Degrees, class sizes and divisors of character values

Abstract

Abstract. In the character table of a finite group there is a tendency either for the character degree to divide the conjugacy class size or the character value to vanish. There is also a partial divisibility where the determinant of the character is not 1. There are versions of these depending on a subgroup, based on an arithmetic property of spherical functions which generalizes the integrality of the values of the characters and the central characters.

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.…”