Multiplicities of representations of compact Lie groups, qualitative properties and some computations
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Mathematical Challenges in Quantum Information
Coauthor: Baldoni Velleda (Roma Tor VergataItaly)
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 quasipolynomial (or the DuistermaatHeckman measure) in some examples, including the function t> c(tlambda,t mu,tnu) for ClebschGordan coefficients (in low rank) and the function t> k(talpha,tbeta,tgamma) for Kroneckercoefficients (with number of rows less or equal to 3). Our method is based on a multidimensional residue theorem (JeffreyKirwan residues).
This talk is part of the Isaac Newton Institute Seminar Series series.
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