Root strings in type $A_2$

The roots of ${\mathfrak{s}\mathfrak{l}}_3(\mathbb{C})$ are the six functionals ${\epsilon }_i-{\epsilon }_j$, $i\mathrm{\ne }j$, so $\beta$, $\beta +\alpha ={\epsilon }_1-{\epsilon }_3$, $\alpha$ and $-\alpha$ are roots while $2\alpha =2{\epsilon }_1-2{\epsilon }_2$ is not, by the reducedness statement that the only scalar multiples of a root that are roots are $\pm$ themselves.

The Cartan integer evaluates as $\beta (h_{\alpha })=({\epsilon }_2-{\epsilon }_3)(E_{11}-E_{22})$: the diagonal matrix $E_{11}-E_{22}$ has coordinate vector $(1,-1,0)$, so ${\epsilon }_2-{\epsilon }_3$ gives $(-1)-0=-1$, matching $p-q=0-1=-1$.

For the $\alpha$-string through $\beta$: the indices $k\in \mathbb{Z}$ with $\beta +k\alpha$ a root or $0$ are $k=0,1$; indeed $\beta +\alpha ={\epsilon }_1-{\epsilon }_3$ is a root, while $\beta -\alpha =2{\epsilon }_2-{\epsilon }_1-{\epsilon }_3$ and $\beta +2\alpha =2{\epsilon }_1-{\epsilon }_2-{\epsilon }_3$ are not among the six roots and are nonzero. Hence $p=0$, $q=1$.

For the $\alpha$-string through $\beta =\alpha$, the terms are $\beta +k\alpha =(k+1)\alpha$. They are roots or zero exactly for $k\in \{-2,-1,0\}$, giving the terms $-\alpha ,0,\alpha$ and hence $p=2$, $q=0$, and $p-q=2$. Also $\alpha (h_{\alpha })=({\epsilon }_1-{\epsilon }_2)(E_{11}-E_{22})=1-(-1)=2$, so the string identity $p-q=\beta (h_{\alpha })$ holds.