SU(3) quadratic Casimir eigenvalue ($C_2(a,b)=(a^2+ab+b^2+3a+3b)/3$)

For generators normalized by ${\mathrm{tr}}_3(t_a{t}_b)={\delta }_{ab}/2$, the quadratic Casimir operator on the irreducible SU(3) representation of Dynkin labels $(a,b)$ has the displayed scalar value. In SU(3) highest-weight coordinates $(p,q)$ it is $p^2+q^2+p+\sqrt{3}q$.

To derive this, write the operator as $H_1^2+H_2^2+{\sum }_{i=1}^3(E_{+}^i{E}_{-}^i+E_{-}^i{E}_{+}^i)$. On a highest-weight vector, the raising operators vanish, and the three root commutators sum to $H_1+\sqrt{3}{H}_2$. Centrality and the Schur lemma extend this value to the representation. The singlet, fundamental, adjoint and decuplet values are $0,4/3,3,6$, respectively. Tensor labels and multiplet dimensions are summarized in the Particle Data Group's SU(n) multiplet guide.