#8581: Implement an uniform set for index infinite variable algebraic structure,
implement polynomial ring in infinite set of indeterminate with categories,
implement the Schubert base ring in y1, y2, y3, ...
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Reporter: nborie | Owner: nborie
Type: task | Status: needs_review
Priority: major | Milestone: sage-5.0
Component: combinatorics | Keywords: polynomial, infinite, Schubert,
category
Author: | Upstream: N/A
Reviewer: | Merged:
Work_issues: |
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Changes (by nborie):
* type: enhancement => task
Old description:
New description:
This patch Implement a set with categories to index monomials of infinite
indeterminate algebraic structure. There is 4 versions of this set : power
of indeterminate can be negative or only positive and there is a dense and
sparse implementation of the set (list/dictionary).
From this set, there is an example of graded_algebras_with_basis which is
the polynomial ring in infinite indeterminate.
The Schubert base ring inherit from this example and changing 3 thing:
* the name of the ring
* the name of variable (y1, y2, y3, ...) for Schubert
* A __call __ method for element which correspond to the specialization
With this patch, one can do for example:
{{{
sage: from sage.combinat.multivariate_polynomials.schubert_base_ring
import SchubertBaseRing
sage: from sage.categories.examples.graded_algebras_with_basis import
PolynomialRingInfiniteIndeterminate
sage: S = SchubertBaseRing(ZZ); S
Base ring for Schubbert polynomials in the variables y1, y2, y3, ... over
Integer Ring
sage: A = PolynomialRingInfiniteIndeterminate(S); A
An example of graded algebra with basis: the polynomial ring in infinite
indeterminate over Base ring for Schubbert polynomials in the variables
y1, y2, y3, ... over Integer Ring
sage: A.base_ring().an_element()*A.an_element()
(1+2*y1+y1*y3^2*y4^3+3*y1^2) + (2+4*y1+2*y1*y3^2*y4^3+6*y1^2)*X1 +
(1+2*y1+y1*y3^2*y4^3+3*y1^2)*X1*X3^2*X4^3 +
(3+6*y1+3*y1*y3^2*y4^3+9*y1^2)*X1^2
}}}
This built the ambient space for Schubert polynomials which are
polynomials in two infinite alphabet of indeterminate indexed by
PositiveIntegers() (or NonNegativeIntegers()... setting this is easy...)
--
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Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/8581#comment:2>
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