# Which linear combinations of simple roots are roots

Let $\Delta$ be the root system of a complex simple Lie algebra, $\Delta^+$ be positive roots and $\Pi$ be simple roots. We view $\Pi$ as nodes of the Dynkin diagram.

Then for any two simple roots $\alpha$ and $\beta$, whether a linear combination $n\alpha+m\beta$ is a root can be judged easily from the Dynkin diagram.

My question is how to do this further. Precisely,

Question:

1. Given non-negative integers $(n_\alpha)_{\alpha\in\Pi}$, is there a combinatorial criterion to judge whether $\sum_{\alpha\in\Pi}n_\alpha\alpha$ is a root?

2. Is there a recursive way to enumerate all the $(n_\alpha)$'s such that $\sum_{\alpha\in\Pi}n_\alpha\alpha$ is a root?

3. In particular, how to determine combinatorially the $(n_\alpha)$ corresponding to the highest root?

• Using the action of the Weyl group you can move your $\lambda$ into the dominant chamber. And in the dominant chamber there are at most two roots --- the dominant long root and the dominant short root which can be easily written down. Another "algorithm" (which is much faster) is to look at tables of roots, say in Bourbaki. Jun 17 '14 at 7:52
• Or, if you don't like tables, you can memorize (or print out) the weight diagrams of adjoint representations, where roots correspond to chains of edges starting from some zero weight node. For example, you can find the pictures and descriptions here: citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.30.5052 Jun 17 '14 at 19:07