Roots, Weyl Group, and Dominant Weights

Fix a split reductive $G/k$ and a split maximal torus $T$ (see maximal torus and $X^*(T)$ ).

The roots $\Phi\subset X^*(T)$ are the nonzero weights for the adjoint action of $T$ on $\mathrm{Lie}(G)$; the root lattice is $\mathbb{Z}\Phi\subset X^*(T)$.

A choice of Borel subgroup $B\supset T$ determines positive roots $\Phi^+$ and simple roots $\Delta\subset \Phi^+$.

The Weyl group is $W:=N_G(T)/T$; it acts on $X^*(T)$, and a Weyl chamber is a connected component cut out by the hyperplanes $\{\alpha=0\}$.

A weight $\lambda\in X^*(T)$ is dominant if $\langle \lambda,\alpha^\vee\rangle\ge 0$ for all $\alpha\in\Delta$; irreducible algebraic representations of $G$ are classified by their highest dominant weight.

Example: For $G=\mathrm{GL}_n$, $\Phi=\{e_i-e_j\}$ and $W\cong S_n$ permutes the $e_i$.