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Multiplicities of representations of compact Lie groups, qualitative properties and some computations

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If you have a question about this talk, please contact Mustapha Amrani.

Mathematical Challenges in Quantum Information

Co-author: Baldoni Velleda (Roma Tor Vergata-Italy)

Let V be a representation space for a compact connected Lie group G decomposing as a sum of irreductible representations pi of G with finite multiplicity m(pi,V).

When V is constructed as the geometric quantization of a symplectic manifold with proper moment map, the multiplicity function pi-> m(pi,V)$ is piecewise quasi polynomial on the cone of dominant weights. In particular, the function t-> m(t pi,V) is a quasipolynomial,alonf the ray tpi, when t runs over the non negative integers. We will explain how to compute effectively this quasi-polynomial (or the Duistermaat-Heckman measure) in some examples, including the function t-> c(tlambda,t mu,tnu) for Clebsch-Gordan coefficients (in low rank) and the function t-> k(talpha,tbeta,tgamma) for Kronecker-coefficients (with number of rows less or equal to 3). Our method is based on a multidimensional residue theorem (Jeffrey-Kirwan residues).

This talk is part of the Isaac Newton Institute Seminar Series series.

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