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 $$ \begin{align*}Z^*_{\ell}(n)\sim \alpha_{\ell}^2 \cdot \sigma_{\ell}( n+\delta_{\ell})^2 \gg_{\ell} n^{\ell-3}. \end{align*} $$