Characters of symmetric groups: sharp bounds on virtual degrees and the Witten zeta function

Abstract

We prove sharp bounds on the virtual degrees introduced by Larsen and Shalev. This leads to improved bounds on characters of symmetric groups. We then sharpen bounds of Liebeck and Shalev concerning the Witten zeta function. Our main application is a characterization of the fixed-point free conjugacy classes whose associated random walk mixes in 2 steps.

arXiv commentv4: 50 pages, 13 figures. Asymptotic equivalents of the Witten zeta function were added. v3: 50 pages, 12 figures. More general bounds and an application to random maps were added. v2: 44 pages, 12 figures. We added character bounds in terms of the number of cycles and two applications. v1: 37 pages, 11 figures. Comments welcome!