# Compute formal character of semisimple Lie algebras.

Let $\mathfrak{g}$ be a semisimple Lie algebra and $V_{\lambda}$ be the irreducible $\mathfrak{g}$-module with highest weight $\lambda$. Are there some softwares which can compute the formal character $ch(V_{\lambda})=\sum_{\mu}dim(V_{\mu})e(\mu)$ explicitly? We can use Weyl formula. But it is difficult to compute the result in practice. Thank you very much.

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Maybe it's worth pointing out that Weyl's conceptually elegant character formula is almost never useful for actual computations of weight multiplicities. Algorithms for weight multiplicities are typically based on some variant of Freudenthal's formula, which doesn't require working simultaneously with all elements of the Weyl group. (Still, it's a problem to track down easily usable software, since people tend to develop systems and then move on to other things.) – Jim Humphreys Aug 26 '11 at 21:30

The software package LiE is good for this. It is no longer maintained, but it has a lot of functionality, and the documentation is good. There is an online demonstration here. There is a bit of a learning curve, but it can do pretty much anything you want. Also you can write your own programs using the previously defined commands, so it is quite customizable.

If you use Ubuntu or Debian, the software is already in the repositories and you just have to install it using the package manager. Otherwise you have to download the source files from the page I linked to and compile them yourself.

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@MTS, thank you very much. Should I use the command sudo apt-get install LiE(or some other names) to install the software in my Ubuntu system? Can a C program use LiE to do computation? Or can a LiE program to use results from a C program directly? Where could I find a reference about using LIE to program (not just using commands). – Jianrong Li Aug 26 '11 at 22:44
yeah, I think just sudo apt-get install lie will work. I don't know about calling it from a C program or vice versa, sorry! – MTS Aug 26 '11 at 23:27
@MTS, thank you very much. – Jianrong Li Aug 26 '11 at 23:42

You can use sage for this (and for many other things)

See the following manual page:

The Weyl character ring

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