Restructuring Diagram/Partition Algebras to match category structure

[ghseeli]

Currently, the Partition/Diagram Algebra implementations in Sage need to be redone. This problem was identified at Sage Days 38. The documentation is not very clear on how it should be used, and although it is supposed to be an algebra, it does not follow the standard form for algebras in Sage (most likely because these algebras have not been modified since 2007.)

This attached program seeks to provide an alternate implementation for, and eventually replace once dependencies are resolved, the existing PartionAlgebra package. More detail about these specific algebras can be found in a 2005 paper by Halverson and Ram titled "Partition Algebras." This new implementation restructures the Partition/Diagram Algebras to use the category structure in Sage, so that they are actually implemented as Algebras_with_basis. The new implementation also provides much more detailed documentation on how to use the Partition Algebras, what they actually are, and provides an easier and more standard usage pattern (inherited from CombinatorialFreeModule.)

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Apply:

• attachment: trac_14234-folded-ts.patch

Depends on #10630

CC: @alauve @srdoty @saliola

Component: algebra

Keywords: Partition algebra, diagram algebra, days38, days40

Author: Stephen Doty, Aaron Lauve, George H. Seelinger

Reviewer: Travis Scrimshaw, Darij Grinberg

Merged: sage-5.12.beta3

Issue created by migration from https://trac.sagemath.org/ticket/14234

[ghseeli]

Changed keywords from Partition algebra, diagram algebra to Partition algebra, diagram algebra, days38, days40

[ghseeli]

comment:1

Attachment: 17710.patch.gz

[tscrim]

comment:2

This patch has been useful to some of my colleagues, and I would like to see it get into sage.

Currently I couldn't see how we could display the diagrams pictorially in latex, and I think it would be really cool if we could have the latex version of the elements as sums of the pictorial diagrams. There seemed to be a few functions missing some doctests. Let me know when this gets ready for a formal review. Thanks.

[ghseeli]

comment:3

I have a second revision on my computer that I am adding the finishing touches to. I will try to upload it in the following week. I have some ideas for pictorial representations using tikz, but the standard networkx and other drawing libraries are not powerful enough to make meaningful diagrams the same way sage does with other graphs. I wrote some code at one point in networkx to display them, but this ended up with issues when trying to create edges along the rows of diagrams, such as an edge between vertex 1 and vertex 3 (it looked the same as having edges {1,2},{2,3}). I will be sure to let you know when I need some formal review and I am glad your colleagues have found this patch useful. Thank you!

[tscrim]

comment:4

Even if the diagrams are not too clean pictorially in tikz, as long as the output is moderately easily parseable and the connectivity is there, I think that will be good. Anyways, just let me know when it's ready. Thanks.

[ghseeli]

comment:5

Attachment: trac_14232_latexDrawing.patch.gz

I have added a new patch that I believe may need to be applied after the previous one has been. However, this patch should now allow for the generation of LaTeX code using TikZ to create diagrams for individual, basis diagrams. I have not changed any of the tests, however, so any tests that failed before will probably still fail.

[tscrim]

comment:6

How close is this to being ready review? Thanks.

[ghseeli]

comment:7

I think the patch is ready for review. I'm sorry it took so long! Please let me know if more changes need to be made.

[tscrim]

comment:8

I've set the ticket to needs_review and I'll write a review patch. I would like to know if there is a better name than IdealAlgebra since that will likely cause some confusion (perhaps something that does not begin with Ideal such as PartitionIdealAlgebra)? I'll take care of this in the review patch as well. Also, you'll need to fill in your real names as the authors.

Thank you,

Travis

PS - No need to worry about the time delay. Thanks for getting this done.

[tscrim]

Reviewer: Travis Scrimshaw