On the Divisibility of Character Values of the Symmetric Group

Abstract

Fix a partition μ = ( μ 1 , … , μ m ) of an integer k and positive integer d . For each n > k , let χ λ μ denote the value of the irreducible character of S n at a permutation with cycle type ( μ 1 , … , μ m , 1 n − k ) . We show that the proportion of partitions λ of n such that χ λ μ is divisible by d approaches 1 as n approaches infinity.