refresh the shuffle algebra file

[fchapoton]

fixing a very wrong coproduct

enhancing the category

some pep8 changes

CC: @tscrim @jm58660

Component: combinatorics

Author: Frédéric Chapoton, Travis Scrimshaw

Branch/Commit: 3787e45

Reviewer: Travis Scrimshaw, Frédéric Chapoton

Issue created by migration from https://trac.sagemath.org/ticket/28881

[fchapoton]

Commit: 3631eea

[fchapoton]

Branch: u/chapoton/28881

[fchapoton]

New commits:

3631eea

refreshing the shuffle algebra file

[tscrim]

comment:2

Well, technically the shuffle algebra should be in the category of commutative Hopf algebras, so it might be better to use the category of Bialgebras(R).Commutative(). However, the antipode is easy to implement: reverse the order of the n letters times (-1)ⁿ (see this MathOverflow post). So I would propose changing the category to HopfAlgebras(R).Commutative() and adding

def antipode_on_basis(self, w):
    """
    Return the antipode on the basis element ``w``.
    """
    mone = -self.base_ring().one()
    return self.term(w.reversal(), mone**len(w))

along with in the dual PBW basis

def antipode(self, elt):
    """
    Return the antipode of the element ``elt``.
    """
    return self(self.expansion(elt).antipode())

[sagetrac-git]

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

7056cc1

more work on shuffle Hopf algebra

[sagetrac-git]

Changed commit from 3631eea to 7056cc1

[fchapoton]

comment:4

mmm; not so easy. I am hitting problems in the testsuites.

[tscrim]

comment:5

Hmm...maybe the product and coproduct implemented are not compatible? I will check it on Monday (Australian time).

[sagetrac-git]

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

0722049

some more care for shuffle counit

[sagetrac-git]

Changed commit from 7056cc1 to 0722049

[sagetrac-git]

Changed commit from 0722049 to 065860a

[sagetrac-git]

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

065860a

fix coproduct for shuffle algebra

[fchapoton]

comment:8

The coproduct was indeed broken. There still remains a failure..

[tscrim]

comment:9

It almost seems like the old coproduct is meant to be the inflation product. (The coproduct was added in #15212, but there is little information available there.)

[tscrim]

comment:10

So the problem seems to come from the coproduct in the dual PBW basis. It is only converting basis indices over (whereas the implementation was meant to be treating them as monomials). Here is a fix that doesn't try to be clever and just uses the coercion framework.

[tscrim]

Changed commit from 065860a to abccb82

[tscrim]

Reviewer: Travis Scrimshaw, Frédéric Chapoton

[tscrim]

Changed author from Frédéric Chapoton to Frédéric Chapoton, Travis Scrimshaw

[tscrim]

comment:11

I fixed it, and I also made the shuffle algebra know it is a graded connected Hopf algebra.

---

New commits:

d29de54	Fixing dual PBW coproduct and adding grading.
9d8dcee	A few other little PEP8 tweaks.
abccb82	Added connected to the category.

[tscrim]

Changed branch from u/chapoton/28881 to public/algebras/refresh_shuffle_algebras-28881

[sagetrac-git]

Changed commit from abccb82 to 3787e45

[sagetrac-git]

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

3787e45

trac 28881 fixing the pyflakes warning

[fchapoton]

comment:13

Thanks a lot. Looks good now. I have fixed a pyflakes warning, and added some #long tags on the TestSuites.

Now launching the bot.

[tscrim]

comment:14

Worked locally for me, so I am going to assume the patchbot comes back green. Thank you.

[vbraun]

Changed branch from public/algebras/refresh_shuffle_algebras-28881 to 3787e45