On parity and characters of symmetric groups

On even entries in the character table of the symmetric group

“…In contrast to the situation modulo primes, it appears from the data in [9] that the proportion of zeros in the character table of S n may tend to a positive constant. Our methods do not say anything on this matter, though very recently Larsen and Miller [6] have shown that the proportion of nonzero entries of character tables of groups of Lie type goes to zero as the rank goes to infinity.…”

“…The proofs of Theorems 1.1 and 1.2 will require a couple of basic results about characters of the symmetric group. The first gives us a useful condition for determining character values modulo primes (see, for example, Section 3 of [11] or Proposition 1 of [9]).…”

“…In [9], Miller conjectured that the proportion of odd entries in the character table of S n goes to zero as n goes to infinity, based on computational data for n up to 76. It has been known for a long time, due to work of McKay [8], that only a vanishingly small proportion of characters of the symmetric group have odd degree.…”

“…The second is longer, but generalizes to some other primes, and is quantitatively stronger. While the conjecture resolved by Theorem 1.1 is the main focus of [9], Miller also presents data for the number of entries in the character table of S n (for n up to 38) that are divisible by 3 and 5. Based on this, he conjectured that, for any fixed prime p, the proportion of the character table of S n that is not divisible by p tends to zero as n goes to infinity.…”

Congruences in character tables of symmetric groups

“…There are also general results of Burnside, J. G. Thompson, and P. X. Gallagher, with Burnside proving that zeros exist for nonlinear irreducible characters of a finite group [1], J. G. Thompson modifying Burnside's argument with a result of C. L. Siegel [15] to show that each irreducible character is zero or root of unity on more than a third of the group [7], and P. X. Gallagher proving similarly that more than a third of the irreducible characters are zero or root of unity on a larger than average class [5]. For large symmetric groups S n it was shown a few years ago [11], using estimates of Erdős-Lehner [3] and Goncharoff [6], that χ(σ) = 0 for all but an o(1) proportion of pairs χ, σ ∈ Irr(S n ) × S n , and conjectured [12] that, for any prime p, χ λ (µ) ≡ 0 (mod p) for all but an o(1) proportion of pairs of partitions λ, µ of n. Theorem 1 with d ≥ p implies χ λ (µ) ≡ 0 (mod p) for all pairs of partitions λ, µ of n.…”