We give asymptotic formulas for the multiplicities of weights and irreducible summands in high-tensor powers Vλ ⊗N of an irreducible representation Vλ of a compact connected Lie group G. The weights are allowed to depend on N, and we obtain several regimes of pointwise asymptotics, ranging from a central limit region to a large deviations region. We use a complex steepest descent method that applies to general asymptotic counting problems for lattice paths with steps in a convex polytope.
- Central limit region
- Lattice path with steps in a convex polytope
- Multiplicity of a weight or irreducible
- Strong deviations region
- Tensor power
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