Implement down-up algebras and their Verma modules #35484
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There is a |
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If you have time, please check whether there are any connections to |
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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 |
| \\ du^2 & = q(q+1) udu - q^3 u^2d + r u, | ||
| \end{aligned} | ||
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| or `\alpha = q(q+1)`, `\beta = -q^3`, and `\gamma = r`. Specializing |
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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
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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
(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
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This is really cool! |
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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. |
fix the reference BR1998
fix a typo in the doc
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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! 🎉 |
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ok, let's go |
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Thank you! |
📚 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
⌛ Dependencies