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

TL;DR

This paper proves that for large symmetric groups, the probability that a randomly chosen irreducible character takes the value zero at a random group element approaches one, revealing a typical behavior of character values.

Contribution

It establishes the asymptotic probability that a character value is zero for the symmetric group, a new insight into the distribution of irreducible character values.

Findings

Probability that X(g)=0 tends to 1 as n increases

Character values are almost surely zero for large n

Provides asymptotic behavior of irreducible characters

Abstract

We consider random character values X(g) of the symmetric group on n symbols, where X is chosen at random from the set of irreducible characters and g is chosen at random from the group, and we show that X(g)=0 with probability tending to one as n tends to infinity.