Lower Bound for The Number of Zeros in The Character Table of The Symmetric Group

Abstract

For any two partitions $\lambda$ and $\mu$ of a positive integer $N$, let $\chi_{\lambda}(\mu)$ be the value of the irreducible character of the symmetric group $S_{N}$ associated with $\lambda$, evaluated at the conjugacy class of elements whose cycle type is determined by $\mu$. Let $Z(N)$ be the number of zeros in the character table of $S_N$, and $Z_{t}(N)$ be defined as $$ Z_{t}(N):= \#\{(\lambda,\mu): \chi_{\lambda}(\mu) = 0 \; \text{with $\lambda$ a $t$-core}\}. $$ We prove $$ Z(N) \ge \frac{2\, p(N)^{2}}{\log N} \left(1+O\left(\frac{1}{\sqrt{\log N}}\right)\right), $$ where $p(N)$ denotes the number of partitions of $N$. We also give explicit lower bounds for $Z_t(N)$ in various ranges of $t$.