Integral, dominant, and strictly dominant weights

Assume the Axiom of Choice. Let $\mathfrak{g}$ be a finite-dimensional complex semisimple Lie algebra with Cartan subalgebra $\mathfrak{h}$ and a chosen positive system whose base is the set of simple roots $\mathrm{\Delta }=\{{\alpha }_1,\dots ,{\alpha }_r\}$ (Positive systems and simple roots, The roots form a reduced crystallographic Euclidean root system). For $\lambda \in {\mathfrak{h}}^{\ast }$ write $⟨\lambda ,{\alpha }_i^{\vee }⟩:=\lambda (h_{{\alpha }_i})$ for the pairing with the simple coroots (Coroot of a Lie-algebra root). Then $\lambda$ is:

• integral if $⟨\lambda ,{\alpha }_i^{\vee }⟩\in \mathbb{Z}$ for every $i$;
• dominant integral if $⟨\lambda ,{\alpha }_i^{\vee }⟩\in {\mathbb{Z}}_{\ge 0}$ for every $i$;
• strictly dominant if $⟨\lambda ,{\alpha }_i^{\vee }⟩>0$ for every $i$;
• antidominant if $⟨\lambda ,{\alpha }_i^{\vee }⟩\le 0$ for every $i$.

Dominance, strict dominance, and antidominance depend on the chosen base $\mathrm{\Delta }$ and hence on the positive system: the same functional may be dominant for one choice and antidominant for another. Integrality is independent of that choice, because, as verified below, it is exactly membership in the weight lattice $P$. Only the integer and the sign of the finitely many simple-coroot pairings enter the displayed tests, so they are finite verifications.

Integral weights are the weight lattice. By Simple roots form a signed integral basis and step (v) of The roots form a reduced crystallographic Euclidean root system the simple roots form a basis of $E={\mathrm{span}}_{\mathbb{R}}\mathrm{\Phi }$ and the simple coroots form a basis of $\mathfrak{h}$, so a functional $\lambda \in E$ is determined by its pairings with the simple coroots. The fundamental weights ${\omega }_1,\dots ,{\omega }_r$ are dual to the simple coroots, $({\omega }_i,{\alpha }_j^{\vee })={\delta }_{ij}$ (Fundamental weights), and they form a basis of the weight lattice $P$ (Root, coroot, weight, and coweight lattices). Hence every $\lambda \in E$ has the unique expansion $\lambda ={\sum }_{i=1}^r⟨\lambda ,{\alpha }_i^{\vee }⟩{\omega }_i,$ so for $\lambda \in E$ integrality is equivalent to $\lambda \in P$. Conversely an integral $\lambda \in {\mathfrak{h}}^{\ast }$ automatically lies in $E$: the simple coroots form a real basis of ${\mathfrak{h}}_{\mathbb{R}}$ by loc. cit., the values $\lambda (h_{{\alpha }_i})$ are real, hence $\lambda$ is real valued on ${\mathfrak{h}}_{\mathbb{R}}$, which is exactly the condition defining $E$ inside ${\mathfrak{h}}^{\ast }$. Thus the integral weights are the elements of $P\subseteq E$, and the dominant integral weights are those elements of $P$ whose fundamental-weight coefficients are nonnegative integers. By contrast, the strictly dominant weights are all elements of $E$ whose fundamental-weight coefficients are positive real numbers; they need not be integral or belong to $P$.