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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?

roi_saumon
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  • What are your thoughts and attempts? What background knowledge of root systems can we use? – Torsten Schoeneberg Jun 15 '19 at 20:14
  • If I am given the Cartan matrix, I would try to compute the reflections $s_{\alpha_i}(\alpha_j)$ for every $\alpha_i$ in the basis. But I don't know if this is the fastest way or even if it does give me the entire string. (I don't know much about root systems) – roi_saumon Jun 15 '19 at 21:15
  • Check this answer: https://math.stackexchange.com/a/1467486/96384 – Torsten Schoeneberg Jun 16 '19 at 05:55
  • @TorstenSchoeneberg, thank you, I checked it out and edited my question – roi_saumon Jun 16 '19 at 10:51
  • I think your latest justification is correct. Of course to fully understand what's going on, you can and maybe should aim to understand that fact (that the length of those chains is given by those numbers) as well. It basically comes down to a few cases of what the angles between two roots can be, and what sums turn out to be roots then: a little geometry and combinatoric in the $2$-dimesnional plane. – Torsten Schoeneberg Jun 17 '19 at 18:41

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