 Shell multiplicities in an SU(3) weight diagram

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