#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, ...
-----------------------------+----------------------------------------------
Reporter: nborie | Owner: nborie
Type: enhancement | Status: needs_review
Priority: major | Milestone: sage-5.0
Component: combinatorics | Keywords: polynomial, infinite, Schubert,
category
Author: | Upstream: N/A
Reviewer: | Merged:
Work_issues: |
-----------------------------+----------------------------------------------
Description changed by nborie:
Old 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...)
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...)
Will be followed by the Adrien's work on #6629
--
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Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/8581#comment:4>
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