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Polynomial functions on the set of Young diagrams. (English. Abridged French version) Zbl 0830.20028
Summary: The character $$\chi^\lambda_\rho$$ of a symmetric group is usually considered as a function of the conjugacy class $$\rho$$ with a Young diagram $$\lambda$$ as a parameter. Here we study $$\chi^\lambda_\rho$$ as a function of $$\lambda$$, the class $$\rho$$ being fixed. We show that after a suitable normalization, the character becomes a supersymmetric function in modified Frobenius coordinates of $$\lambda$$. This result has an important application to the classification of the characters of the infinite symmetric group. Our method relies on a Capelli type identity for the Schur-Weyl duality and on the study of “polynomial functions” on the set $$\mathbf Y$$ of the Young diagrams.

##### MSC:
 20C32 Representations of infinite symmetric groups 05E05 Symmetric functions and generalizations 05E10 Combinatorial aspects of representation theory