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An integral weight $\lambda$ is dominant when $⟨\lambda ,{\alpha }_i^{\vee }⟩\ge 0$ for every simple coroot. Equivalently, it is a nonnegative integer combination of the fundamental weights.

This is the lattice-restricted dominant condition on a Weight.

A weight is dominant when all its Dynkin labels are nonnegative.

Dominance is a nonnegativity condition on a Weight; integrality additionally requires lattice membership.

A highest-weight representation is generated by a weight vector killed by every positive-root subspace.

The generating vector has a highest Weight relative to the chosen positive root system.

The multiplicity of a weight $\mu$ is the dimension of its weight space.

Thus it is a dimension attached to a Weight and its weight space.

The weight space $V_{\mu }$ consists of all weight vectors of weight $\mu$, together with zero.

This is the simultaneous eigenspace belonging to a Weight.

A vector $v$ has weight $\mu \in {\mathfrak{t}}^{\ast }$ when $hv=\mu (h)v$ for every $h$ in the Cartan subalgebra.

A Weight is the eigenvalue functional; a weight vector is a nonzero vector carrying it.