(3) + G.monomial(4) sage: h = H.monomial(5) + H.monomial(6) sage: tensor([f, g]) 2*B[1] # B[3] + B[1] # B[4] + 4*B[2] # B[3] + 2*B[2] # B[4]
>>> from sage.all import * >>> f = F.monomial(Integer(1)) + Integer(2) * F.monomial(Integer(2)) >>> g = Integer(2)*G.monomial(Integer(3)) + G.monomial(Integer(4)) >>> h = H.monomial(Integer(5)) + H.monomial(Integer(6)) >>> tensor([f, g]) 2*B[1] # B[3] + B[1] # B[4] + 4*B[2] # B[3] + 2*B[2] # B[4]
Again, the tensor product is associative on elements:
sage: tensor([f, tensor([g, h])]) == tensor([f, g, h]) True sage: tensor([tensor([f, g]), h]) == tensor([f, g, h]) True
>>> from sage.all import * >>> tensor([f, tensor([g, h])]) == tensor([f, g, h]) True >>> tensor([tensor([f, g]), h]) == tensor([f, g, h]) True
Note further that the tensor product spaces need not preexist:
sage: t = tensor([f, g, h]) sage: t.parent() F # G # H
>>> from sage.all import * >>> t = tensor([f, g, h]) >>> t.parent() F # G # H
- tensor_constructor(modules)[source]¶
INPUT:
modules
– tuple \((F_1,\dots,F_n)\) of free modules whose tensor product is self
Returns the canonical multilinear morphism from \(F_1 \times \dots \times F_n\) to \(F_1 \otimes \dots \otimes F_n\)
EXAMPLES:
sage: F = CombinatorialFreeModule(ZZ, [1,2]); F.rename('F') sage: G = CombinatorialFreeModule(ZZ, [3,4]); G.rename('G') sage: H = CombinatorialFreeModule(ZZ, [5,6]); H.rename('H') sage: f = F.monomial(1) + 2*F.monomial(2) sage: g = 2*G.monomial(3) + G.monomial(4) sage: h = H.monomial(5) + H.monomial(6) sage: FG = tensor([F, G]) sage: phi_fg = FG.tensor_constructor((F, G)) sage: phi_fg(f, g) 2*B[1] # B[3] + B[1] # B[4] + 4*B[2] # B[3] + 2*B[2] # B[4] sage: FGH = tensor([F, G, H]) sage: phi_fgh = FGH.tensor_constructor((F, G, H)) sage: phi_fgh(f, g, h) 2*B[1] # B[3] # B[5] + 2*B[1] # B[3] # B[6] + B[1] # B[4] # B[5] + B[1] # B[4] # B[6] + 4*B[2] # B[3] # B[5] + 4*B[2] # B[3] # B[6] + 2*B[2] # B[4] # B[5] + 2*B[2] # B[4] # B[6] sage: phi_fg_h = FGH.tensor_constructor((FG, H)) sage: phi_fg_h(phi_fg(f, g), h) 2*B[1] # B[3] # B[5] + 2*B[1] # B[3] # B[6] + B[1] # B[4] # B[5] + B[1] # B[4] # B[6] + 4*B[2] # B[3] # B[5] + 4*B[2] # B[3] # B[6] + 2*B[2] # B[4] # B[5] + 2*B[2] # B[4] # B[6]
>>> from sage.all import * >>> F = CombinatorialFreeModule(ZZ, [Integer(1),Integer(2)]); F.rename('F') >>> G = CombinatorialFreeModule(ZZ, [Integer(3),Integer(4)]); G.rename('G') >>> H = CombinatorialFreeModule(ZZ, [Integer(5),Integer(6)]); H.rename('H') >>> f = F.monomial(Integer(1)) + Integer(2)*F.monomial(Integer(2)) >>> g = Integer(2)*G.monomial(Integer(3)) + G.monomial(Integer(4)) >>> h = H.monomial(Integer(5)) + H.monomial(Integer(6)) >>> FG = tensor([F, G]) >>> phi_fg = FG.tensor_constructor((F, G)) >>> phi_fg(f, g) 2*B[1] # B[3] + B[1] # B[4] + 4*B[2] # B[3] + 2*B[2] # B[4] >>> FGH = tensor([F, G, H]) >>> phi_fgh = FGH.tensor_constructor((F, G, H)) >>> phi_fgh(f, g, h) 2*B[1] # B[3] # B[5] + 2*B[1] # B[3] # B[6] + B[1] # B[4] # B[5] + B[1] # B[4] # B[6] + 4*B[2] # B[3] # B[5] + 4*B[2] # B[3] # B[6] + 2*B[2] # B[4] # B[5] + 2*B[2] # B[4] # B[6] >>> phi_fg_h = FGH.tensor_constructor((FG, H)) >>> phi_fg_h(phi_fg(f, g), h) 2*B[1] # B[3] # B[5] + 2*B[1] # B[3] # B[6] + B[1] # B[4] # B[5] + B[1] # B[4] # B[6] + 4*B[2] # B[3] # B[5] + 4*B[2] # B[3] # B[6] + 2*B[2] # B[4] # B[5] + 2*B[2] # B[4] # B[6]
- tensor_factors()[source]¶
Return the tensor factors of this tensor product.
EXAMPLES:
sage: F = CombinatorialFreeModule(ZZ, [1,2]) sage: F.rename('F') sage: G = CombinatorialFreeModule(ZZ, [3,4]) sage: G.rename('G') sage: T = tensor([F, G]); T F # G sage: T.tensor_factors() (F, G)
>>> from sage.all import * >>> F = CombinatorialFreeModule(ZZ, [Integer(1),Integer(2)]) >>> F.rename('F') >>> G = CombinatorialFreeModule(ZZ, [Integer(3),Integer(4)]) >>> G.rename('G') >>> T = tensor([F, G]); T F # G >>> T.tensor_factors() (F, G)