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?



Which linear combinations of simple roots are roots
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](http://pi.math.cornell.edu/%7Eallenk/elek_reflection_groups_2016.pdf). 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.


