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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
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