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 $\ell \ge 5,$ we prove an asymptotic formula of the form $$Z_{\ell }(n)\sim {\alpha }_{\ell }\cdot {\sigma }_{\ell }(n+{\delta }_{\ell })p(n){\gg }_{\ell }{n}^{\frac{\ell -5}{2}}{e}^{\pi \sqrt{2n/3}},$$ 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 $Z_{\ell }^{\ast }(n)$ is the number of zero entries indexed by two $\ell$-cores, then for $\ell \ge 5$ we obtain the asymptotic $$Z_{\ell }^{\ast }(n)\sim {\alpha }_{\ell }^2\cdot {\sigma }_{\ell }(n+{\delta }_{\ell }{)}^2{\gg }_{\ell }{n}^{\ell -3}.$$

8 pages, 1 figure

Cited by in corpus (2)

• Common zeros of irreducible characters
• Asymptotic Properties of Maximal $p$-Core $p^{\prime }$-Partitions