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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.

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