The probability that a character value is zero for the symmetric group

Abstract

Let be chosen at random from the irreducible characters of the symmetric group Sn and let g be chosen at random from the group itself. What is the probability that .g/ D 0? In this short note we give a remarkable asymptotic answer of one. Throughout the paper “at random” means uniformly at random. Theorem 1. If is chosen at random from the irreducible characters of Sn and g is chosen at random from Sn, then .g/ D 0 with probability P.Sn/ ! 1 as n ! 1. It will follow that the same must be true for the alternating group An. Theorem 2. If is chosen at random from the irreducible characters ofAn and g is chosen at random from An, then .g/ D 0 with probability P.An/ ! 1 as n ! 1. We prove these results in Section 1 and make some remarks in Section 2.