Zeros in the character tables of symmetric groups with an 
$\ell $
 -core index

Abstract Let 
$\mathcal {C}_n =\left [\chi _{\lambda }(\mu )\right ]_{\lambda , \mu }$
 be the character table for 
$S_n,$
 where the indices 
$\lambda $
 and 
$\mu $
 run over the 
$p(n)$
 many integer partitions of 
$n.$
 In this note, we study 
$Z_{\ell }(n),$
 the number of zero entries 
$\chi _{\lambda }(\mu )$
 in 
$\mathcal {C}_n,$
 where 
$\lambda $
 is an 
$\ell $
 -core partition of 
$n.$
 For every prime 
$\ell \geq 5,$
 we prove an asymptotic formula of the form 
$$ \begin{align*}Z_{\ell}(n)\sim \alpha_{\ell}\cdot \sigma_{\ell}(n+\delta_{\ell})p(n)\gg_{\ell} n^{\frac{\ell-5}{2}}e^{\pi\sqrt{2n/3}}, \end{align*} $$
 where 
$\sigma _{\ell }(n)$
 is a twisted Legendre symbol divisor function, 
$\delta _{\ell }:=(\ell ^2-1)/24,$
 and 
$1/\alpha _{\ell }>0$
 is a normalization of the Dirichlet L-value 
$L\left (\left ( \frac {\cdot }{\ell } \right ),\frac {\ell -1}{2}\right ).$
 For primes 
$\ell $
 and 
$n>\ell ^6/24,$
 we show that 
$\chi _{\lambda }(\mu )=0$
 whenever 
$\lambda $
 and 
$\mu $
 are both 
$\ell $
 -cores. Furthermore, if 
$Z^*_{\ell }(n)$
 is the number of zero entries indexed by two 
$\ell $
 -cores, then, for 
$\ell \geq 5$
 , we obtain the asymptotic 
$$ \begin{align*}Z^*_{\ell}(n)\sim \alpha_{\ell}^2 \cdot \sigma_{\ell}( n+\delta_{\ell})^2 \gg_{\ell} n^{\ell-3}. \end{align*} $$


Introduction and statement of results
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 conjectured that the same conclusion holds for arbitrary prime powers, a claim which remains open.
In recent papers [8,9], Miller raised the problem of determining the limiting behavior of Z(n), the number of zero entries in C n . Despite the remarkable theorem of Peluse and Soundararajan, little is known. Moreover, due to the rapid growth of p(n), it is computationally infeasible to compute many values of Z(n). Consequently, there are no conjectures that are supported with substantial numerics. For example, is there a limiting proportion for the zeros in C n ? Such a proportion would be given by the limit lim n→+∞ Z(n) p(n) 2 . Limited numerics suggest that such a limit might exist, and might be ≈ 0.36 (see Table 3 of [8]). However, this is a dubious guess at best. What's more, the simpler problem of determining whether lim inf n→+∞ Z(n)/p(n) 2 > 0 also seems to be out of reach. In view of these difficulties, McKay [5] posed a less ambitious problem; he asked for lower bounds arising from ℓ-cores that illustrate the rapid growth of Z(n). Here we answer this question, and for primes ℓ ≥ 5, we obtain asymptotic formulas for Z ℓ (n) := # {(λ, µ) : χ λ (µ) = 0 with λ an ℓ-core} . (1.1)

ELEANOR MCSPIRIT AND KEN ONO
To this end, suppose that λ = (λ 1 , λ 2 , . . . , λ s ) is a partition of n. As is typical in the representation theory of symmetric groups, we make use of ℓ-core partitions, which are defined using Young diagrams of partitions, the left-justified arrays of cells where the row lengths are the parts. The hook for the cell in position (k, j) is the set of cells below or to the right of that cell, including the cell itself, and so its hook length h λ (k, j) := (λ k − k) + (λ ′ j − j) + 1. Here λ ′ j is the number of boxes in the jth column of the diagram. We say that λ is an ℓ-core partition if none of its hook lengths are multiples of ℓ. If c ℓ (n) denotes the number of ℓ-core partitions of n, then we have (for example, see [2,7]) the generating function Example. The Young diagram of the partition λ := (5, 4, 1), where each cell is labelled with its hook length, is given in Figure 1. By inspection, we see that λ is an ℓ-core for every prime ℓ > 7.
To state the asymptotics formulas, we let L · ℓ , s be the Dirichlet L-function for the Legendre symbol · ℓ , and let By the functional equations of these Dirichlet L-functions and the theory of generalized Bernoulli numbers, we have that 1/α ℓ is always a positive integer (see p. 339 of [3]). For example, we have 1/α 5 = 1, 1/α 7 = 8, 1/α 11 = 1275, and 1/α 13 = 33463. In addition, we require the integers δ ℓ := (ℓ 2 − 1)/24, and the twisted Legendre symbol divisor functions In terms of these quantities and functions, we obtain the following asymptotics for Z ℓ (n).
To obtain these results, we use the well-known vanishing result that follows from the Murnaghan-Nakayama rule and says that χ λ (µ) = 0 whenever µ has a part that is not the length of any hook in λ. Therefore, our goal is reduced to counting pairs of partitions (λ, µ) of large n, where µ has a part that is a multiple of ℓ, and where λ is an ℓ-core. Theorem 1.1 is obtained by estimating these counts using asymptotics and lower bounds for various partition functions due to Hardy and Ramanujan, Hagis, and Granville and the second author. Theorem 1.3 concerns the cases where (λ, µ) are both ℓ-cores, and is a consequence of the fact (see Theorem 4.1) that every large ℓ-core has a part that is a multiple of ℓ. This fact is proved using the "abacus theory" of ℓ-cores, and is a generalization of Section 3 of [10] by Sze and the second author in the case of 4-core partitions. Corollary 1.4 then follows from the asymptotics for c ℓ (n) due to Granville and the second author.
This paper is organized as follows. Section 3 recalls well-known vanishing result and bounds, as well as the asymptotics and estimates for the relevant partition functions. Section 4 gives the abacus theory of ℓ-cores and the statement and proof of Theorem 4.1. In Section 5 we employ these results to prove Theorems 1.1 and 1.3, and Corollaries 1.2 and 1.4.

Acknowledgements
The authors thank Sarah Peluse and Richard Stanley as well as the referee for helpful comments that improved this paper.

Nuts and Bolts
In this section we recall essential facts that we require for the proofs of our results. We first state a criterion that guarantees the vanishing of character values, and then we give estimates for the relevant partition functions.
3.1. Criterion for the vanishing of χ λ (µ). Here we recall a standard partition theoretic criterion that guarantees the vanishing of a character value χ λ (µ). Suppose that λ = (λ 1 , . . . , λ s ) and µ = (µ 1 , . . . , µ t ) are partitions of size n, and let {h λ (i, j)} be the multiset of hook lengths for λ. Thanks to the Murnaghan-Nakayama formula (for example, see Theorem 2.4.7 of [6]), we have that Given a prime ℓ, this immediately gives natural families of vanishing character table entries indexed by pairs of partitions (λ, µ) of n, where µ has a part that is a multiple of ℓ, and λ is an ℓ-core partition. To make use of this observation, we recall that a partition µ is A-regular if none of its parts µ i are 4 ELEANOR MCSPIRIT AND KEN ONO multiples of A. If p A (n) denotes the number of A-regular partitions of n, then one easily confirms the generating function which shows that p A (n) also is the number of partitions of n where parts appear at most A − 1 times. In terms of p(n), p ℓ (n) and c ℓ (n), we have the following lower bounds for Z(n).
Lemma 3.1. If ℓ is prime, then the following are true.
(2) If n is a positive integer, then we have Proof.
(1) By hypothesis, µ is not ℓ-regular, meaning that it has a part that is a multiple of ℓ. As λ is an ℓ-core, none of its hook lengths are multiples of ℓ. Therefore, χ λ (µ) = 0 by Murnaghan-Nakayama. (2) The number of partitions of n that are not ℓ-regular is p(n) − p ℓ (n). Therefore, (1) gives the conclusion that Z ℓ (n) ≥ (p(n) − p ℓ (n))c ℓ (n).
where C := π 2/3 and C A : Finally, we recall facts about c t (n), the number of t-core partitions of n that were obtained by Granville and the second author in [3]. In terms of α ℓ defined in (1.2), and the twisted Legendre symbol divisor functions σ ℓ (n) defined in (1.3), we have the following theorem. In particular, c 3 (n) = 0 for almost all n.
(3) If t ≥ 4 and n is a non-negative integer, then c t (n) > 0.

Abaci and large ℓ-core partitions
Throughout this section, suppose that ℓ is prime. The main result here is the following theorem which shows that every sufficiently large ℓ-core partition has a part that is a multiple of ℓ.
Remark. We note that N ℓ < ℓ 6 /24 is not optimal. Indeed, if we let N max ℓ be the largest n admitting an ℓ-regular ℓ-core partition, then it turns out that N max 3 = 10 and N 3 = 16.
Using these structure numbers, we represent the partition λ as an ℓ-abacus A λ , consisting of beads placed on rods numbered 0, 1, . . . , ℓ − 1. For each B i , there is a unique pair of integers (r i , c i ) for which B i = ℓ(r i − 1) + c i and 0 ≤ c i ≤ ℓ − 1. The abacus A λ then consists of s beads, where for each i, one places a bead in position (r i , c i ). In view of this lemma, we may represent an abacus of an ℓ-core partition by ℓ-tuples of nonnegative integers, say (b 0 , . . . , b ℓ−1 ), where b i denotes the number of beads in column i. However, such representations are not unique as they generally allow for parts of size zero. We have the following elementary lemma.  Lemma 1, [10]). The following abaci both represent the same ℓ-core partition: By repeatedly applying this lemma, we may canonically define the unique abacus representation for an ℓ-core to be the one with zero beads in the first column. Thus, when we talk about the abacus representation of an ℓ-core λ, we will always mean the abacus of the form A λ = (0, b 1 , · · · , b ℓ−1 ). 1 Using these abaci, we offer the following lemma that will allow us to rule out the existence of partitions that are simultaneously ℓ-core and ℓ-regular for all but finitely many n.
Lemma 4.4. Suppose that A λ = (0, b 1 , . . . , b ℓ−1 ) is the abacus corresponding to an ℓ-core λ, and suppose that there is an integer k ≥ 0 such that for each 1 ≤ i ≤ ℓ − 1 we have either b i ≤ k or b i ≥ k + ℓ. If there is at least one j for which b j ≥ k + ℓ, then λ is not an ℓ-regular partition.
Remark. Let A λ = (0, b 1 , · · · , b ℓ−1 ) be the abacus of an ℓ-core λ. If min(b 1 , · · · b ℓ−1 ) ≥ ℓ, then the proof of the lemma will show that λ has a part of exact size ℓ. These are the cases where one can choose k = 0 in the lemma.
Proof. By our hypothesis, we may fix j for which b j ≥ k + ℓ. Let δ denote the total number of columns with length at least k + ℓ. Note that if B i and B i ′ are structure numbers corresponding to consecutive beads in column j between rows k + 1 and k + ℓ, then |i − i ′ | = δ. Further, we have In particular, the difference between parts corresponding to consecutive beads in column j between rows k + 1 and k + ℓ is fixed and coprime to ℓ. As a consequence, these parts form a modulus ℓ − δ arithmetic progression consisting of ℓ values. Thus, the parts cover all residue classes modulo ℓ, and so includes a part that is a multiple of ℓ. □ Example. Let ℓ = 3, and consider the 3-core abacus (0, 4, 1) as shown below.
Finally, consider the abacus with the bead in row 4 removed. One easily checks that the corresponding partition is 3-regular, demonstrating that the condition on the size of the gap in column lengths cannot be relaxed.
With two more observations, we will be able to construct an abacus which gives an upper bound for the size of an ℓ-regular ℓ-core partition.
For the purpose of obtaining N ℓ , the following lemma allows us to restrict our attention to those abaci where the b i are weakly increasing.
Proof. We may write and likewise n ′ = s k=1 B ′ k + s k=1 (k − s), where s remains the same because we have not changed the total number of beads. Since the second sum is the same in both expressions, it suffices to prove as desired, where the inequality holds since i < j and b i > b j . □

Proofs of our results
We are now in a position to prove Theorems 1.1 and 1.3, and Corollaries 1.2 and 1.4.