Fundamental representation

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

For traceless Hermitian matrices representing the Lie algebra of the special unitary group $SU(N)$, normalized by $\mathrm{tr}(T^a{T}^b)={\delta }^{ab}/2$, completeness of $\{I/\sqrt{2N},T^a\}$ in the space of Hermitian matrices gives the displayed identity. In particular ${\sum }_a{T}^{a}X{T}^a=\frac{1}{2}\mathrm{tr}(X)I-X/(2N)$. For traceless $X=T^b$, this becomes ${\sum }_a{T}^a{T}^b{T}^a=-T^b/(2N)$. These identities reduce QCD color charge sums without choosing explicit matrices.

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.

For a representation $R$ of $SU(N)$ with $N\ge 3$, choose the Hermitian quantum generator convention and normalize the cubic anomaly coefficient byHere the anticommutator symmetrizes the last two generators; cyclic property of the trace makes the whole tensor symmetric. This coefficient multiplies the local four-dimensional gauge anomaly from a chiral fermion in $R$. It adds under direct sums.

The conjugate representation has generators $-(T^a{)}^T$, so transposition and the three minus signs give $A(\overline{R})=-A(R)$. Hence a real representation or pseudoreal representation has zero cubic anomaly coefficient. For $SU(3)$, the two-index antisymmetric representation is the conjugate triplet and has coefficient $-1$. For $SU(2)$ the symmetric cubic invariant itself vanishes, so normalization to a nonzero fundamental cubic coefficient is unavailable; this statement concerns the local cubic anomaly and does not settle global anomalies.^[ref]

The two-index symmetric representation of $SU(N)$ is the symmetric square of its fundamental representation. For $N\ge 3$, 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. For $N=2$, the Dynkin index formula still applies in the stated normalization, but every symmetric cubic trace vanishes, so the stated fundamental-normalized cubic anomaly coefficient is not defined.

The two-index antisymmetric representation of $SU(N)$ is the exterior square of its fundamental representation. For $N\ge 3$, it has dimension $N(N-1)/2$, Dynkin index $N-2$, and cubic anomaly coefficient $N-4$ in the same normalization. For $N=2$, the Dynkin index formula still applies in the stated normalization, but every symmetric cubic trace vanishes, so the stated fundamental-normalized cubic anomaly coefficient is not defined.

For a complex antisymmetric tensor $A\in {\wedge }^2{\mathbb{C}}^4$, the SU(4) action is $A\mapsto UA{U}^T$. The unitary skew-diagonalization of an antisymmetric matrix gives $A=\mathrm{diag}(aϵ,bϵ)$ in suitable coordinates, where $ϵ=\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)$. If $|a|\ne |b|$, preserving $A{A}^{†}$ forces a stabilizing unitary matrix to preserve the two coordinate planes. The identity $VϵV^T=(\det V)ϵ$ then forces both two-by-two blocks to have determinant one, using $\det U=1$ also when one of $a,b$ vanishes. The exact stabilizer is $SU(2)\times SU(2)$. If $|a|=|b|\ne 0$, the stabilizer is the larger compact symplectic group $USp(4)$. If $a=b=0$, it is all of $SU(4)$. The real structure of the SU4 exterior square imposes $b=\bar{a}$ in this form, so a nonzero real six-dimensional field has the larger symplectic stabilizer, whereas unequal paired singular values are generic for a complex six-dimensional field.

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.