A vector partition function for the multiplicities of sl_k(C)

Details A vector partition function for the multiplicities of sl_k(C) A vector partition function for the multiplicities of sl_k(C)

2003-07-16

arxiv.org/abs/math/0307227v1

Mathematics

Combinatorics

34 pages, 11 figures and diagrams; submitted to Journal of Algebra

Scientific paper

We use Gelfand-Tsetlin diagrams to write down the weight multiplicity function for the Lie algebra sl_k(C) (type A_{k-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 sl_4(C) (A_3).

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