Abstract root data and their Weyl groups

A root datum is a quadruple $(X,\mathrm{\Phi },X^{\vee },{\mathrm{\Phi }}^{\vee })$ consisting of free $\mathbb{Z}$-modules $X,X^{\vee }$ of finite rank in perfect duality $⟨,⟩:X\times X^{\vee }\to \mathbb{Z}$, finite subsets $\mathrm{\Phi }\subseteq X$ and ${\mathrm{\Phi }}^{\vee }\subseteq X^{\vee }$, and a bijection $\alpha \mapsto {\alpha }^{\vee }$ from $\mathrm{\Phi }$ to ${\mathrm{\Phi }}^{\vee }$, subject to (rd1) $⟨\alpha ,{\alpha }^{\vee }⟩=2$ for all $\alpha \in \mathrm{\Phi }$; (rd2) the reflections $s_{\alpha }(x)=x-⟨x,{\alpha }^{\vee }⟩\alpha$ and $s_{{\alpha }^{\vee }}(y)=y-⟨\alpha ,y⟩{\alpha }^{\vee }$ satisfy $s_{\alpha }(\mathrm{\Phi })\subseteq \mathrm{\Phi }$ and $s_{{\alpha }^{\vee }}({\mathrm{\Phi }}^{\vee })\subseteq {\mathrm{\Phi }}^{\vee }$; (rd3) the group $W(R)$ generated by the $s_{\alpha }$ acting on $X$ is finite. The root datum is reduced if $\mathbb{Q}\alpha \cap \mathrm{\Phi }=\{\pm \alpha \}$ for every $\alpha \in \mathrm{\Phi }$. The group $W(R)$ is the Weyl group, $\mathrm{\Phi }$ the root system and ${\mathrm{\Phi }}^{\vee }$ the coroot system; $W(R)$ acts dually on $X^{\vee }$ by the $s_{{\alpha }^{\vee }}$, and $s_{\alpha }$ fixes the hyperplane $⟨,{\alpha }^{\vee }⟩=0$. For $V_{\mathbb{R}}=X{\otimes }_{\mathbb{Z}}\mathbb{R}$, the Weyl chambers are the connected components of $V_{\mathbb{R}}\setminus {\bigcup }_{\alpha \in \mathrm{\Phi }}\{x:⟨x,{\alpha }^{\vee }⟩=0\}$, and a base is a linearly independent subset $\mathrm{\Delta }\subseteq \mathrm{\Phi }$ such that every root is a $\mathbb{Z}$-combination of $\mathrm{\Delta }$ with all coefficients of one sign. A root datum is semisimple if $Z\mathrm{\Phi }$ has finite index in $X$.

The reflection $s_{\alpha }$ depends only on the pair $(\alpha ,{\alpha }^{\vee })$: it fixes the hyperplane $⟨,{\alpha }^{\vee }⟩=0$, sends $\alpha$ to $-\alpha$ by (rd1), and is determined by those two conditions on the rational direct sum $\mathbb{Q}\alpha \oplus \ker (⟨,{\alpha }^{\vee }⟩:X\otimes \mathbb{Q}\to \mathbb{Q})$. The lattice form above is related to the Euclidean root-system theory of Weyl group and Positive systems and simple roots through a $W$-invariant form on $V_{\mathbb{R}}$, but no choice of invariant inner product belongs to the definition. The perfect character–cocharacter pairing for a split torus is supplied by Character and cocharacter lattices of a split torus; roots and coroots are additional data, not supplied by that lattice lemma.

An isomorphism of root data $R=(X,\mathrm{\Phi },X^{\vee },{\mathrm{\Phi }}^{\vee })\to R_1=(X_1,{\mathrm{\Phi }}_1,X_1^{\vee },{\mathrm{\Phi }}_1^{\vee })$ is a pair of $\mathbb{Z}$-linear isomorphisms $f:X\to X_1$ and $f^{\vee }:X_1^{\vee }\to X^{\vee }$ that are transpose to one another, $⟨f(x),y_1⟩=⟨x,f^{\vee }(y_1)⟩$ for all $x\in X$, $y_1\in X_1^{\vee }$, and that carry $\mathrm{\Phi }$ bijectively onto ${\mathrm{\Phi }}_1$ with $f(\alpha {)}^{\vee }=f^{\vee }{}^{-1}({\alpha }^{\vee })$ for every $\alpha \in \mathrm{\Phi }$. Composites of isomorphisms are isomorphisms, with the evident inverses, so root data form a category whose isomorphisms are these pairs; this is the notion of isomorphism used when separating the root data of SL_2 and PGL_2.