Finite Weyl root system, lattice and chamber conventions

Let $E$ be a finite-dimensional real vector space with a positive definite symmetric bilinear form $(,)$. A finite reduced crystallographic root system is a finite subset $\mathrm{\Phi }\subseteq E\setminus \{0\}$ spanning $E$, such that for every $\alpha ,\beta \in \mathrm{\Phi }$ the integer $⟨\beta ,{\alpha }^{\vee }⟩=2(\beta ,\alpha )\mathrm{/}(\alpha ,\alpha )$ is integral, the reflection $s_{\alpha }(x)=x-⟨x,{\alpha }^{\vee }⟩\alpha$ preserves $\mathrm{\Phi }$, and the only scalar multiples of $\alpha$ in $\mathrm{\Phi }$ are $\alpha ,-\alpha$. The Weyl group $W$ is the subgroup of linear isometries of $E$ generated by these reflections. A root reflection means one of the $s_{\alpha }$. The empty root system on $E=0$ is allowed, with $W=\{1\}$.

Choose $t\in E$ such that $(t,\alpha )\mathrm{\ne }0$ for every root. Put ${\mathrm{\Phi }}^{+}=\{\alpha :(t,\alpha )>0\}$ and ${\mathrm{\Phi }}^{-}=-{\mathrm{\Phi }}^{+}$. A positive root is simple if it is not the sum of two positive roots. The following positive-root lemma proves that the simple roots form a basis ${\alpha }_1,\dots ,{\alpha }_r$ and that the simple reflections $s_i=s_{{\alpha }_i}$ generate $W$; no such assertion is assumed by this definition. Once proved, let $\mathrm{\ell }(w)$ be the least number of simple reflections in a word for $w$ and let $\mathrm{Inv}(w)=\{\alpha \in {\mathrm{\Phi }}^{+}:w\alpha \in {\mathrm{\Phi }}^{-}\}$. A word is reduced if its length equals $\mathrm{\ell }(w)$.

Identify $E$ with its dual by the displayed form, so ${\alpha }^{\vee }=2\alpha \mathrm{/}(\alpha ,\alpha )$. The root and coroot groups are $Q={\sum }_{\alpha \in \mathrm{\Phi }}\mathbb{Z}\alpha$ and $Q^{\vee }={\sum }_{\alpha \in \mathrm{\Phi }}\mathbb{Z}{\alpha }^{\vee }$. The weight group is $P=\{\lambda \in E:(\lambda ,{\alpha }^{\vee })\in \mathbb{Z}\text{ for all }\alpha \in \mathrm{\Phi }\}$; the positive-root lemma proves it is a lattice with basis the vectors dual to the simple coroots. Put $Q_{+}={\sum }_i{\mathbb{Z}}_{\ge 0}{\alpha }_i$, and write $\mu \le \lambda$ when $\lambda -\mu \in Q_{+}$. The open and closed positive chambers are $\mathcal{C}=\{x:(x,{\alpha }_i)>0\text{ for all }i\}$ and $\bar{\mathcal{C}}=\{x:(x,{\alpha }_i)\ge 0\text{ for all }i\}$. A weight is dominant if it belongs to $P\cap \bar{\mathcal{C}}$. More generally chambers are the connected components of $E\setminus {\bigcup }_{\alpha \in \mathrm{\Phi }}{\alpha }^{\perp }$. The closed-chamber representative and stabilizer assertions are proved later, not included as axioms.