Fundamental weight (${\omega }_i$)

The fundamental weights are dual to the simple coroots: $⟨{\omega }_i,{\alpha }_j^{\vee }⟩={\delta }_{ij}$.

The fundamental representation associated with a fundamental weight ${\omega }_i$ is the irreducible highest-weight representation $V({\omega }_i)$.

The Dynkin index of a representation $R$ fixes the trace normalization ${\mathrm{tr}}_R(T^a{T}^b)=I(R){\delta }^{ab}$ up to the convention for the generators. It controls matter contributions to gauge beta functions and mixed anomalies.

The cubic anomaly coefficient is defined by the symmetrized trace of three representation matrices. A conjugate representation has the opposite coefficient, while a real or pseudoreal representation has coefficient zero.

The two-index symmetric representation of $SU(N)$ is the symmetric square of its fundamental representation. It has dimension $N(N+1)/2$, Dynkin index $N+2$, and cubic anomaly coefficient $N+4$ when the fundamental index and anomaly are normalized to one.

The two-index antisymmetric representation of $SU(N)$ is the exterior square of its fundamental representation. It has dimension $N(N-1)/2$, Dynkin index $N-2$, and cubic anomaly coefficient $N-4$ in the same normalization.

On ${\bigwedge }^2{\mathbb{C}}^4$, the invariant Hermitian form and volume tensor define the displayed antilinear map with square one, commuting with SU(4). Its fixed space satisfies $T^{12}=\overline{T^{34}}$, $T^{13}=-\overline{T^{24}}$, and $T^{14}=\overline{T^{23}}$. Three complex choices give six real coordinates, and this fixed space has the original exterior square as its complexification. It is therefore a real representation of dimension six, rather than merely the twelve-dimensional realification of an arbitrary six-dimensional complex representation.