On divisibility by primes in columns of character tables of symmetric groups

TL;DR

This paper proves that as the size of the symmetric group grows, the proportion of entries divisible by a prime in specific character table columns approaches 100%, using bounds on the number of large $k$-cores.

Contribution

It establishes a new asymptotic result on divisibility properties of character table entries for symmetric groups, linking prime divisibility to combinatorial core counts.

Findings

Proportion of entries divisible by prime p tends to 1 as n increases.

Lower bounds on the number of large k-cores are derived.

Asymptotic behavior of divisibility in character tables is characterized.

Abstract

For an arbitrary prime $p$ we prove that the proportion of entries divisible by $p$ in certain columns of the character table of the symmetric group $S_n$ tends to 1 as $n\to\infty$. This is done by finding lower bounds on the number of $k$-cores, for $k$ large enough with respect to $n$.

Equations

$${\Omega }_p(n):=\{\text{mbox}partitionsofn\text{mbox}intopartsnotdivisiblebyp\}.$$

$$K_p(\lambda ):=\{\mu \in P(n)|{\mu }^{\ast }=\lambda \}.$$

$$\frac{|\{\alpha \in P(n):{\chi }_{\mu }^{\alpha }\equiv 0\mathrm{mod}p\}|}{p(n)}\ge 1-\frac{c^{4}d\log (n)}{n^{c\pi /\sqrt{6}-1/2}}$$

$$p(n)-c_k(n)$$

$$\le (k+1)\sum _{m=1}^{\lfloor n/k\rfloor }{c}_k(n-k-(m-1)k)p_k(m-1)$$

$$=(k+1)p(n-k).$$

$$\frac{c_k(n)}{p(n)}\ge 1-\frac{c^{4}d\log (n)}{n^{c\pi /\sqrt{6}-1/2}}$$

$$\frac{p(n)-c_k(n)}{p(n)}\le \frac{(k+1)p(n-k)}{p(n)}.$$

$$\frac{d_1}{m}{e}^{\pi \sqrt{\frac{2m}{3}}}\le p(m)\le \frac{d_2}{m}{e}^{\pi \sqrt{\frac{2m}{3}}}$$

$$\frac{(k+1)p(n-k)}{p(n)}\le \frac{d_2(k+1)n}{d_1(n-k)}{e}^{-\pi \sqrt{\frac{2n}{3}}(1-\sqrt{1-\frac{k}{n}})}\le \frac{2d_{2}kn}{d_1(n-k)}{e}^{-\frac{\pi k}{\sqrt{6n}}}.$$

$$\frac{p(n)-c_k(n)}{p(n)}\le \frac{2d_2{n}^2}{d_1}{e}^{-\frac{4c\pi \log (n)}{\sqrt{6}}}=\frac{2d_2}{d_1{n}^{4(c\pi /\sqrt{6}-1/2)}}$$

$$\frac{p(n)-c_k(n)}{p(n)}\le \frac{4d_2\overline{c}\sqrt{n}\log (n)}{d_1}{e}^{-\frac{\pi \overline{c}\log (n)}{\sqrt{6}}}\le \frac{16c{d}_2\log (n)}{d_1{n}^{c\pi /\sqrt{6}-1/2}}$$

$$\frac{p(n)-c_k(n)}{p(n)}\le \frac{2d_2{d}_3{\overline{c}}^4\sqrt{n}\log (n)}{d_1}{e}^{-\frac{\pi \overline{c}\log (n)}{\sqrt{6}}}\le \frac{512c^4{d}_2{d}_3\log (n)}{d_1{n}^{c\pi /\sqrt{6}-1/2}}.$$

$$\frac{|\{\alpha \in P(n):{\chi }_{\mu }^{\alpha }\equiv 0\mathrm{mod}p\}|}{p(n)}=\frac{|\{\alpha \in P(n):{\chi }_{\overline{\lambda }}^{\alpha }\equiv 0\mathrm{mod}p\}|}{p(n)}.$$

Full text

1. Introduction

In [3] Miller formulated the following conjecture about the character table of symmetric groups:

Conjecture 1**.**

Let $p$ be a prime and $E_{p}(n)$ be the number of entries divisible by $p$ in the character table of $S_{n}$. Then $E_{p}(n)/(p(n))^{2}\to 1$ as $n\to\infty$.

Here, as in the rest of the paper, $p(n)$ is the number of partitions of $n$.

In [1] Gluck proved that on certain columns of the character table of $S_{n}$, the proportion of even entries tends to 1. The main results of this paper extend this to a larger set of columns of the character table of $S_{n}$ and hold for any prime $p$. These results however are not sufficient to prove Conjecture 1. In order to state our main results we need the following notation. Let $P(n)$ be the set of all partitions of $n$ and

[TABLE]

Further for any partition $\mu$ of $n$ let $\mu^{*}\in\Omega_{p}(n)$ be obtained from $\mu=(\mu_{1},\mu_{2},\ldots)$ by replacing each part $\mu_{i}=p^{k_{i}}a_{i}$ with $p\nmid a_{i}$ by $p^{k_{i}}$ parts $a_{i}$. Moreover, for $\lambda\in\Omega_{p}(n)$, let

[TABLE]

For $\alpha,\beta\in P(n)$ let $\chi^{\alpha}$ be the irreducible character of $S_{n}$ indexed by $\alpha$ and $\chi^{\alpha}_{\beta}$ be the value that $\chi^{\alpha}$ takes on the conjugacy class with cycle partition $\beta$.

Theorem 2**.**

Let $p$ be a prime, $n\geq 2$, $c>\frac{\sqrt{3/2}}{\pi}$ and $\lambda=(a_{1}^{b_{1}},\ldots,a_{h}^{b_{h}})\in\Omega_{p}(n)$. Assume that for some $1\leq i\leq h$ there exists $s$ with $p^{s}\leq b_{i}$ and $a_{i}p^{s}\geq c\sqrt{n}\log(n)$. Then for any $\mu\in K_{p}(\lambda)$ we have that

[TABLE]

for some constant $d$.

Note that $\frac{\log(n)}{n^{c\pi/\sqrt{6}-1/2}}\to 0$ since $c>\frac{\sqrt{3/2}}{\pi}$.

Corollary 3**.**

Let $p$ be a prime, $n\geq 2$, $c>\frac{\sqrt{3/2}}{\pi}$ and $\lambda=(a_{1}^{b_{1}},\ldots,a_{h}^{b_{h}})\in\Omega_{p}(n)$. If $h\leq\frac{\sqrt{n}}{cp\log(n)}$, then for any $\mu\in K_{p}(\lambda)$ we have that

[TABLE]

for some constant $d$.

Corollary 3 easily follows from Theorem 2, since under the assumptions of Corollary 3 there exists $i$ with $a_{i}b_{i}\geq cp\sqrt{n}\log(n)$ and then the assumptions of Theorem 2 are satisfied.

If $\mu,\nu\in K_{p}(\lambda)$ then the two columns of the character table of $S_{n}$ corresponding to conjugacy classes with cycle partitions $\mu$ and $\nu$ are congruent modulo $p$ (see [3, Proposition 1]). In particular the numbers of character values divisible by $p$ in the two columns are equal. This explains why Theorem 2 and Corollary 3 only have assumptions on $\lambda=\mu^{*}$ and not on $\mu$.

2. Proof of Theorem 2

Given a positive integer $k$ and a partition $\gamma$, we say that $\gamma$ is a $k$-core if $\gamma$ has no hook of length divisible by $k$. For any partition $\beta$ of $n$ and a positive integer $k$, one can define its $k$-core partition $\gamma$ to be the partition obtained from $\beta$ by recursively removing as many $k$-hooks as possible ($\gamma$ does not depend on which maximal sequence of $k$-hooks is removed from $\beta$), thus $|\beta|=|\gamma|+mk$ for a certain non-negative integer $m$ (see for example [4, Section 3]).

For any integer $m\geq 0$ let $p_{k}(m)$ be the number of multipartitions of $m$ into $k$ partitions. For any non-negative integer $m$ and any $k$-core partition $\gamma$ of $n-km$, the number of partitions of $n$ with $k$-core $\gamma$ is always equal to $p_{k}(m)$ (see for example [4, Proposition 3.7]).

We start by finding bounds on the number of $k$-core partitions of $n$ when $k$ is large enough. To obtain these bounds we will need bounds on the growth of the number of multipartitions, which will allow us to find lower bounds on $c_{k}(n)$, the number of $k$-core partitions of $n$. These results will then allow us to prove Theorem 2 at the end of this section.

Lemma 4**.**

Let $k\geq 1$ and $m\geq 1$. Then $p_{k}(m)\leq(k+1)p_{k}(m-1)$.

Proof.

For $\lambda=(\lambda^{1},\ldots,\lambda^{k})$ a multipartition of $m-1$ let $h$ be maximal such that $|\lambda^{h}|>0$ (set $h=0$ if $m=1$) and let $A(\lambda)$ be the set of multipartitions of $m$ which can be obtain by adding a node either to $\lambda^{h}$ on the last row or the first column or by adding one node to some $\lambda^{i}$ with $i>h$. Note that $|A(\lambda)|\leq k+1$ for each $\lambda$ and any multipartition of $m$ is contained in $A(\lambda)$ for some multipartition $\lambda$ of $m-1$. The result follows. ∎

Lemma 5**.**

For any $1\leq k\leq n$ we have $p(n)-c_{k}(n)\leq(k+1)p(n-k)$.

Proof.

It follows from Lemma 4 and the classification of partitions with the same $k$-core (see for example [4, Proposition 3.7]), since

[TABLE]

∎

Lemma 6**.**

Let $n\geq 2$ and $c>\frac{\sqrt{3/2}}{\pi}$. If $k\geq c\sqrt{n}\log(n)$ then

[TABLE]

for some constant $d$.

Proof.

From Lemma 5 we have that

[TABLE]

Note that there exist constants $d_{1},d_{2}>0$ such that for any $m\geq 1$

[TABLE]

(see [2, (1.41)]). Using the inequalities displayed above, we see that the statement holds for $k=n$, so we may assume that $n-k\geq 1$. Then

[TABLE]

If $k\geq 4c\sqrt{n}\log(n)$ then

[TABLE]

so in this case the lemma holds, since $\frac{c\pi}{\sqrt{6}}-\frac{1}{2}>0$ by assumption on $c$.

If $k=\overline{c}\sqrt{n}\log(n)$ with $c\leq\overline{c}<4c$ and $k\leq n/2$ then

[TABLE]

so that also in this case the lemma holds.

If $k=\overline{c}\sqrt{n}\log(n)$ with $c\leq\overline{c}<4c$ and $k>n/2$ then $\overline{c}>\sqrt{n}/(2\log(n))$. Since $d_{3}\sqrt{n}>8(\log(n))^{3}$ for $d_{3}$ large enough, there exists a constant $d_{3}$ such that $n<d_{3}\overline{c}^{3}$. It then follows that

[TABLE]

∎

We are now ready to prove Theorem 2.

Proof of Theorem 2.

Let $\lambda=(a_{1}^{b_{1}},\ldots,a_{h}^{b_{h}})\in\Omega_{p}(n)$ and assume that there exists $i$ and $s$ with $p^{s}\leq b_{i}$ and $a_{i}p^{s}\geq c\sqrt{n}\log(n)$. For any $1\leq j\leq h$ let $b_{j}=f_{j,0}p^{0}+\ldots+f_{j,g_{j}}p^{g_{j}}$ be the $p$-adic decomposition of $b_{j}$ and set $\overline{\delta}_{j}:=((p^{g_{j}})^{f_{j,g_{j}}},\ldots,1^{f_{j,0}})$ and $\overline{\lambda}:=a_{1}\overline{\delta}_{1}\cup\ldots\cup a_{h}\overline{\delta}_{h}$ (if $\phi=(\phi_{1},\ldots,\phi_{r})$ and $\psi=(\psi_{1},\ldots,\psi_{s})$ are partitions and $t$ is a non-negative integer, then $t\phi=(t\phi_{1},\ldots,t\phi_{r})$ and $\phi\cup\psi$ is the partition obtained by rearranging the parts of $(\phi_{1},\ldots,\phi_{r},\psi_{1},\ldots,\psi_{s})$). Then $\overline{\lambda}\in K_{p}(\lambda)$ and by assumption $\overline{\lambda}_{1}\geq c\sqrt{n}\log(n)$.

Note that for any partition $\mu\in K_{p}(\lambda)$ we have from [3, Proposition 1] that

[TABLE]

Since $\overline{\lambda}_{1}\geq c\sqrt{n}\log(n)$, the theorem holds for $\overline{\lambda}$ by Lemma 6 and the Murnaghan-Nakayama formula. So the statement of the theorem holds also for $\mu$. ∎

Acknowledgements

The author thanks Alexander Miller for bringing this problem to her attention and for some discussion.

The author was supported by the DFG grant MO 3377/1-1.