Highest weight of a representation ($\Lambda$)

The highest weight of a highest-weight representation is the weight of a generating vector annihilated by every positive-root operator. Finite-dimensional irreducible representations of a complex semisimple Lie algebra are classified by dominant integral highest weights.

The coordinates of a highest weight record its eigenvalues on a chosen basis of a Cartan subalgebra. They depend on that basis. Dynkin labels instead use the simple coroots; orthonormal Cartesian coordinates and Dynkin labels must be converted before applying a dimension or Casimir eigenvalue formula.

With $H_1=\mathrm{diag}(1,-1,0)/2$ and $H_2=\mathrm{diag}(1,1,-2)/(2\sqrt{3})$, the simple roots are $(1,0)$ and $(-1/2,\sqrt{3}/2)$. Their coroots give Dynkin labels $a=2p$ and $b=\sqrt{3}q-p$ for Cartesian highest-weight coordinates $(p,q)$. Finite-dimensional irreducible representations require $a,b\in {\mathbb{Z}}_{\ge 0}$. The vertical coordinate is $\sqrt{3}/2$ times flavor hypercharge.