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I'm pretty sure many a mathematician has longed for such a tool but I wasn't able to find such a question here, so here we go.

Is there, by any chance, an existing package or program that allows one to conveniently carry out explicit computations in finite-dimensional irreps of classical Lie algebras in terms of the action of the universal enveloping algebra?

To be more specific, the task I have at hand right now is to check whether the set of vectors obtained from the HWV by the action of certain elements of the UEA comprises a basis in the representation.

As far as I can tell, the software I've been able to find (such as LiE and LieART) does not provide such functionality.

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    $\begingroup$ This is a reasonable question, but keep in mind that software often gets developed and then abandoned. It's essential to formulate your question as precisely as possible. You might get some advice from the Dutch mathematician Willlem de Graaf (email degraaf [at] science.unitn.it) or possibly from some of the people involved with the Lie groups Atlas project: liegroups.org $\endgroup$ Jun 9, 2016 at 15:56
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    $\begingroup$ I have such a package, written in mathematica, for quantum enveloping algebras. It is essentially undocumented, except for an ancient outline. The package itself is part of the KnotTheory package available from katlas.org. I'm happy to answer questions by email. If the representations are at all large (dimension > 30, say) it starts to struggle. $\endgroup$ Jun 9, 2016 at 16:38
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    $\begingroup$ To complement what Jim Humphreys says about abandoned software, the (somewhat related to this question) packages for computations with Hecke algebras and Coxeter groups in GAP (named chevie) has not been updated for version 4, so one needs to run it on a outdated version of GAP. $\endgroup$ Jun 10, 2016 at 6:39
  • $\begingroup$ I realise this might not be suitable for you, as a paid-for piece of software so depending on your/your institution's access, but have you had a look at Magma? I think it should have most of the tools necessary to do what you would like, though you might well have to do some work to join them together in the right way. The documentation is at least freely available on the internet, so it's possible to get some idea without having bought it. The Lie algebras bit is here: magma.maths.usyd.edu.au/magma/handbook/lie_algebras $\endgroup$ Jun 13, 2016 at 10:49

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