Weight diagram

A diagram placing the weights of a representation in the real span of its weight lattice, recording each weight multiplicity. Coincident weights must retain their multiplicity: for an Adjoint representation the zero-weight multiplicity is the dimension of its Cartan subalgebra.

Let $P_{a,b}$ be the convex hull of the Weyl group orbit of the highest weight $a{\omega }_1+b{\omega }_2$ for the SU(3) Lie algebra. Restrict the displayed formula to the weight lattice coset $p{\omega }_1+q{\omega }_2+$ root lattice. It describes successive inward shells: the outer boundary has weight multiplicity one, each inward shell increases it by one, and the final triangular region has the constant value $\min (p,q)+1$. When $p=q$ that region is a single point. A triangular diagram with $p=0$ or $q=0$ has multiplicity one throughout. This is a geometric form of the dominant weight multiplicity formula for sl3, transported by Weyl reflections.

// Target: lie-theory.bigb

Using $I_3=\mathrm{diag}(1/2,-1/2,0)$ and flavour $Y=\mathrm{diag}(1/3,1/3,-2/3)$, the three coordinate vectors of the SU(3) group defining group representation have the displayed weights of a representation. The complex-conjugate group representation has their negatives. These coordinates use flavor hypercharge; another Cartan normalization rescales the vertical axis.

For a tensor product of Lie algebra representations with weight-space decompositions, the weights add and their weight multiplicities convolve. A diagram must sum contributions landing at the same weight, rather than treating coincident points as distinct positions.