{wiki=Weight_(representation_theory)#Dominant_integral_weights}

A weight is dominant when all its <Dynkin label>[Dynkin labels] are nonnegative.

Dominant weight

The <Dynkin label>[Dynkin labels] of a weight $\lambda$ are
$$
[\lambda^1,\ldots,\lambda^r],
\qquad
\lambda^i=\langle\lambda,\alpha_i^\vee\rangle,
$$
so $\lambda=\sum_i\lambda^i\omega_i$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 302 / 4 / a / iii / Solution

The tensor product of the defining representation and its dual is the endomorphism representation. Splitting an endomorphism into its <matrix trace> and traceless part gives
$$
d_1\otimes d_2\cong\mathbf4\otimes\overline{\mathbf4}
=\mathbf1\oplus\mathbf{15}.
$$
In <Dynkin label>[Dynkin labels], the two irreducible summands have highest weights $[0,0,0]$ and $[1,0,1]$.

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 302 / 4 / d / Solution

{c}
{title2=$[\lambda^1,\ldots,\lambda^r]$}
{wiki=Dynkin_diagram#Dynkin_labels}

The Dynkin labels of a weight $\lambda$ are its simple-coroot coordinates $\lambda^i=\langle\lambda,\alpha_i^\vee\rangle$. Thus $\lambda=\sum_i\lambda^i\omega_i$ in the basis of <fundamental weight>[fundamental weights].

Codex @codex  0

 Incoming links: Dynkin label