Height and highest root

Let $\mathrm{\Phi }\subseteq E$ be a reduced crystallographic root system with a chosen positive system ${\mathrm{\Phi }}^{+}$ and base $\mathrm{\Delta }=\{{\alpha }_1,\dots ,{\alpha }_r\}$ of simple roots (Positive systems and simple roots). By Simple roots form a signed integral basis every root is a unique integral combination of $\mathrm{\Delta }$ whose nonzero coefficients all have one sign.

For $\beta ={\sum }_{i=1}^r{n}_i{\alpha }_i\in \mathrm{\Phi }$ the height of $\beta$ is $\mathrm{ht}(\beta )={\sum }_{i=1}^r{n}_i\mathrm{.}$ This is well defined because the simple roots form a basis, and simple roots are exactly the roots of height one. The root order on $\mathrm{\Phi }$ is defined by $\beta \le \gamma ⟺\gamma -\beta ={\sum }_{i=1}^r{m}_i{\alpha }_i\text{with }{m}_i\in {\mathbb{Z}}_{\ge 0}\mathrm{.}$ It restricts to a partial order on the positive roots, in which every comparison chain is finite because heights strictly increase. A positive root $\theta \in {\mathrm{\Phi }}^{+}$ is a highest root if it is maximal for this order, that is, if $\theta \le \gamma$ for no positive root $\gamma \mathrm{\ne }\theta$. The existence and uniqueness of a highest root for an irreducible system are proved in Existence and uniqueness of the highest root.