On the Divisibility of Character Values of the Symmetric Group

Fix a partition $\mu=(\mu_1,\dotsc,\mu_m)$ of an integer $k$ and positive integer $d$. For each $n>k$, let $\chi^\lambda_\mu$ denote the value of the irreducible character of $S_n$ at a permutation with cycle type $(\mu_1,\dotsc,\mu_m,1^{n-k})$. We show that the proportion of partitions $\lambda$ of $n$ such that $\chi^\lambda_\mu$ is divisible by $d$ approaches $1$ as $n$ approaches infinity.

Let k be a positive integer, and µ = (µ 1 , . . . , µ m ) a partition of k. For a partition λ of an integer n ≥ k, let χ λ µ denote the value of the irreducible character of S n corresponding to λ at an element with cycle type (µ 1 , . . . , µ m , 1 n−k ). The purpose of this article is to prove: Main Theorem. For any positive integers k and d, and any partition µ of k, lim n→∞ #{λ ⊢ n | χ λ µ is divisible by d} p(n) = 1.
Here p(n) denotes the number of partitions of n.
In particular, for any integer d, the probability that an irreducible character of S n has degree divisible by d converges to 1 as n → ∞.
Recall the theorem of Lassalle [4,Theorem 6], which implies that there exists an integer A λ µ such that Here (n) k = n(n−1) · · · (n−k+1), and f λ is the degree of the irreducible character of S n corresponding to λ. Therefore, in order to prove the main theorem, we focus on the divisibility properties of f λ . For each prime number q, let v q (m) denote the q-adic valuation of an integer m, in other words, q vq(m) is the largest power of q that divides m. Also write log n = log q n. The main theorem will follow from the following result: Theorem A. For every prime number q and non-negative integer m, In the rest of this article, we first prove Theorem A, and then show that it implies the main theorem.

Proof of Theorem A
The proof of Theorem A is based on the theory of q-core towers. This construction originated in the seminal paper [5] of Macdonald, and was developed further by Olsson in [6]. We now recall the relevant aspects.
Let [q] denote the set {0, . . . , q − 1}, and consider the disjoint union The set T q can be regarded as a rooted q-ary tree with root ∅. The children of a vertex (a 1 , . . . , a i ) ∈ [q] i are the vertices (a 1 , . . . , a i , a i+1 ), where a i+1 ∈ [q]. A partition λ is said to be a q-core if no cell in its Young diagram has hook length divisible by q. Denote the set of all q-core partitions by C q . The q-core tower construction associates to each partition λ of n a function T λ q : T q → C q known as the q-core tower of λ. For a partition λ, define: Then the q-core tower satisfies the following constraint: In particular, T λ q (x) = ∅ for all i > log q n. This function λ → T λ q is a bijection from the set of partitions of n onto the set of q-core towers satisfying the condition (2).
Define a(n) = r i=0 a i . Recall the following Theorem: . For a partition λ, let w(λ) = r i=0 w i (λ). For any partition λ of n and any prime q, v q (f λ ) = w(λ) − a(n) q − 1 .

Theorem 1 can be used to find constraints on partitions with small values of
The expansion ( * ) implies that r ≤ log n < r + 1, so that a(n) Thus an upper bound for the number p b (n) of partitions λ of n such that v q (f λ ) ≤ b can be obtained by counting the number of q-core towers with (q − 1)(log n + 1 + b) or fewer cells. The total number of vertices in the first r + 1 rows of T q , i.e., in r i=0 [q] i , is: since q r ≤ n. Let c q (n) denote the number of q-core partitions of n. Set N b = (q − 1)(log n + b + 1). Letc q (n) denote max{c q (i) | 1 ≤ i ≤ n}.
There are w+N −1 w ways to distribute w cells into N nodes. Thus It is known that, for every integer q, there exists a polynomial f q (n) such thatc q (n) ≤ f q (n) for all n ≥ 0. Indeed, for q = 2, it is wellknown that c 2 (n) ≤ 1, and for q = 3, using a formula of Granville and Ono [2, Section 3, p. 340], c 3 (n) ≤ 3n + 1. For q ≥ 4, the existence of f q (n) follows from Anderson [1, Corollary 7].
From this the main theorem follows.