Blog on HomogeneousTools

Representations with the same degree

Thu, 04 Jun 2026 00:00:00 +0000

Time for the next installment of the arXiv series. Today’s preprint is from earlier this year, and it’s Frank Lübeck: Representations with the same degree. In fact, whilst preparing this blogpost, I discovered that this question was also already addressed by Andy Huchala in his 2018 undergraduate thesis. In what follows, I will (mostly) refer to the arXiv preprint.

As before, we will use Semisimple.jl to play with the constructions in the paper.

Verifying Kostant's conjecture using Semisimple.jl

Fri, 29 May 2026 00:00:00 +0000

Here is a blogpost that is the first installment of something I would like to try.

Looking at today’s arXiv feed, I noticed a new and interesting preprint: Rekha Biswal, Sam Jeralds: Components of $\mathrm{V}(m\rho)\otimes\mathrm{V}(n\rho)$), for which Semisimple.jl can be used to experiment and verify the conjectures in some explicit cases. Because I think it is useful to see how to do such experiments, let us quickly discuss how we can verify Kostant’s conjecture using Semisimple.jl.

Lie.jl is now Semisimple.jl

Tue, 12 May 2026 00:00:00 +0000

The Julia package previously known as Lie.jl has been renamed to Semisimple.jl. This was needed, because the Julia registry rejected the name Lie.jl as being too short. I doubt many people have started using the package, I just have some renaming to do in the (not yet public) downstream packages.

The package should within the next 3 days be part of the Julia registry, so that you can install it in the usual way for Julia packages. Stay tuned for an update on that!

Lie.jl 1.0.0 is out

Mon, 11 May 2026 00:00:00 +0000

Note: This package was originally released under the name Lie.jl and has since been renamed to Semisimple.jl. See the rename announcement for details.

Semisimple.jl 1.0.0 is now available!

This first stable release focuses on efficient computations with semisimple Lie algebras, of the type that are most useful to an algebraic geometry who wants to do computations for completely reducible vector bundles on partial flag varieties. A large part of that efficiency comes from leaning into Julia’s type system and specialization, so fairly high-level code can still run very fast.

HomogeneousTools is now live

Fri, 08 May 2026 00:00:00 +0000

We are happy to announce that the HomogeneousTools website is now live. HomogeneousTools is a collection of software for working with homogeneous varieties and homogeneous vector bundles, with applications to sheaf cohomology, Fano geometry, and related areas.

More tools, documentation, and posts are to come soon. Stay tuned.

As a small example of the kind of computation one can do with Semisimple.jl (formerly Lie.jl, see the rename announcement), here is a short session using the root system of type $\mathrm{A}_2$ (i.e. $\mathfrak{sl}_3$):