A technical question about root systems

Question

I'm studying root systems and coming up with an observation:

Let $\Phi$ be an irreducible root system and $\Phi^+$ be a positive root system. Denote by $\Delta=\{\alpha_1, \alpha_2,\ldots,\alpha_n\}$ be the corresponding base. For $\beta, \gamma \in \Phi^+$, $\alpha_1, \alpha_2 \in \Delta$ with the properties that $\beta=\gamma +\alpha_1+\alpha_2$, $\gamma +\alpha_1 \in \Phi^+, \gamma +\alpha_2\in \Phi^+, \alpha_1+ \alpha_2 \notin \Phi^+$ then

Statement: There exists $\alpha_3 \in \Delta\setminus\{\alpha_1,\alpha_2\}$ such that either $\gamma +\alpha_1-\alpha_3 \in \Phi^+$ or $\gamma +\alpha_2-\alpha_3 \in \Phi^+$.

The motivation is to investigate the relationship between a positive positive root and other lower positive roots. That is to understand the so-called Hasse diagram root systems (for instance see definition of Hasse diagram in here)

Any help would be much appreciated.

Answer 1

Consider the root system $\mathsf{B}_n$ with the standard numbering of the fundamental roots (that is, $\alpha_n$ is short). Take $\alpha_{n-2}$ and $\alpha_n$ as a pair of orthogonal roots (indeed, $\alpha_{n-2}+\alpha_n$ is not a root), and take $\gamma=\alpha_{n-1}+\alpha_n$. Then $\beta_1=\alpha_{n-2}+\alpha_{n-1}+\alpha_n$, $\beta_2=\alpha_{n-1}+2\alpha_n$ and $\alpha_{n-2}+\alpha_{n-1}+2\alpha_n$ are all roots. But among the differences of the form $\beta_i-\alpha_j$ the only roots are $\beta_1-\alpha_{n-2}$, $\beta_1-\alpha_n$ and $\beta_2-\alpha_n$, so this gives a counter-example.

The beautiful pictures of Hasse diagrams you refer to provide a good way to spot such examples, but for this one should draw them in a different way, which is easier to read. The keyword here is a weight diagram. For examples, see this collection of the diagrams along with a description of their various usages.

Namely, Figure 14 on page 35 is the weight diagram of the adjoint representation of a group (or of a Lie algebra) of type $\mathsf{B}_n$, which also describes the structure of the root system (its vertices are the roots, plus the "zero weights" corresponding to the fundamental roots). One looks for a square which has non-consecutive label on its sides (say $i$ and $j$), such that the bonds joining its top or bottom vertex to something on the right are also labeled by either $i$ or $j$. Such a square is immediately found on the very bottom of the picture slightly to the left from the middle.

By the way, the quick inspection of the weight diagrams shows that there is no counter-examples besides the one above (I'm not sure, but I haven't spotted any).