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A vector partition function for the multiplicities of \(\mathfrak{sl}_k\mathbb C\). (English) Zbl 1116.17005

Summary: We use Gelfand-Tsetlin diagrams to write down the weight multiplicity function for the Lie algebra \(\mathfrak{sl}_k\mathbb C\) (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 \(\mathfrak{sl}_k\mathbb C(A_3)\).

MSC:

17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
05E05 Symmetric functions and generalizations

Software:

Convex
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References:

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