= Fundamental-antifundamental decomposition for SU(N)
{title2=$\mathbf N\otimes\overline{\mathbf N}=\mathbf1\oplus\mathbf{Adj}$}
Identify the product with endomorphisms of the defining space. <Special unitary group> conjugation fixes scalar matrices and preserves the complementary traceless subspace. For $N\ge2$, the latter is the complex <Adjoint representation of a Lie algebra>, which is irreducible because $\mathfrak{sl}_N$ is simple. With fundamental <Dynkin index> $1/2$, the trace-Casimir identity gives $C_2(\mathbf N)=(N^2-1)/(2N)$; the <tensor-product Casimir trace identity> then gives $C_2(\mathbf{Adj})=N$.
Back to article page
