If I have a set of simple roots $\{\alpha_1,\ldots, \alpha_n\}$ for a root system $\Phi$. What would be the way to compute the $\alpha_i$ string through $\alpha_j$?
Okay, I read this answer proposed by Torsten Schoeneberg. So say I have a $10 \times10$ Cartan matrix $A$. Then the $\alpha_8$-string through $\alpha_2$ is given by $\alpha_2, \alpha_2+\alpha_8,\ldots, \alpha_2-A_{28}\alpha_8$? But why is that? Why can't it be longer? Why isn't there negative terms (I mean like $\alpha_2-\alpha_8, \alpha_2-2\alpha_8,\ldots$?)
Maybe I just understood why... First I noticed that the $\alpha_8$ chain through $\alpha_2$ has to be of the form $\alpha_2,\alpha_2+\alpha_8, \alpha_2+2\alpha_8,\ldots$ Indeed elements of the form $\alpha_2-\alpha_8$ or $\alpha_2-2\alpha_8$ (etc...) are not roots since they are not a all-positive or all-negative linear combination of basis elements. Then I just went across a theorem that states that the length of the $\alpha$ chain through $\beta$ is $\frac{2(\beta,\alpha)}{(\alpha,\alpha)}$ which is the number in the Cartan matrix up to sign. Is this the correct justification?