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Asymptotics of matrix integrals and tensor invariants of compact Lie groups. (English) Zbl 1156.22015

Summary: We give an asymptotic formula for a matrix integral which plays a crucial role in the approach of Diaconis et al. to random matrix eigenvalues. The choice of parameter for the asymptotic analysis is motivated by an invariant-theoretic interpretation of this type of integral. For arbitrary regular irreducible representations of arbitrary connected semisimple compact Lie groups, we obtain an asymptotic formula for the trace of permutation operators on the space of tensor invariants, thus extending a result of Biane on the dimension of these spaces.

MSC:

22E46 Semisimple Lie groups and their representations
43A99 Abstract harmonic analysis
15B52 Random matrices (algebraic aspects)
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