Weight lattice ($P$)

The weight lattice is the set of vectors $\lambda$ for which $⟨\lambda ,{\alpha }^{\vee }⟩$ is an integer for every root $\alpha$.

Restrict a finite-dimensional representation of a complex semisimple Lie algebra to the sl2 subalgebra associated with a root $\alpha$. Its Cartan element is the coroot $h_{\alpha }$. The classification of finite-dimensional sl2 representations gives only integer eigenvalues for this element, so every weight of a representation has integral value on every coroot. This proves containment of all such representation weights in the weight lattice; it does not hold for unrestricted infinite-dimensional modules.

The group ring of a weight lattice $P$ consists of finite sums ${\sum }_{\lambda }{a}_{\lambda }e(\lambda )$ with integer coefficients, where $e(\lambda )e(\mu )=e(\lambda +\mu )$. A finite-dimensional Lie algebra representation has formal character ${\sum }_{\lambda }(\dim V_{\lambda })e(\lambda )$ in this ring. The Weyl group acts by $e(\lambda )\mapsto e(w\lambda )$. The ambient ring need not consist of genuine characters; the representation ring of a semisimple Lie algebra of finite-dimensional modules for a complex semisimple Lie algebra identifies with the invariant subring $\mathbb{Z}[P{]}^W$.

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For a root reflection on the weight lattice, $s_{\alpha }\lambda =\lambda -m\alpha$ with $m=\lambda (H_{\alpha })\in \mathbb{Z}$. A finite anti-invariant element of the group ring of a weight lattice is a sum of terms $a_{\lambda }[e(\lambda )-e(s_{\alpha }\lambda )]$; fixed weights have zero coefficient. Each pair is divisible by $1-e(\alpha )$, because $1-t^m$ is divisible by $1-t$ in $\mathbb{Z}[t,t^{-1}]$ for every integer $m$. Thus the quotient has finite support.

The formal expression $({\sum }_{n\ge 0}e(n\alpha ))u$ is interpreted coefficientwise. Along each coset of $\mathbb{Z}\alpha$, only finitely many coefficients of $u$ are nonzero and their total is zero by reflection anti-invariance. Partial coefficient sums therefore vanish before and after that finite interval. The result has finite support, even though the individual summands $e(n\alpha )u$ do not eventually vanish.

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The Dynkin labels of a weight $\lambda$ are its simple-coroot coordinates ${\lambda }^i=⟨\lambda ,{\alpha }_i^{\vee }⟩$. Thus $\lambda ={\sum }_i{\lambda }^i{\omega }_i$ in the basis of fundamental weights.

A weight is dominant when all its Dynkin labels are nonnegative.

Dominance is a nonnegativity condition on a Weight; integrality additionally requires lattice membership.

An integral weight $\lambda$ is dominant when $⟨\lambda ,{\alpha }_i^{\vee }⟩\ge 0$ for every simple coroot. Equivalently, it is a nonnegative integer combination of the fundamental weights.

This is the lattice-restricted dominant condition on a Weight.

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.