TITLE: Which linear combinations of simple roots are roots
QUESTION [5 upvotes]: 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: 

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? 
Is there a recursive way to enumerate all the $(n_\alpha)$'s such that $\sum_{\alpha\in\Pi}n_\alpha\alpha$ is a root?
In particular, how to determine combinatorially the $(n_\alpha)$ corresponding to the highest root?

REPLY [6 votes]: My favorite answer to #2 and #3 is Kostant's "Find the highest root game", which is written up in detail in section 5.4 of Balázs Elek's notes on reflection groups. It is not hard to show that all plays of the game (from all starting positions, i.e., simple roots) take you through all the positive roots.

REPLY [2 votes]: I guess the following is not exactly what you are looking for but is somewhat revealing. A former colleague of mine tackled these type of questions in his PhD thesis. He describes something that he calls "minimal relations" i.e. relations of minimal length between roots. He gives a complete description of these minimal relations in Section 4.2.3 of his thesis which I linked below
http://math.jacobs-university.de/penkov/papers/PhD_Milev.pdf
As far as I know, he also proves a number of criterions of when an expression of roots is again a root.