Implement down-up algebras and their Verma modules

[tscrim]

📚 Description

Down-up algebras arose from certain operators on posets. In particular, this is a generalization of the algebra of those operators for $r$-differential posets. They have a PBW-type basis and corresponding Verma modules with a triangular-type decomposition. We provide an implementation of these.

📝 Checklist

• The title is concise, informative, and self-explanatory.
• The description explains in detail what this PR is about.
• I have linked a relevant issue or discussion.
• I have created tests covering the changes.
• I have updated the documentation accordingly.

⌛ Dependencies

[fchapoton]

There is a BR1988 instead of the correct BR1998

[mantepse]

If you have time, please check whether there are any connections to growth.py that make sense to be written down!

[tscrim]

Thanks @fchapoton for the fixes.

@mantepse I added an example and discussion about the relationship with differential posets. This was all I could figure out connecting this with growth diagrams. There might be more to be said, but I don't know where to look for it.

Additionally, I decided to change the degree() to weight() for Verma modules since it is not graded as a DU-module (which could be added later). This also reflects the terminology used.

[mantepse]

replacing "or" with "Put differently, we set" (i.e., starting a new sentence and making it clear, that we are just saying the same thing differently) would make it possibly easier to read. (Another possibility:

    For a `(q,r)`-differential poset,
    we have `\alpha = q(q+1)`, `\beta = -q^3`, and `\gamma = r`, or, explicitly

    .. MATH::

        \begin{aligned}
        d^2u & = q(q+1) dud - q^3 ud^2 + r d,
        \\ du^2 & = q(q+1) udu - q^3 u^2d + r u.
        \end{aligned}

Another thing: are you saying that the Weyl relation is satisfied in $DU(0,1,2)$?

[tscrim]

Will change.

Another thing: are you saying that the Weyl relation is satisfied in $DU(0,1,2)$?

Yes, and you can explicitly see that in the doctest on Young's lattice. The proof of this was given in Benkart-Roby.

[mantepse]

Are you sure it's not the other way round? I can see that if I have $du-ud=rI$ I have a down-up algebra $DU(0,1,2)$, but not conversely.

(sorry I cannot check properly right now, I didn't even get to the other ticket yet)

[tscrim]

I thought I copied it correctly from Benkart-Roby. Assuming I did, then there might be a minor misprint in the paper (swapping what d and u mean in the Weyl algebra). I also did not check carefully. I will do so tomorrow morning.

(No problem; I am going to write my longer response now, must later than the "tomorrow" I had promised...)Perhaps a slightly better phrasing would be "...and it affords a representation..."

[tscrim]

Indeed, there was a problem. The Weyl algebra is supposed to only be a quotient of $DU(0, 1, 2)$. I have updated the doc accordingly.

[mantepse]

This is really cool!

[tscrim]

Thanks. This was something that came up in something else I was looking at (although it turned out to be not so useful). However, it was quick to implement and seemed like a nice addition further connecting Lie algebras and classical partition/tableaux combinatorics.

[fchapoton]

• please use https in the header for the link to the GNU license
• codecov is not totally happy in the new file, some tests should be added to fix that

[tscrim]

Done and should be done (although I am not fully convinced the codecov is smart enough to detect it, nor that it is that useful to have tests for every line of code).

[github-actions]

Documentation preview for this PR is ready! 🎉
Built with commit: a932b10

[fchapoton]

ok, let's go

[fchapoton]

ok

[tscrim]

Thank you!