On the Fourier transform on the infinite symmetric group

A. M. Vershik ; N. V. Tsilevich

Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial and algoritmic methods. Part XII, Tome 325 (2005), pp. 61-82

Résumé

We present a sketch of the Fourier theory on the infinite symmetric group ${\mathfrak S}_\infty$. As a dual space to ${\mathfrak S}_\infty$, we suggest the space (groupoid) of Young bitableaux $\mathcal B$. The Fourier transform of a function on the infinite symmetric group is a martingale with respect to the so-called full Plancherel measure on the groupoid of bitableaux. The Plancherel formula determines an isometry of the space $l^2({\mathfrak S}_\infty,m)$ of square integrable functions on the infinite symmetric group with the counting measure and the space $L^2({\mathcal B},\tilde\mu)$ of square integrable functions on the groupoid of bitableaux with the full Plancherel measure.