Filtered Modules With Basis#

A filtered module with basis over a ring \(R\) means (for the purpose of this code) a filtered \(R\)-module \(M\) with filtration \((F_i)_{i \in I}\) (typically \(I = \NN\)) endowed with a basis \((b_j)_{j \in J}\) of \(M\) and a partition \(J = \bigsqcup_{i \in I} J_i\) of the set \(J\) (it is allowed that some \(J_i\) are empty) such that for every \(n \in I\), the subfamily \((b_j)_{j \in U_n}\), where \(U_n = \bigcup_{i \leq n} J_i\), is a basis of the \(R\)-submodule \(F_n\).

See the class documentation FilteredModulesWithBasis for further details.

class sage.categories.filtered_modules_with_basis.FilteredModulesWithBasis(base_category)#

Bases: FilteredModulesCategory

The category of filtered modules with a distinguished basis.

For every \(i \in I\), the \(R\)-submodule of \(M\) spanned by \((b_j)_{j \in J_i}\) is called the \(i\)-th graded component (aka the \(i\)-th homogeneous component) of the filtered module with basis \(M\); the elements of this submodule are referred to as homogeneous elements of degree \(i\). The \(R\)-module \(M\) is the direct sum of its \(i\)-th graded components over all \(i \in I\), and thus becomes a graded \(R\)-module with basis. Conversely, any graded \(R\)-module with basis canonically becomes a filtered \(R\)-module with basis (by defining \(F_n = \bigoplus_{i \leq n} G_i\) where \(G_i\) is the \(i\)-th graded component, and defining \(J_i\) as the indexing set of the basis of the \(i\)-th graded component). Hence, the notion of a filtered \(R\)-module with basis is equivalent to the notion of a graded \(R\)-module with basis.

However, the category of filtered \(R\)-modules with basis is not the category of graded \(R\)-modules with basis. Indeed, the morphisms of filtered \(R\)-modules with basis are defined to be morphisms of \(R\)-modules which send each \(F_n\) of the domain to the corresponding \(F_n\) of the target; in contrast, the morphisms of graded \(R\)-modules with basis must preserve each homogeneous component. Also, the notion of a filtered algebra with basis differs from that of a graded algebra with basis.

Note

Currently, to make use of the functionality of this class, an instance of FilteredModulesWithBasis should fulfill the contract of a CombinatorialFreeModule (most likely by inheriting from it). It should also have the indexing set \(J\) encoded as its _indices attribute, and _indices.subset(size=i) should yield the subset \(J_i\) (as an iterable). If the latter conditions are not satisfied, then basis() must be overridden.

Note

One should implement a degree_on_basis method in the parent class in order to fully utilize the methods of this category. This might become a required abstract method in the future.

EXAMPLES:

sage: C = ModulesWithBasis(ZZ).Filtered(); C
Category of filtered modules with basis over Integer Ring
sage: sorted(C.super_categories(), key=str)
[Category of filtered modules over Integer Ring,
 Category of modules with basis over Integer Ring]
sage: C is ModulesWithBasis(ZZ).Filtered()
True

class ElementMethods#

Bases: object

degree()#

The degree of a nonzero homogeneous element self in the filtered module.

Note

This raises an error if the element is not homogeneous. To compute the maximum of the degrees of the homogeneous summands of a (not necessarily homogeneous) element, use maximal_degree() instead.

EXAMPLES:

sage: # needs sage.combinat sage.modules
sage: A = ModulesWithBasis(ZZ).Filtered().example()
sage: x = A(Partition((3,2,1)))
sage: y = A(Partition((4,4,1)))
sage: z = A(Partition((2,2,2)))
sage: x.degree()
6
sage: (x + 2*z).degree()
6
sage: (y - x).degree()
Traceback (most recent call last):
...
ValueError: element is not homogeneous

An example in a graded algebra:

sage: # needs sage.combinat sage.modules
sage: S = NonCommutativeSymmetricFunctions(QQ).S()
sage: (x, y) = (S[2], S[3])
sage: x.homogeneous_degree()
2
sage: (x^3 + 4*y^2).homogeneous_degree()
6
sage: ((1 + x)^3).homogeneous_degree()
Traceback (most recent call last):
...
ValueError: element is not homogeneous

Let us now test a filtered algebra (but remember that the notion of homogeneity now depends on the choice of a basis):

sage: A = AlgebrasWithBasis(QQ).Filtered().example()
sage: x,y,z = A.algebra_generators()
sage: (x*y).homogeneous_degree()
2
sage: (y*x).homogeneous_degree()
Traceback (most recent call last):
...
ValueError: element is not homogeneous
sage: A.one().homogeneous_degree()
0

degree_on_basis(m)#

Return the degree of the basis element indexed by m in self.

EXAMPLES:

sage: A = GradedModulesWithBasis(QQ).example()                          # needs sage.combinat sage.modules
sage: A.degree_on_basis(Partition((2,1)))                               # needs sage.combinat sage.modules
3
sage: A.degree_on_basis(Partition((4,2,1,1,1,1)))                       # needs sage.combinat sage.modules
10

homogeneous_component(n)#

Return the homogeneous component of degree n of the element self.

Let \(m\) be an element of a filtered \(R\)-module \(M\) with basis. Then, \(m\) can be uniquely written in the form \(m = \sum_{i \in I} m_i\), where each \(m_i\) is a homogeneous element of degree \(i\). For \(n \in I\), we define the homogeneous component of degree \(n\) of the element \(m\) to be \(m_n\).

EXAMPLES:

sage: # needs sage.combinat sage.modules
sage: A = ModulesWithBasis(ZZ).Filtered().example()
sage: x = A.an_element(); x
2*P[] + 2*P[1] + 3*P[2]
sage: x.homogeneous_component(-1)
0
sage: x.homogeneous_component(0)
2*P[]
sage: x.homogeneous_component(1)
2*P[1]
sage: x.homogeneous_component(2)
3*P[2]
sage: x.homogeneous_component(3)
0

sage: # needs sage.combinat sage.modules
sage: A = ModulesWithBasis(ZZ).Graded().example()
sage: x = A.an_element(); x
2*P[] + 2*P[1] + 3*P[2]
sage: x.homogeneous_component(-1)
0
sage: x.homogeneous_component(0)
2*P[]
sage: x.homogeneous_component(1)
2*P[1]
sage: x.homogeneous_component(2)
3*P[2]
sage: x.homogeneous_component(3)
0

sage: A = AlgebrasWithBasis(ZZ).Filtered().example()
sage: G = A.algebra_generators()
sage: g = A.an_element() - 2 * G['x'] * G['y']; g
U['x']^2*U['y']^2*U['z']^3 - 2*U['x']*U['y']
 + 2*U['x'] + 3*U['y'] + 1
sage: g.homogeneous_component(-1)
0
sage: g.homogeneous_component(0)
1
sage: g.homogeneous_component(2)
-2*U['x']*U['y']
sage: g.homogeneous_component(5)
0
sage: g.homogeneous_component(7)
U['x']^2*U['y']^2*U['z']^3
sage: g.homogeneous_component(8)
0

homogeneous_degree()#

The degree of a nonzero homogeneous element self in the filtered module.

Note

This raises an error if the element is not homogeneous. To compute the maximum of the degrees of the homogeneous summands of a (not necessarily homogeneous) element, use maximal_degree() instead.

EXAMPLES:

sage: # needs sage.combinat sage.modules
sage: A = ModulesWithBasis(ZZ).Filtered().example()
sage: x = A(Partition((3,2,1)))
sage: y = A(Partition((4,4,1)))
sage: z = A(Partition((2,2,2)))
sage: x.degree()
6
sage: (x + 2*z).degree()
6
sage: (y - x).degree()
Traceback (most recent call last):
...
ValueError: element is not homogeneous

An example in a graded algebra:

sage: # needs sage.combinat sage.modules
sage: S = NonCommutativeSymmetricFunctions(QQ).S()
sage: (x, y) = (S[2], S[3])
sage: x.homogeneous_degree()
2
sage: (x^3 + 4*y^2).homogeneous_degree()
6
sage: ((1 + x)^3).homogeneous_degree()
Traceback (most recent call last):
...
ValueError: element is not homogeneous

Let us now test a filtered algebra (but remember that the notion of homogeneity now depends on the choice of a basis):

sage: A = AlgebrasWithBasis(QQ).Filtered().example()
sage: x,y,z = A.algebra_generators()
sage: (x*y).homogeneous_degree()
2
sage: (y*x).homogeneous_degree()
Traceback (most recent call last):
...
ValueError: element is not homogeneous
sage: A.one().homogeneous_degree()
0

is_homogeneous()#

Return whether the element self is homogeneous.

EXAMPLES:

sage: # needs sage.combinat sage.modules
sage: A = ModulesWithBasis(ZZ).Filtered().example()
sage: x = A(Partition((3,2,1)))
sage: y = A(Partition((4,4,1)))
sage: z = A(Partition((2,2,2)))
sage: (3*x).is_homogeneous()
True
sage: (x - y).is_homogeneous()
False
sage: (x+2*z).is_homogeneous()
True

Here is an example with a graded algebra:

sage: # needs sage.combinat sage.modules
sage: S = NonCommutativeSymmetricFunctions(QQ).S()
sage: (x, y) = (S[2], S[3])
sage: (3*x).is_homogeneous()
True
sage: (x^3 - y^2).is_homogeneous()
True
sage: ((x + y)^2).is_homogeneous()
False

Let us now test a filtered algebra (but remember that the notion of homogeneity now depends on the choice of a basis, or at least on a definition of homogeneous components):

sage: A = AlgebrasWithBasis(QQ).Filtered().example()
sage: x,y,z = A.algebra_generators()
sage: (x*y).is_homogeneous()
True
sage: (y*x).is_homogeneous()
False
sage: A.one().is_homogeneous()
True
sage: A.zero().is_homogeneous()
True
sage: (A.one()+x).is_homogeneous()
False

maximal_degree()#

The maximum of the degrees of the homogeneous components of self.

This is also the smallest \(i\) such that self belongs to \(F_i\). Hence, it does not depend on the basis of the parent of self.

See also

homogeneous_degree()

EXAMPLES:

sage: # needs sage.combinat sage.modules
sage: A = ModulesWithBasis(ZZ).Filtered().example()
sage: x = A(Partition((3,2,1)))
sage: y = A(Partition((4,4,1)))
sage: z = A(Partition((2,2,2)))
sage: x.maximal_degree()
6
sage: (x + 2*z).maximal_degree()
6
sage: (y - x).maximal_degree()
9
sage: (3*z).maximal_degree()
6

Now, we test this on a graded algebra:

sage: # needs sage.combinat sage.modules
sage: S = NonCommutativeSymmetricFunctions(QQ).S()
sage: (x, y) = (S[2], S[3])
sage: x.maximal_degree()
2
sage: (x^3 + 4*y^2).maximal_degree()
6
sage: ((1 + x)^3).maximal_degree()
6

Let us now test a filtered algebra:

sage: # needs sage.combinat sage.modules
sage: A = AlgebrasWithBasis(QQ).Filtered().example()
sage: x,y,z = A.algebra_generators()
sage: (x*y).maximal_degree()
2
sage: (y*x).maximal_degree()
2
sage: A.one().maximal_degree()
0
sage: A.zero().maximal_degree()
Traceback (most recent call last):
...
ValueError: the zero element does not have a well-defined degree
sage: (A.one()+x).maximal_degree()
1

truncate(n)#

Return the sum of the homogeneous components of degree strictly less than n of self.

See homogeneous_component() for the notion of a homogeneous component.

EXAMPLES:

sage: # needs sage.combinat sage.modules
sage: A = ModulesWithBasis(ZZ).Filtered().example()
sage: x = A.an_element(); x
2*P[] + 2*P[1] + 3*P[2]
sage: x.truncate(0)
0
sage: x.truncate(1)
2*P[]
sage: x.truncate(2)
2*P[] + 2*P[1]
sage: x.truncate(3)
2*P[] + 2*P[1] + 3*P[2]

sage: # needs sage.combinat sage.modules
sage: A = ModulesWithBasis(ZZ).Graded().example()
sage: x = A.an_element(); x
2*P[] + 2*P[1] + 3*P[2]
sage: x.truncate(0)
0
sage: x.truncate(1)
2*P[]
sage: x.truncate(2)
2*P[] + 2*P[1]
sage: x.truncate(3)
2*P[] + 2*P[1] + 3*P[2]

sage: A = AlgebrasWithBasis(ZZ).Filtered().example()
sage: G = A.algebra_generators()
sage: g = A.an_element() - 2 * G['x'] * G['y']; g
U['x']^2*U['y']^2*U['z']^3 - 2*U['x']*U['y']
 + 2*U['x'] + 3*U['y'] + 1
sage: g.truncate(-1)
0
sage: g.truncate(0)
0
sage: g.truncate(2)
2*U['x'] + 3*U['y'] + 1
sage: g.truncate(3)
-2*U['x']*U['y'] + 2*U['x'] + 3*U['y'] + 1
sage: g.truncate(5)
-2*U['x']*U['y'] + 2*U['x'] + 3*U['y'] + 1
sage: g.truncate(7)
-2*U['x']*U['y'] + 2*U['x'] + 3*U['y'] + 1
sage: g.truncate(8)
U['x']^2*U['y']^2*U['z']^3 - 2*U['x']*U['y']
 + 2*U['x'] + 3*U['y'] + 1

class FiniteDimensional(base_category)#

Bases: CategoryWithAxiom_over_base_ring

class ParentMethods#

Bases: object

hilbert_series(prec=None)#

Return the Hilbert series of self as a polynomial.

Let \(R\) be a commutative ring (with unit). Let \(M = \bigcup_{n=0}^{\infty} M_n\) be a filtered \(R\)-module. The Hilbert series of \(M\) is the formal power series

\[H(t) = \sum_{n=0}^{\infty} \ell(M_n / M_{n-1}) t^n,\]

where \(\ell(N)\) is the length of \(N\), which is the longest chain of submodules (over \(R\)), and by convention \(M_{-1} = \{0\}\). By the assumptions of the category, \(M_n / M_{n-1}\) is a free \(R\)-module, and so \(\ell(M_n / M_{n-1})\) is equal to the rank of \(M_n / M_{n-1}\).

EXAMPLES:

sage: OS = hyperplane_arrangements.braid(3).orlik_solomon_algebra(QQ)
sage: OS.hilbert_series()
2*t^2 + 3*t + 1

sage: OS = matroids.Uniform(5, 3).orlik_solomon_algebra(ZZ)
sage: OS.hilbert_series()
t^3 + 3*t^2 + 3*t + 1

sage: OS = matroids.PG(2, 3).orlik_solomon_algebra(ZZ['x','y'])
sage: OS.hilbert_series()
27*t^3 + 39*t^2 + 13*t + 1

class ParentMethods#

Bases: object

basis(d=None)#

Return the basis for (the d-th homogeneous component of) self.

INPUT:

d – (optional, default None) nonnegative integer or None

OUTPUT:

If d is None, returns the basis of the module. Otherwise, returns the basis of the homogeneous component of degree d (i.e., the subfamily of the basis of the whole module which consists only of the basis vectors lying in \(F_d \setminus \bigcup_{i<d} F_i\)).

The basis is always returned as a family.

EXAMPLES:

sage: A = ModulesWithBasis(ZZ).Filtered().example()
sage: A.basis(4)
Lazy family (Term map from Partitions to An example of a
 filtered module with basis: the free module on partitions
 over Integer Ring(i))_{i in Partitions of the integer 4}

Without arguments, the full basis is returned:

sage: A.basis()
Lazy family (Term map from Partitions to An example of a
 filtered module with basis: the free module on partitions
 over Integer Ring(i))_{i in Partitions}
sage: A.basis()
Lazy family (Term map from Partitions to An example of a
 filtered module with basis: the free module on partitions
 over Integer Ring(i))_{i in Partitions}

Checking this method on a filtered algebra. Note that this will typically raise a NotImplementedError when this feature is not implemented.

sage: A = AlgebrasWithBasis(ZZ).Filtered().example()
sage: A.basis(4)
Traceback (most recent call last):
...
NotImplementedError: infinite set

Without arguments, the full basis is returned:

sage: A.basis()
Lazy family (Term map from Free abelian monoid indexed by
 {'x', 'y', 'z'} to An example of a filtered algebra with
 basis: the universal enveloping algebra of Lie algebra
 of RR^3 with cross product over Integer Ring(i))_{i in
 Free abelian monoid indexed by {'x', 'y', 'z'}}

An example with a graded algebra:

sage: E.<x,y> = ExteriorAlgebra(QQ)
sage: E.basis()
Lazy family (Term map from Subsets of {0,1} to
 The exterior algebra of rank 2 over Rational Field(i))_{i in
 Subsets of {0,1}}

from_graded_conversion()#

Return the inverse of the canonical \(R\)-module isomorphism \(A \to \operatorname{gr} A\) induced by the basis of \(A\) (where \(A = ` ``self`\)). This inverse is an isomorphism \(\operatorname{gr} A \to A\).

This is an isomorphism of \(R\)-modules. See the class documentation AssociatedGradedAlgebra.

See also

to_graded_conversion()

EXAMPLES:

sage: # needs sage.combinat sage.modules
sage: A = Modules(QQ).WithBasis().Filtered().example()
sage: p = -2 * A.an_element(); p
-4*P[] - 4*P[1] - 6*P[2]
sage: q = A.to_graded_conversion()(p); q
-4*Bbar[[]] - 4*Bbar[[1]] - 6*Bbar[[2]]
sage: A.from_graded_conversion()(q) == p
True
sage: q.parent() is A.graded_algebra()
True

graded_algebra()#

Return the associated graded module to self.

See AssociatedGradedAlgebra for the definition and the properties of this.

If the filtered module self with basis is called \(A\), then this method returns \(\operatorname{gr} A\). The method to_graded_conversion() returns the canonical \(R\)-module isomorphism \(A \to \operatorname{gr} A\) induced by the basis of \(A\), and the method from_graded_conversion() returns the inverse of this isomorphism. The method projection() projects elements of \(A\) onto \(\operatorname{gr} A\) according to their place in the filtration on \(A\).

Warning

When not overridden, this method returns the default implementation of an associated graded module – namely, AssociatedGradedAlgebra(self), where AssociatedGradedAlgebra is AssociatedGradedAlgebra. But some instances of FilteredModulesWithBasis override this method, as the associated graded module often is (isomorphic) to a simpler object (for instance, the associated graded module of a graded module can be identified with the graded module itself). Generic code that uses associated graded modules (such as the code of the induced_graded_map() method below) should make sure to only communicate with them via the to_graded_conversion(), from_graded_conversion() and projection() methods (in particular, do not expect there to be a conversion from self to self.graded_algebra(); this currently does not work for Clifford algebras). Similarly, when overriding graded_algebra(), make sure to accordingly redefine these three methods, unless their definitions below still apply to your case (this will happen whenever the basis of your graded_algebra() has the same indexing set as self, and the partition of this indexing set according to degree is the same as for self).

EXAMPLES:

sage: A = ModulesWithBasis(ZZ).Filtered().example()
sage: A.graded_algebra()
Graded Module of An example of a filtered module with basis:
 the free module on partitions over Integer Ring

hilbert_series(prec=None)#

Return the Hilbert series of self.

Let \(R\) be a commutative ring (with unit). Let \(M = \bigcup_{n=0}^{\infty} M_n\) be a filtered \(R\)-module. The Hilbert series of \(M\) is the formal power series

\[H(t) = \sum_{n=0}^{\infty} \ell(M_n / M_{n-1}) t^n,\]

INPUT:

prec – (default: \(\infty\)) the precision

OUTPUT:

If the precision is finite, then this returns an element in the PowerSeriesRing over ZZ. Otherwise it returns an element in the LazyPowerSeriesRing over ZZ.

EXAMPLES:

sage: A = GradedModulesWithBasis(ZZ).example()
sage: A.hilbert_series()
1 + t + 2*t^2 + 3*t^3 + 5*t^4 + 7*t^5 + 11*t^6 + O(t^7)
sage: A.hilbert_series(10)
1 + t + 2*t^2 + 3*t^3 + 5*t^4 + 7*t^5 + 11*t^6 + 15*t^7 + 22*t^8 + 30*t^9 + O(t^10)

homogeneous_component(d)#

Return the d-th homogeneous component of self.

EXAMPLES:

sage: A = GradedModulesWithBasis(ZZ).example()
sage: A.homogeneous_component(4)
Degree 4 homogeneous component of An example of a graded module
 with basis: the free module on partitions over Integer Ring

homogeneous_component_basis(d)#

Return a basis for the d-th homogeneous component of self.

EXAMPLES:

sage: A = GradedModulesWithBasis(ZZ).example()
sage: A.homogeneous_component_basis(4)
Lazy family (Term map
 from Partitions
   to An example of a graded module with basis: the free module
      on partitions over Integer Ring(i))_{i in Partitions of the integer 4}

sage: # needs sage.modules
sage: cat = GradedModulesWithBasis(ZZ)
sage: C = CombinatorialFreeModule(ZZ, ['a', 'b'], category=cat)
sage: C.degree_on_basis = lambda x: 1 if x == 'a' else 2
sage: C.homogeneous_component_basis(1)
Finite family {'a': B['a']}
sage: C.homogeneous_component_basis(2)
Finite family {'b': B['b']}

induced_graded_map(other, f)#

Return the graded linear map between the associated graded modules of self and other canonically induced by the filtration-preserving map f : self -> other.

Let \(A\) and \(B\) be two filtered modules with basis, and let \((F_i)_{i \in I}\) and \((G_i)_{i \in I}\) be their filtrations. Let \(f : A \to B\) be a linear map which preserves the filtration (i.e., satisfies \(f(F_i) \subseteq G_i\) for all \(i \in I\)). Then, there is a canonically defined graded linear map \(\operatorname{gr} f : \operatorname{gr} A \to \operatorname{gr} B\) which satisfies

\[(\operatorname{gr} f) (p_i(a)) = p_i(f(a)) \qquad \text{for all } i \in I \text{ and } a \in F_i ,\]

where the \(p_i\) on the left hand side is the canonical projection from \(F_i\) onto the \(i\)-th graded component of \(\operatorname{gr} A\), while the \(p_i\) on the right hand side is the canonical projection from \(G_i\) onto the \(i\)-th graded component of \(\operatorname{gr} B\).

INPUT:

other – a filtered algebra with basis
f – a filtration-preserving linear map from self to other (can be given as a morphism or as a function)

OUTPUT:

The graded linear map \(\operatorname{gr} f\).

EXAMPLES:

Example 1.

We start with the free \(\QQ\)-module with basis the set of all partitions:

sage: A = Modules(QQ).WithBasis().Filtered().example(); A               # needs sage.combinat sage.modules
An example of a filtered module with basis: the free module
 on partitions over Rational Field
sage: M = A.indices(); M                                                # needs sage.combinat sage.modules
Partitions
sage: p1, p2, p21, p321 = [A.basis()[Partition(i)]                      # needs sage.combinat sage.modules
....:                      for i in [[1], [2], [2,1], [3,2,1]]]

Let us define a map from A to itself which acts on the basis by sending every partition \(\lambda\) to the sum of the conjugates of all partitions \(\mu\) for which \(\lambda / \mu\) is a horizontal strip:

sage: # needs sage.combinat sage.modules
sage: def map_on_basis(lam):
....:     def mus(k):
....:         return lam.remove_horizontal_border_strip(k)
....:     return A.sum_of_monomials([Partition(mu).conjugate()
....:                                for k in range(sum(lam) + 1)
....:                                for mu in mus(k)])
sage: f = A.module_morphism(on_basis=map_on_basis,
....:                       codomain=A)
sage: f(p1)
P[] + P[1]
sage: f(p2)
P[] + P[1] + P[1, 1]
sage: f(p21)
P[1] + P[1, 1] + P[2] + P[2, 1]
sage: f(p21 - p1)
-P[] + P[1, 1] + P[2] + P[2, 1]
sage: f(p321)
P[2, 1] + P[2, 1, 1] + P[2, 2] + P[2, 2, 1]
 + P[3, 1] + P[3, 1, 1] + P[3, 2] + P[3, 2, 1]

We now compute \(\operatorname{gr} f\)

sage: # needs sage.combinat sage.modules
sage: grA = A.graded_algebra(); grA
Graded Module of An example of a filtered module with basis:
 the free module on partitions over Rational Field
sage: pp1, pp2, pp21, pp321 = [A.to_graded_conversion()(i)
....:                          for i in [p1, p2, p21, p321]]
sage: pp2 + 4 * pp21
Bbar[[2]] + 4*Bbar[[2, 1]]
sage: grf = A.induced_graded_map(A, f); grf
Generic endomorphism of Graded Module of
 An example of a filtered module with basis:
  the free module on partitions over Rational Field
sage: grf(pp1)
Bbar[[1]]
sage: grf(pp2 + 4 * pp21)
Bbar[[1, 1]] + 4*Bbar[[2, 1]]

Example 2.

We shall now construct \(\operatorname{gr} f\) for a different map \(f\) out of the same A; the new map \(f\) will lead into a graded algebra already, namely into the algebra of symmetric functions:

sage: # needs sage.combinat sage.modules
sage: h = SymmetricFunctions(QQ).h()
sage: def map_on_basis(lam):  # redefining map_on_basis
....:     def mus(k):
....:         return lam.remove_horizontal_border_strip(k)
....:     return h.sum_of_monomials([Partition(mu).conjugate()
....:                                for k in range(sum(lam) + 1)
....:                                for mu in mus(k)])
sage: f = A.module_morphism(on_basis=map_on_basis,
....:                       codomain=h)  # redefining f
sage: f(p1)
h[] + h[1]
sage: f(p2)
h[] + h[1] + h[1, 1]
sage: f(A.zero())
0
sage: f(p2 - 3*p1)
-2*h[] - 2*h[1] + h[1, 1]

The algebra h of symmetric functions in the \(h\)-basis is already graded, so its associated graded algebra is implemented as itself:

sage: # needs sage.combinat sage.modules
sage: grh = h.graded_algebra(); grh is h
True
sage: grf = A.induced_graded_map(h, f); grf
Generic morphism:
  From: Graded Module of An example of a filtered
   module with basis: the free module on partitions
   over Rational Field
  To:   Symmetric Functions over Rational Field
   in the homogeneous basis
sage: grf(pp1)
h[1]
sage: grf(pp2)
h[1, 1]
sage: grf(pp321)
h[3, 2, 1]
sage: grf(pp2 - 3*pp1)
-3*h[1] + h[1, 1]
sage: grf(pp21)
h[2, 1]
sage: grf(grA.zero())
0

Example 3.

After having had a graded module as the codomain, let us try to have one as the domain instead. Our new f will go from h to A:

sage: # needs sage.combinat sage.modules
sage: def map_on_basis(lam):  # redefining map_on_basis
....:     def mus(k):
....:         return lam.remove_horizontal_border_strip(k)
....:     return A.sum_of_monomials([Partition(mu).conjugate()
....:                                for k in range(sum(lam) + 1)
....:                                for mu in mus(k)])
sage: f = h.module_morphism(on_basis=map_on_basis,
....:                       codomain=A)  # redefining f
sage: f(h[1])
P[] + P[1]
sage: f(h[2])
P[] + P[1] + P[1, 1]
sage: f(h[1, 1])
P[1] + P[2]
sage: f(h[2, 2])
P[1, 1] + P[2, 1] + P[2, 2]
sage: f(h[3, 2, 1])
P[2, 1] + P[2, 1, 1] + P[2, 2] + P[2, 2, 1]
 + P[3, 1] + P[3, 1, 1] + P[3, 2] + P[3, 2, 1]
sage: f(h.one())
P[]
sage: grf = h.induced_graded_map(A, f); grf
Generic morphism:
  From: Symmetric Functions over Rational Field
   in the homogeneous basis
  To:   Graded Module of An example of a filtered
   module with basis: the free module on partitions
   over Rational Field
sage: grf(h[1])
Bbar[[1]]
sage: grf(h[2])
Bbar[[1, 1]]
sage: grf(h[1, 1])
Bbar[[2]]
sage: grf(h[2, 2])
Bbar[[2, 2]]
sage: grf(h[3, 2, 1])
Bbar[[3, 2, 1]]
sage: grf(h.one())
Bbar[[]]

Example 4.

The construct \(\operatorname{gr} f\) also makes sense when \(f\) is a filtration-preserving map between graded modules.

sage: # needs sage.combinat sage.modules
sage: def map_on_basis(lam):  # redefining map_on_basis
....:     def mus(k):
....:         return lam.remove_horizontal_border_strip(k)
....:     return h.sum_of_monomials([Partition(mu).conjugate()
....:                                for k in range(sum(lam) + 1)
....:                                for mu in mus(k)])
sage: f = h.module_morphism(on_basis=map_on_basis,
....:                       codomain=h)  # redefining f
sage: f(h[1])
h[] + h[1]
sage: f(h[2])
h[] + h[1] + h[1, 1]
sage: f(h[1, 1])
h[1] + h[2]
sage: f(h[2, 1])
h[1] + h[1, 1] + h[2] + h[2, 1]
sage: f(h.one())
h[]
sage: grf = h.induced_graded_map(h, f); grf
Generic endomorphism of
 Symmetric Functions over Rational Field in the homogeneous basis
sage: grf(h[1])
h[1]
sage: grf(h[2])
h[1, 1]
sage: grf(h[1, 1])
h[2]
sage: grf(h[2, 1])
h[2, 1]
sage: grf(h.one())
h[]

projection(i)#

Return the \(i\)-th projection \(p_i : F_i \to G_i\) (in the notations of the class documentation AssociatedGradedAlgebra, where \(A = ` ``self`\)).

This method actually does not return the map \(p_i\) itself, but an extension of \(p_i\) to the whole \(R\)-module \(A\). This extension is the composition of the \(R\)-module isomorphism \(A \to \operatorname{gr} A\) with the canonical projection of the graded \(R\)-module \(\operatorname{gr} A\) onto its \(i\)-th graded component \(G_i\). The codomain of this map is \(\operatorname{gr} A\), although its actual image is \(G_i\). The map \(p_i\) is obtained from this map by restricting its domain to \(F_i\) and its image to \(G_i\).

EXAMPLES:

sage: # needs sage.combinat sage.modules
sage: A = Modules(ZZ).WithBasis().Filtered().example()
sage: p = -2 * A.an_element(); p
-4*P[] - 4*P[1] - 6*P[2]
sage: q = A.projection(2)(p); q
-6*Bbar[[2]]
sage: q.parent() is A.graded_algebra()
True
sage: A.projection(3)(p)
0

to_graded_conversion()#

Return the canonical \(R\)-module isomorphism \(A \to \operatorname{gr} A\) induced by the basis of \(A\) (where \(A = ` ``self`\)).

This is an isomorphism of \(R\)-modules. See the class documentation AssociatedGradedAlgebra.

See also

from_graded_conversion()

EXAMPLES:

sage: # needs sage.combinat sage.modules
sage: A = Modules(QQ).WithBasis().Filtered().example()
sage: p = -2 * A.an_element(); p
-4*P[] - 4*P[1] - 6*P[2]
sage: q = A.to_graded_conversion()(p); q
-4*Bbar[[]] - 4*Bbar[[1]] - 6*Bbar[[2]]
sage: q.parent() is A.graded_algebra()
True

class Subobjects(category, *args)#

Bases: SubobjectsCategory

class ElementMethods#

Bases: object

degree()#

Return the degree of self.

EXAMPLES:

sage: # needs sage.modules
sage: E.<x,y,z> = ExteriorAlgebra(QQ)
sage: S = E.submodule([x + y, x*y - y*z, y])
sage: B = S.basis()
sage: [B[0].lift(), B[1].lift(), B[2].lift()]
[x, y, x*y - y*z]
sage: B[0].degree()
1
sage: B[1].degree()
1
sage: (B[0] + 3*B[1]).degree()
1

The degree of inhomogeneous elements is not defined (following the behavior of the exterior algebra):

sage: (B[0] + B[2]).degree()                                        # needs sage.modules
Traceback (most recent call last):
...
ValueError: element is not homogeneous

We can still get the maximal degree:

sage: (B[0] + B[2]).maximal_degree()                                # needs sage.modules
2

maximal_degree()#

The maximum of the degrees of the homogeneous components of self.

This is also the smallest \(i\) such that self belongs to \(F_i\). Hence, it does not depend on the basis of the parent of self.

See also

homogeneous_degree()

EXAMPLES:

sage: # needs sage.modules
sage: E.<x,y,z> = ExteriorAlgebra(QQ)
sage: F = E.submodule([x + 1, x*y - 1])
sage: B = F.basis()
sage: [B[0].lift(), B[1].lift()]
[-x*y + 1, x*y + x]
sage: B[0].maximal_degree()
2
sage: B[1].maximal_degree()
2

class ParentMethods#

Bases: object

degree_on_basis(m)#

Return the degree of the basis element indexed by m in self.

EXAMPLES:

sage: # needs sage.modules
sage: E.<x,y,z> = ExteriorAlgebra(QQ)
sage: S = E.submodule([x + y, x*y - y*z, y])
sage: B = S.basis()
sage: [B[0].lift(), B[1].lift(), B[2].lift()]
[x, y, x*y - y*z]
sage: S.degree_on_basis(0)
1
sage: S.degree_on_basis(1)
1
sage: S.degree_on_basis(2)
2