Graded modules with basis¶
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class
sage.categories.graded_modules_with_basis.
GradedModulesWithBasis
(base_category)¶ Bases:
sage.categories.graded_modules.GradedModulesCategory
The category of graded modules with a distinguished basis.
EXAMPLES:
sage: C = GradedModulesWithBasis(ZZ); C Category of graded modules with basis over Integer Ring sage: sorted(C.super_categories(), key=str) [Category of filtered modules with basis over Integer Ring, Category of graded modules over Integer Ring] sage: C is ModulesWithBasis(ZZ).Graded() True
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class
ElementMethods
¶ Bases:
object
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degree_negation
()¶ Return the image of
self
under the degree negation automorphism of the graded module to whichself
belongs.The degree negation is the module automorphism which scales every homogeneous element of degree \(k\) by \((-1)^k\) (for all \(k\)). This assumes that the module to which
self
belongs (that is, the moduleself.parent()
) is \(\ZZ\)-graded.EXAMPLES:
sage: E.<a,b> = ExteriorAlgebra(QQ) sage: ((1 + a) * (1 + b)).degree_negation() a*b - a - b + 1 sage: E.zero().degree_negation() 0 sage: P = GradedModulesWithBasis(ZZ).example(); P An example of a graded module with basis: the free module on partitions over Integer Ring sage: pbp = lambda x: P.basis()[Partition(list(x))] sage: p = pbp([3,1]) - 2 * pbp([2]) + 4 * pbp([1]) sage: p.degree_negation() -4*P[1] - 2*P[2] + P[3, 1]
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class
ParentMethods
¶ Bases:
object
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degree_negation
(element)¶ Return the image of
element
under the degree negation automorphism of the graded moduleself
.The degree negation is the module automorphism which scales every homogeneous element of degree \(k\) by \((-1)^k\) (for all \(k\)). This assumes that the module
self
is \(\ZZ\)-graded.INPUT:
element
– element of the moduleself
EXAMPLES:
sage: E.<a,b> = ExteriorAlgebra(QQ) sage: E.degree_negation((1 + a) * (1 + b)) a*b - a - b + 1 sage: E.degree_negation(E.zero()) 0 sage: P = GradedModulesWithBasis(ZZ).example(); P An example of a graded module with basis: the free module on partitions over Integer Ring sage: pbp = lambda x: P.basis()[Partition(list(x))] sage: p = pbp([3,1]) - 2 * pbp([2]) + 4 * pbp([1]) sage: P.degree_negation(p) -4*P[1] - 2*P[2] + P[3, 1]
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class