Past exam of the groups and representations course of the University of Oxford / 2022 / 2 / b / Solution

Write $a=a_1$, $b=a_2$ with $a,b\ge 0$, and let $\lambda =a{\omega }_1+b{\omega }_2$ be the highest weight. The positive roots are ${\alpha }_1$, ${\alpha }_2$ and ${\alpha }_1+{\alpha }_2$, and the Weyl vector is $\rho ={\omega }_1+{\omega }_2$. The Weyl dimension formula givesFor the quadratic Casimir operator, use the root-length normalization $C(\lambda )=(\lambda ,\lambda +2\rho )$. Substituting the Gram matrix in solution yieldsTaking the trace of the defining quadratic Casimir operator identity gives $d(a,b)C(a,b)=\dim (\mathfrak{su}(3))c(a,b)=8c(a,b)$, where $c$ is the Dynkin index in the matching normalization. ThereforeIn particular $c(3)=1$ and $c(8)=6$. The common particle-physics convention uses $C_2=C/2$ and $T=c/2$, giving $C_2(3)=4/3$ and $T(3)=1/2$. The subsequent discrete gauge anomaly congruences use the integer Dynkin index $c$, while the sign test for the supplied beta function is independent of a common rescaling of all indices.