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We give a short proof, in the case of representations over the complex numbers, of G.D. James characterization of the irreducible representations of the symmetric group as intersection of the kernels of suitable invariant operators. Following E. Bolker, P. Diaconis and S. Sternberg, those operators are interpreted as Radon transforms. Our proof is based on the harmonic analysis of the invariant Laplacian that describes the Bernoulli-Laplace diffusion model with many urns.

Radon transforms on the symmetric group and harmonic analysis of a class of invariant Laplacians