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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?

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    $\begingroup$ 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. $\endgroup$ – Sasha Jun 17 '14 at 7:52
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    $\begingroup$ 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 $\endgroup$ – Andrei Smolensky Jun 17 '14 at 19:07
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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.

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  • $\begingroup$ I assume it's named after Kostant since he discussed it somewhere; do you happen to have a reference? $\endgroup$ – LSpice Jul 17 at 17:53
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    $\begingroup$ I don't know that he wrote it anywhere (but I won't say he didn't); he told me about it in person at some point. $\endgroup$ – Allen Knutson Sep 24 at 20:41
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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.

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