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Implement down-up algebras and their Verma modules #35484

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merged 8 commits into from
May 22, 2023

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tscrim
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@tscrim tscrim commented Apr 12, 2023

📚 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.

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@fchapoton
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There is a BR1988 instead of the correct BR1998

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

@tscrim
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tscrim commented Apr 18, 2023

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.

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

or `\alpha = q(q+1)`, `\beta = -q^3`, and `\gamma = r`. Specializing
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@mantepse mantepse Apr 18, 2023

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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 $DU(0,1,2)$?

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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.

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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)

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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..."

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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
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This is really cool!

@tscrim
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tscrim commented Apr 18, 2023

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.

@tscrim tscrim force-pushed the algebras/down_up_algebras branch from 39d58f8 to c962e4f Compare May 8, 2023 03:05
@fchapoton
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  • 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
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tscrim commented May 11, 2023

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).

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

@fchapoton
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ok, let's go

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ok

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tscrim commented May 13, 2023

Thank you!

@vbraun vbraun merged commit c3ed171 into sagemath:develop May 22, 2023
@mkoeppe mkoeppe added this to the sage-10.1 milestone May 22, 2023
@tscrim tscrim deleted the algebras/down_up_algebras branch May 23, 2023 00:31
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5 participants