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 $$Z_t(N):=\#\{(\lambda ,\mu ):{\chi }_{\lambda }(\mu )=0\text{with λ a t-core}\}.$$ We prove $$Z(N)\ge \frac{2p(N{)}^2}{\log N}\left(1+O\left(\frac{1}{\sqrt{\log N}}\right)\right),$$ where $p(N)$ denotes the number of partitions of $N$. We also give explicit lower bounds for $Z_t(N)$ in various ranges of $t$.

19 pages