A central limit theorem for star-generators of $S_{\infty}$, which relates to traceless CCR-GUE matrices

Abstract

We prove a limit theorem concerning the sequence of star-generators of $S_{\infty}$, where the expectation functional is provided by a character of $S_{\infty}$ with weights $(w_1, \ldots , w_d, 0,0, \ldots )$ in the Thoma classification. The limit law turns out to be the law of a "traceless CCR-GUE" matrix, an analogue of the traceless GUE where the off-diagonal entries $g_{i,j}$ satisfy the commutation relation $g_{i,j}g_{j,i} = g_{j,i}g_{i,j} + (w_j - w_i)$. The special case $w_1 = \cdots = w_d = 1/d$ yields the law of a bona fide traceless GUE matrix, and we retrieve a result of K\"ostler and Nica from 2021, which in turn extended a result of Biane from 1995.