On the Divisibility of Character Values of the Symmetric Group

TL;DR

This paper proves that for large symmetric groups, the majority of irreducible characters evaluated at fixed cycle types are divisible by a given integer, revealing a deep asymptotic divisibility property.

Contribution

It establishes that as the size of the symmetric group grows, the proportion of partitions with character values divisible by a fixed integer approaches one.

Findings

Proportion of partitions with divisible character values tends to 1 as n increases.

Provides asymptotic behavior of character divisibility in symmetric groups.

Connects partition structure with divisibility properties of character values.

Abstract

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.

Equations

$$\underset{n\to \infty }{\lim }\frac{\#\{\lambda ⊢n\mid {\chi }_{\mu }^{\lambda }\text{ is divisible by d}\}}{p(n)}=1.$$

$${\chi }_{\mu }^{\lambda }=\frac{f_{\lambda }}{(n{)}_k}{A}_{\mu }^{\lambda }.$$

$$\underset{n\to \infty }{\lim }\frac{\#\{\lambda ⊢n\mid v_q(f_{\lambda })\le m+(q-1)\log n\}}{p(n)}=0.$$

$$T_q=\coprod _{i=0}^{\infty }[q{]}^i.$$

$$w_i(\lambda )=\sum _{x\in [q{]}^i}|{\mathcal{T}}_q^{\lambda }(x)|.$$

$$\sum _{i=0}^{\infty }{w}_i(\lambda )q^i=n.$$

$$\ast$$

$$v_q(f_{\lambda })=\frac{w(\lambda )-a(n)}{q-1}.$$

$$w(\lambda )\le a(n)+b(q-1).$$

$$w(\lambda )\le (q-1)(\log n+1+b).$$

$$1+q+\cdots +q^r=\frac{q^{r+1}-1}{q-1}<qn,$$

$$p_b(n)$$

$$\le {\tilde{c}}_q(N_b{)}^{N_b}(qn+N_b{)}^{N_b}$$

$$p_b(n)\le f_q(N_b{)}^{N_b}(qn+N_b{)}^{N_b},$$

$$\log p_b(n)\le N_b[\log f_q(N_b)+\log (qn+N_b)].$$

$$v_q({\chi }_{\mu }^{\lambda })\ge v_q(f_{\lambda })-v_q((n{)}_k).$$

$$v_q((n{)}_k)=v_q\left(\frac{n!}{(n-k)!}\right)=\frac{k+a(n-k)-a(n)}{q-1}\le k+(q-1)\log n.$$

$$\underset{n\to \infty }{\lim }\frac{\#\{\lambda ⊢n\mid v_p({\chi }_{\mu }^{\lambda })\le b\}}{p(n)}=0.$$

Full text

Let $k$ be a positive integer, and $\mu=(\mu_{1},\dotsc,\mu_{m})$ a partition of $k$. For a partition $\lambda$ of an integer $n\geq k$, let $\chi^{\lambda}_{\mu}$ denote the value of the irreducible character of $S_{n}$ corresponding to $\lambda$ at an element with cycle type $(\mu_{1},\dotsc,\mu_{m},1^{n-k})$. The purpose of this article is to prove:

[TABLE]

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 $n\to\infty$.

Recall the theorem of Lassalle [4, Theorem 6], which implies that there exists an integer $A^{\lambda}_{\mu}$ such that

[TABLE]

Here $(n)_{k}=n(n-1)\dotsb(n-k+1)$, and $f_{\lambda}$ is the degree of the irreducible character of $S_{n}$ corresponding to $\lambda$. Therefore, in order to prove the main theorem, we focus on the divisibility properties of $f_{\lambda}$. For each prime number $q$, let $v_{q}(m)$ denote the $q$-adic valuation of an integer $m$, in other words, $q^{v_{q}(m)}$ 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:

[TABLE]

In the rest of this article, we first prove Theorem A, and then show that it implies the main theorem.

1. 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 $\{0,\dotsc,q-1\}$, and consider the disjoint union

[TABLE]

The set $T_{q}$ can be regarded as a rooted $q$-ary tree with root $\emptyset$. The children of a vertex $(a_{1},\dotsc,a_{i})\in[q]^{i}$ are the vertices $(a_{1},\dotsc,a_{i},a_{i+1})$, where $a_{i+1}\in[q]$. A partition $\lambda$ 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 $\lambda$ of $n$ a function $\mathcal{T}_{q}^{\lambda}:T_{q}\to C_{q}$ known as the $q$-core tower of $\lambda$. For a partition $\lambda$, define:

[TABLE]

Then the $q$-core tower satisfies the following constraint:

[TABLE]

In particular, $\mathcal{T}^{\lambda}_{q}(x)=\emptyset$ for all $i>\log_{q}n$. This function $\lambda\mapsto\mathcal{T}_{q}^{\lambda}$ is a bijection from the set of partitions of $n$ onto the set of $q$-core towers satisfying the condition (2).

Let $n$ be a positive integer with $q$-ary expansion:

[TABLE]

Define $a(n)=\sum_{i=0}^{r}a_{i}$.

Recall the following Theorem:

Theorem 1** ([5, Equation (3.3)]).**

For a partition $\lambda$, let $w(\lambda)=\sum_{i=0}^{r}w_{i}(\lambda)$. For any partition $\lambda$ of $n$ and any prime $q$,

[TABLE]

Theorem 1 can be used to find constraints on partitions with small values of $v_{q}(f_{\lambda})$. Suppose that $v_{q}(f_{\lambda})\leq b$. By Theorem 1, this is equivalent to

[TABLE]

The expansion ($*$ ‣ 1) implies that $r\leq\log n<r+1$, so that $a(n)\leq(q-1)(r+1)\leq(q-1)(\log n+1)$. So if $v_{q}(f_{\lambda})\leq b$, then

[TABLE]

Thus an upper bound for the number $p_{b}(n)$ of partitions $\lambda$ of $n$ such that $v_{q}(f_{\lambda})\leq b$ 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 $\coprod_{i=0}^{r}[q]^{i}$, is:

[TABLE]

since $q^{r}\leq n$. Let $c_{q}(n)$ denote the number of $q$-core partitions of $n$. Set $N_{b}=(q-1)(\log n+b+1)$. Let $\tilde{c}_{q}(n)$ denote $\max\{c_{q}(i)\mid 1\leq i\leq n\}$. There are $\binom{w+N-1}{w}$ ways to distribute $w$ cells into $N$ nodes. Thus

[TABLE]

It is known that, for every integer $q$, there exists a polynomial $f_{q}(n)$ such that $\tilde{c}_{q}(n)\leq f_{q}(n)$ for all $n\geq 0$. Indeed, for $q=2$, it is well-known that $c_{2}(n)\leq 1$, and for $q=3$, using a formula of Granville and Ono [2, Section 3, p. 340], $c_{3}(n)\leq 3n+1$. For $q\geq 4$, the existence of $f_{q}(n)$ follows from Anderson [1, Corollary 7].

We get:

[TABLE]

whence

[TABLE]

Taking $b=m+(q-1)\log n$ gives $N_{b}=(q-1)(q\log n+m+1)$. Thus $\log p_{b}(n)=o(n^{\epsilon})$ for every $\epsilon>0$. On the other hand, the Hardy-Ramanujan asymptote [3] for $p(n)$ implies that $\log p(n)$ grows faster than $n^{\frac{1}{2}-\epsilon}$ for any $\epsilon>0$. Thus Theorem A follows.

2. Proof of the Main Theorem

The identity (1) implies that

[TABLE]

Using Legendre’s formula on the valuation of a factorial, that $v_{q}(n!)=\frac{n-a(n)}{q-1}$, we have:

[TABLE]

Hence if $v_{q}(f_{\lambda})\geq m+(q-1)\log n$, then $v_{q}(\chi^{\lambda}_{\mu})\geq(m-k)$. Thus taking $m=k+b$ in Theorem A tells us that

[TABLE]

From this the main theorem follows.