Almost all entries in the character table of the symmetric group are multiples of any given prime

Abstract

We show that almost every entry in the character table of S N {S_{N}} is divisible by any fixed prime as N → ∞ {N\to\infty} . This proves a conjecture of Miller.

Zeros in the Character Tables of Symmetric Groups with an $\ell$-Core Index

“…Let C n = [χ λ (µ)] λ,µ be the usual character table (for example, see [6,13,14]) for the symmetric group S n , where the indices λ and µ both vary over the p(n) many integer partitions of n. Confirming conjectures of Miller [8], Peluse and Soundararajan [11,12] recently proved that if ℓ is prime, then almost all of the p(n) 2 entries in C n , as n → +∞, are multiples of ℓ. We note that Miller conjectures that the same conclusion holds for arbitrary prime powers, a claim which remains open.…”

Divisibility of character values of the symmetric group by prime powers

“…Proof of Lemma 2.4. The proof is essentially identical to that of Proposition 1 of [16], but we include the short argument for completeness. Since every partition of n is a t-core for t > n, we may naturally assume that t ≤ n. From Lemma 5 of [14], we know that at most (t + 1)p(n − t) partitions of n are not t-cores.…”

“…Based on the second observation, Miller [11,13] also conjectured, more generally, that for any fixed prime p, almost every entry of the character table of S n is a multiple of p as n goes to infinity. We proved this conjecture in [16], with a uniform upper bound for the number of entries not divisible by a fixed prime. Recently, Miller [12] conjectured, even more generally, that for any fixed prime power q, almost every entry of the character table of S n is a multiple of q as n goes to infinity.…”

“…For any partitions λ and µ of n, let χ λ µ denote the value of the irreducible character of S n corresponding to λ on the conjugacy class of elements with cycle type corresponding to µ. In [16], our argument proceeded by combining two key facts: (i) if µ contains a part substantially larger than the typical largest part of a random partition, then χ λ µ = 0 for almost every λ, and (ii) if ν is another partition of n that is obtained from µ by combining p parts of the same size m into one part of size pm, then χ λ µ ≡ χ λ ν (mod p) for every λ. We showed that, for almost every µ, repeatedly combining p parts of the same size in this manner produces a partition µ containing a very large part, large enough so that χ λ µ must be zero for almost every λ.…”

“…However, when r > 1, it is no longer the case that starting from a typical partition µ of n and repeatedly combining p r parts of the same size m into p r−1 parts of size pm produces a partition µ containing a part substantially larger than the largest part of a typical partition of n. The argument from [16] that worked for primes thus breaks down for all other prime powers.…”