A vector partition function for the multiplicities of

We use Gelfand–Tsetlin diagrams to write down the weight multiplicity function for the Lie algebra (type Ak−1) as a single partition function. This allows us to apply known results about partition functions to derive interesting properties of the weight diagrams. We relate this description to that of the Duistermaat–Heckman measure from symplectic geometry, which gives a large-scale limit way to look at multiplicity diagrams. We also provide an explanation for why the weight polynomials in the boundary regions of the weight diagrams exhibit a number of linear factors. Using symplectic geometry, we prove that the partition of the permutahedron into domains of polynomiality of the Duistermaat–Heckman function is the same as that for the weight multiplicity function, and give an elementary proof of this for (A3).