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 $(w_1,\dots ,w_d,0,0,\dots )$ 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 Köstler and Nica from 2021, which in turn extended a result of Biane from 1995.

36 pages, 3 figures