Length of root strings is at most 4

Solution 1:

I had trouble with exactly the same sentence in Humphrey's book (last sentence before the exercise on page 45 --- there was no "at once" for me). After googling, I ended up on this webpage, but I found the exchange of remarks tricky to follow. For the record, an easy way to prove that a root string has length at most 4 is to note that the sequence $$ \langle \beta + i \alpha, \alpha \rangle, \enspace \hbox {where $i$ is an integer}, $$ forms an AP with difference 2. If $\phi$ and $\psi$ are any two linearly independent roots, then $$ \langle \psi, \phi \rangle =-3, -2, -1, 0, 1, 2 \hbox { or } 3, $$ from which it follows that the root string has length at most 4.

The formula $r - q = \langle \beta, \alpha \rangle$ becomes really useful slightly later when constructing the Hasse diagram of the positive roots from a base of simple roots using root strings: $$ \hbox {string length} = r + q + 1 = 2r + 1 - \langle \beta, \alpha \rangle. $$ In particular, when $r=0$ (that is, when $\beta - \alpha$ is not a root, as happens when $\alpha$ and $\beta$ are simple roots), then the string length is $1 - \langle \beta, \alpha \rangle$.

Solution 2:

By positive definiteness $0 \leq (\alpha, \beta)^2 < (\alpha, \alpha)( \beta, \beta )$ or, equivalently, $0 \leq (\alpha, \beta^\vee) (\beta, \alpha^\vee ) < 4$. Note the strict inequality.

Edit: I replaced $\langle \, , \, \rangle$ by $(\, , \,)$ to denote the inner product because of their conflicting interpretations.