Implement the abstract ring of multivariate polynomials, with several bases (Schur schubert, ...)

[nthiery]

We build an implantation of polynomials as formal sum of exponents. This allows to work on any number of variables with some easy morphism. We use some actions of Weyl group on the exponents to create operators (especially divided difference operators).

These operators allow us to build different bases that comes from geometry and have nice combinatorial description.

CC: @sagetrac-sage-combinat @VivianePons @tscrim @opechenik

Component: combinatorics

Keywords: multivariate polynomials, schubert polynomials, non symmetric polynomials, days54, fpsac2019

Author: Viviane Pons

Branch/Commit: public/combinat/polynomial_bases-6629 @ d6e1201

Issue created by migration from https://trac.sagemath.org/ticket/6629

[VivianePons]

Author: Viviane Pons

[VivianePons]

Description changed:

--- 
+++ 
@@ -1 +1,3 @@
-See: http://wiki.sagemath.org/combinat/MultivariatePolynomials
+We build an implantation of polynomials as formal sum of exponents. This allows to work on any number of variables with some easy morphism. We use some actions of Weyl group on the exponents to create operators (especially divided difference operators). 
+
+These operators allow us to build different bases that comes from geometry and have nice combinatorial description.

[VivianePons]

comment:2

the patch apply on 5.6 and is mostly working. It is not yet ready for review though, it needs more tests and docs and possibly a few fixes.

[jeffpferreira]

comment:3

What is the status of this ticket? Is it still in development?

[VivianePons]

comment:4

Yes, sorry for taking such a long time but this is a big implementation and I work on it only when I have time (which is not so often). My plan is to finish a first version of it on the upcoming Sage days in Paris and find someone to work on the review. In the meantime, the patch is working already and some people use it.

[VivianePons]

comment:5

Attachment: trac_6629-multivariate-polynomials-on-several-bases-vp.patch.gz

Just attached the last version of the patch to apply on Sage 5.9

[VivianePons]

Commit: 066a1b9

[VivianePons]

New commits:

[changeset:066a1b9]	typo fix
[changeset:2149df9]	#6629 Implement the abstract ring of multivariate polynomials, with several bases (monomials, Schur schubert, Demazure ...)

[VivianePons]

Branch: public/combinat/6629-Schubert-Grothendieck-Key-polynomials

[tscrim]

Changed keywords from multivariate polynomials, schubert polynomials, non symmetric polynomials to multivariate polynomials, schubert polynomials, non symmetric polynomials, days54

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Changed commit from 066a1b9 to e011845

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Branch pushed to git repo; I updated commit sha1. New commits:

e011845

Merge branch 'master' into 6629-Schubert-Grothendieck-Key-polynomials

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Branch pushed to git repo; I updated commit sha1. New commits:

933a033

Merge remote-tracking branch 'origin/develop' into polynomials

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Changed commit from e011845 to 933a033

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Branch pushed to git repo; I updated commit sha1. New commits:

b92a96e

Merge branch 'public/combinat/6629-Schubert-Grothendieck-Key-polynomials' of trac.sagemath.org:sage into public/combinat/6629-Schubert-Grothendieck-Key-polynomials

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Changed commit from 933a033 to b92a96e

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Changed commit from b92a96e to 0271f67

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Branch pushed to git repo; I updated commit sha1. New commits:

b70d426	Merge branch 'public/combinat/6629-Schubert-Grothendieck-Key-polynomials' of trac.sagemath.org:sage into public/combinat/6629-Schubert-Grothendieck-Key-polynomials
0271f67	Trying to make things work again. Not there yet.

[tscrim]

comment:17

Viviane, when you get a chance, could you push the code that you currently have or are you still working on this?

[VivianePons]

Changed branch from public/combinat/6629-Schubert-Grothendieck-Key-polynomials to public/combiat/6629-new-version

[VivianePons]

Changed commit from 0271f67 to 46b6841

[VivianePons]

comment:18

Just pushed my latest code. Mostly everything is working but the tests and examples are all messed up (and, for some reason, I still cannot run any tests).

I have simplified many things, removing useless classes and so on. Now, I consider the monomial basis to be a "generator", I call it x and write:

sage: A.<x> = MultivariatePolynomialAlgebra(QQ)
sage: x[1,2,3] + x[2,3]
x[2, 3, 0] + x[1, 2, 3]

I think it makes more sense and it is more natural to use. The goal is to hide the inner mechanics as much as possible. "New example" are written in this manner, old ones still use the m notation.

All basis are combinatorial free module using a common "key wrapper" class that handles lists and ambient space basis elements.

In terms of mathematical meaning, remember that all bases are indexed by vectors, for Grothendieck and Schubert, they correspond to the Lehmer code of the permutation.

The only part that is not working is the "Double polynomial" one, but it's quite independent from the rest.

I let you have a look at it, it's not finished yet. Feel free to change it if you feel like it. I will be on holidays for the next 2 weeks.

---

New commits:

e0d9682	Merge remote-tracking branch 'origin/public/combinat/6629-Schubert-Grothendieck-Key-polynomials' into local-6629
d9c033a	Work in progress: simplifying the structure, cleaning stuff, writting docs
1db580d	Refactoring / commenting, Work in progress
7805a1e	Merge branch 'develop' into local-6629-wip
43d86b0	Merge branch 'local-6629-wip' into local-6629-wip2
a40f3a8	small fixes
f777a78	Merge branch 'local-6629-wip2' into local-6629-7.3
46b6841	Fixing some broken things

[tscrim]

comment:19

Thank you Viviane, I appreciate it. I will take a look at it soon and pass it along to the REU students. Enjoy your holidays!

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Branch pushed to git repo; I updated commit sha1. New commits:

f7094ec	Merge branch 'develop' into t/6629/public/combiat/6629-new-version
6ae9767	some documentation typos

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Changed commit from 46b6841 to 6ae9767

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Changed commit from 6ae9767 to 8189b01

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Branch pushed to git repo; I updated commit sha1. New commits:

b709b8b	got Schuberts mostly working again
8189b01	fixed things

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Changed commit from 8189b01 to fb5b566

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Branch pushed to git repo; I updated commit sha1. New commits:

fb5b566

fixed broken an_element method

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Branch pushed to git repo; I updated commit sha1. New commits:

5c955e2

somehow finite_basis is failing to pass along the basis keys, trying to debug

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Changed commit from fb5b566 to 5c955e2

[17b1eba5-ac2a-4aa9-b51b-fa077a4f050d]

Changed keywords from multivariate polynomials, schubert polynomials, non symmetric polynomials, days54 to multivariate polynomials, schubert polynomials, non symmetric polynomials, days54, fpsac2019

[tscrim]

Changed branch from public/combiat/6629-new-version to public/combinat/polynomial_bases-6629

[tscrim]

Changed commit from 5c955e2 to d6e1201

[tscrim]

comment:26

A version that works on Python3.

---

New commits:

1c1fdf1	Merge branch 'public/combiat/6629-new-version' of git://trac.sagemath.org/sage into public/combiat/6629-new-version
d6e1201	Convert code to Python3.