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In the classification of root systems, we have four families $A_n,B_n,C_n$, and $D_n$, and six exceptionals $E_6,E_7,E_8, F_4$, and $G_2$. For every non-exceptional case except $A_n$, the root system can be written $(V,\Delta)$ can be written in a "symmetric"/elegant manner as a set of vectors living in $V = \mathrm{R}^k$. However, for the $A$-series, we need to choose our vectors to live in $\mathrm{R}^{n+1}$ and the real vector system of our root system is the $n$-dim vector space spanned by these elements. Writing them directly as elements of $\mathrm{R}^n$ produces a bit of a mess (see this question).

So is there a conceptual explanation of why the $A$-series behaves differently, an way to see why we have to work in a vector space of one dimension higher?

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  • $\begingroup$ I'm not sure there is a good "conceptual" explanation of why $A$ is different in this way; and one thing to note is that there are not really canonical coordinates for the exceptional types either. $\endgroup$ Sep 7, 2021 at 13:44
  • $\begingroup$ However if you want to see one place where $BCD$ having nice coordinates simplifies life greatly (or conversely $A$ not having nice coordinates makes things a pain), check out: arxiv.org/abs/1204.1760. $\endgroup$ Sep 7, 2021 at 14:30

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