= SU(3) quadratic Casimir eigenvalue
{c}
{title2=$C_2(a,b)=(a^2+ab+b^2+3a+3b)/3$}
For generators normalized by $\operatorname{tr}_{\mathbf3}(t_at_b)=\delta_{ab}/2$, 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 $p^2+q^2+p+\sqrt3q$.
To derive this, write the operator as $H_1^2+H_2^2+\sum_{i=1}^3(E_+^iE_-^i+E_-^iE_+^i)$. On a <highest-weight vector>, the raising operators vanish, and the three root commutators sum to $H_1+\sqrt3H_2$. 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 https://pdg.lbl.gov/2024/reviews/rpp2024-rev-young-diagrams.pdf[the Particle Data Group's SU(n) multiplet guide].
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