In the weight diagram of an irreducible (finite dimensional, complex) representation of $\mathfrak{sl}(3,\mathbb{C})$, there are 'rings' of weights in the shapes of triangles or hexagons. Is there an easy way to show that the multiplicities on these rings must be the same without using complicated multiplicity formulas?
Thoughts: I understand that in some cases it suffices to consider $\mathfrak{sl}(2,\mathbb{C})$ submodules and the fact that multiplicities of weights of $\mathfrak{sl}(2,\mathbb{C})$ modules are symmetric about zero. But this seems to only work when the ring considered is small.
Thanks in advance.