Implement Hopf algebras on ordered multiset partitions

alauve commented 6 years ago

The free (noncommutative) algebra generated by finite nonempty sets is given a Hopf algebra structure, with coproduct on finite sets K given by \Delta(K) = \sum_{I \sqcup J = K} I \otimes J.

This Hopf algebra gives formulas for primitives and antipode in other (combinatorial) Hopf algebras via "transfer of structure" as described in [LM2018]. This transfer was already implemented (in the special case of NCSYM) via :meth:`

It would be nice to implement this transfer for other cases. Getting this Hopf algebra and its dual within SageMath would be a good first step.

REFERENCE:

[LM2018] Lauve, Mastnak. Bialgebra coverings and transfer of structure. :arxiv:1803.02691.

Depends on #25148

CC: @tscrim @darijgr @zabrocki @alauve @saliola @amypang

Component: combinatorics

Keywords: IMA coding sprint, CHAs, ordered multiset partitions

Author: Aaron Lauve

Branch/Commit: public/combinat/implement_hopf_algebra_on_OMPs-25543 @ 5b8cab9

Issue created by migration from https://trac.sagemath.org/ticket/25543

alauve commented 6 years ago

Branch: public/combinat/implement_hopf_algebra_on_OMPs-25543

7ed8c4ca-6d56-4ae9-953a-41e42b4ed313 commented 6 years ago

Commit: 85d2d72

7ed8c4ca-6d56-4ae9-953a-41e42b4ed313 commented 6 years ago

Branch pushed to git repo; I updated commit sha1. Last 10 new commits:

`b144c19`	`removing AssertionError calls`
`dad7e7b`	`moving from Set to frozenset; zapping doctest errors that creeped in from earlier changes`
`306b3fe`	modified subset method to behave the way expected from `__init__` method in CombinatorialFreeModule
`c68b518`	`updated minimaj crystal reference`
`4af474f`	`mostly adding doc tests and examples; improved a few methods`
`89284be`	`finished adding tests/examples for all functions; but found some errors to be corrected (see new TODO)`
`830f5ae`	zapping bugs within `__init__` handling of constraints
`d3e375a`	`create module for the Hopf algebras on ordered multiset partitions`
`d18f780`	`revamped tests for satisfying constraints`
`85d2d72`	`Merge branch 'public/combinat/implement-ordered-multiset-partitions-25148' into public/combinat/implement_hopf_algebra_on_OMPs-25543`

7ed8c4ca-6d56-4ae9-953a-41e42b4ed313 commented 6 years ago

Changed commit from 85d2d72 to 8e5aa52

7ed8c4ca-6d56-4ae9-953a-41e42b4ed313 commented 6 years ago

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

`c7dff31`	`implement the dual of the Hopf algebra on ordered multiset partitions`
`659132f`	`fix merge conflicts in references`
`8e5aa52`	`Merge branch 'public/combinat/implement-ordered-multiset-partitions-25148' into public/combinat/implement_hopf_algebra_on_OMPs-25543`

7ed8c4ca-6d56-4ae9-953a-41e42b4ed313 commented 6 years ago

Changed commit from 8e5aa52 to 0ec37d7

7ed8c4ca-6d56-4ae9-953a-41e42b4ed313 commented 6 years ago

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

0ec37d7 work on _antipode; added temporary test code at EOF

7ed8c4ca-6d56-4ae9-953a-41e42b4ed313 commented 6 years ago

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

bd3a383 importing cartesian_product

7ed8c4ca-6d56-4ae9-953a-41e42b4ed313 commented 6 years ago

Changed commit from 0ec37d7 to bd3a383

7ed8c4ca-6d56-4ae9-953a-41e42b4ed313 commented 6 years ago

Changed commit from bd3a383 to 9942bf2

7ed8c4ca-6d56-4ae9-953a-41e42b4ed313 commented 6 years ago

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

`4598426`	`modified .split(), which was ignoring repeats`
`ad3a3e7`	`Merge branch 'public/combinat/implement-ordered-multiset-partitions-25148' into public/combinat/implement_hopf_algebra_on_OMPs-25543`
`9942bf2`	`improved antipode on H basis; moved duality_pairing; other small changes`

7ed8c4ca-6d56-4ae9-953a-41e42b4ed313 commented 5 years ago

Changed commit from 9942bf2 to b112d43

7ed8c4ca-6d56-4ae9-953a-41e42b4ed313 commented 5 years ago

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

b112d43 Merge branch 'public/combinat/implement-ordered-multiset-partitions-25148' into public/combinat/implement_hopf_algebra_on_OMPs-25543

fchapoton commented 5 years ago

comment:8

Please use python3 syntax for print (and for the rest also)

https://wiki.sagemath.org/Python3-compatible%20code

7ed8c4ca-6d56-4ae9-953a-41e42b4ed313 commented 5 years ago

Changed commit from b112d43 to 8184fb8

7ed8c4ca-6d56-4ae9-953a-41e42b4ed313 commented 5 years ago

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

`c43a926`	`Merge branch 'develop' into public/combinat/implement_hopf_algebra_on_OMPs-25543`
`8184fb8`	`refactored some code, and other large changes`

7ed8c4ca-6d56-4ae9-953a-41e42b4ed313 commented 5 years ago

Changed commit from 8184fb8 to fb35251

7ed8c4ca-6d56-4ae9-953a-41e42b4ed313 commented 5 years ago

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

fb35251 a bit of cleaning up

alauve commented 5 years ago

comment:11

N.B. I still have a lot more work to do in this file. I will give an alert when it's ready for a closer look.

7ed8c4ca-6d56-4ae9-953a-41e42b4ed313 commented 5 years ago

Changed commit from fb35251 to 1a66fcd

7ed8c4ca-6d56-4ae9-953a-41e42b4ed313 commented 5 years ago

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

1a66fcd implementing powersum basis and its dual

7ed8c4ca-6d56-4ae9-953a-41e42b4ed313 commented 5 years ago

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

`fe2042b`	`add .finer_pred; improve .fatter`
`76d95fc`	`add ribbon/fundamental bases; clean up docstrings`
`f7196a0`	`cleaning up docstrings and tests`

7ed8c4ca-6d56-4ae9-953a-41e42b4ed313 commented 5 years ago

Changed commit from 1a66fcd to f7196a0

alauve commented 5 years ago

comment:14

Just a remark that this ticket is still not ready for a close look. (Nearly ready, but there are still a few big errors to track down, as well as some coercions/morphisms to add.)

7ed8c4ca-6d56-4ae9-953a-41e42b4ed313 commented 5 years ago

Changed commit from f7196a0 to 77e123c

7ed8c4ca-6d56-4ae9-953a-41e42b4ed313 commented 5 years ago

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

`1c3ca3d`	`add antipode formulas, where obvious`
`77e123c`	`added maps to sym, nsym; left speculative maps as TODO comments`

alauve commented 5 years ago

comment:16

I still have to track down some degree issues, but I think the ticket merits a closer look now.

tscrim commented 5 years ago

comment:17

Some more superficial comments:

I would rename OMPBasis_OMPNSym to OMPNSymBasis_abstract (at least, that is my understanding of its role).
Remove the period here: around which the primitive is built.

This is overindented:

    .. TODO::

        - sort out the degree issues above. (search for "wrong!!!")

For this:
```
.. TODO: Do we need these tests, or are they checked within TestSuite?::
```
the answer is yes, they are checked by the _test_antipode unless I am misremembering something.
.. MATH:: needs a blankline before it (see coproduct_on_basis).
I would put the TestSuite tests in the __init__ method rather than the class-level doc.
*Warning:* -> .. WARNING::.
The __classcall_private__ will need doctests.

alauve commented 5 years ago

comment:18

Thanks, Travis! Extremely helpful, as usual.

7ed8c4ca-6d56-4ae9-953a-41e42b4ed313 commented 5 years ago

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

`19375ac`	`fixing/improving docstrings`
`5b8cab9`	`added combinatorial_character and character_polynomial methods`

7ed8c4ca-6d56-4ae9-953a-41e42b4ed313 commented 5 years ago

Changed commit from 77e123c to 5b8cab9

sagemath / sage

Implement Hopf algebras on ordered multiset partitions #25543