Minuscule weights

Assume the Axiom of Choice. Let $\mathfrak{g}$ be a finite-dimensional complex simple Lie algebra, with Cartan subalgebra $\mathfrak{h}$, root system $\mathrm{\Phi }\subseteq {\mathfrak{h}}^{\ast }$, positive system ${\mathrm{\Phi }}^{+}$, set of dominant integral weights ${\mathrm{\Lambda }}^{+}=\{\lambda \in P:⟨\lambda ,{\alpha }^{\vee }⟩\ge 0\mathrm{\forall }\alpha \in {\mathrm{\Phi }}^{+}\}$ and coroots ${\alpha }^{\vee }\in \mathfrak{h}$ of roots $\alpha$ (Integral, dominant, and strictly dominant weights, Coroot of a Lie-algebra root, Finite Weyl root system, lattice and chamber conventions). Here $⟨\lambda ,{\alpha }^{\vee }⟩=\lambda (h_{\alpha })$ is the natural pairing, and the coroots of $\mathrm{\Phi }$ form the dual root system ${\mathrm{\Phi }}^{\vee }=\{{\alpha }^{\vee }:\alpha \in \mathrm{\Phi }\}$ (The root set is a reduced crystallographic root system, Fundamental weights).

A dominant integral weight $\omega \in {\mathrm{\Lambda }}^{+}$ is minuscule if $⟨\omega ,{\beta }^{\vee }⟩\le 1\text{for every }\beta \in {\mathrm{\Phi }}^{+}\mathrm{.}$ Since the coroot of a negative root is the negative of the coroot of the positive root, ${\mathrm{\Phi }}^{\vee }=-{\mathrm{\Phi }}^{\vee }$ at the level of the sets $\{{\beta }^{\vee }\}$, and since $\omega$ is dominant integral the pairing $⟨\omega ,{\beta }^{\vee }⟩$ is a nonnegative integer for $\beta \in {\mathrm{\Phi }}^{+}$; hence the displayed condition is equivalent to $\mathrm{\mid }⟨\omega ,{\beta }^{\vee }⟩\mathrm{\mid }\le 1\text{for every }\beta \in \mathrm{\Phi }\mathrm{.}$ The zero weight is minuscule, and the Weyl group acts on weights by the reflection action $\lambda \mapsto w\lambda$ (Root reflections and the Weyl group action).