具有$\ell$核心索引的对称群的字符表中的零

Pub Date : 2022-05-19 DOI:10.4153/S0008439522000443
Eleanor Mcspirit, K. Ono
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引用次数: 3

摘要

抽象Let$\mathcal{C}_n=\left[\chi{\lambda}(\mu)\right]{\lLambda,\mu}$是$S_n,$的字符表,其中索引$\lamba$和$\mu$在$n的$p(n)$多个整数分区上运行。在本文中,我们研究$Z_{\ell}(n),$\mathcal中零项$\chi{\lambda}(\mu)$的数量{C}_n,$,其中$\lambda$是$n.$的$\ell$核心分区。对于每一个素数$\ell\geq5,$我们证明了形式为$$\boot{align*}Z_{\ell}{n)\sim\alpha_{ell}\cdot\sigma\{ell}{n+\delta_{el}p(n)\gg_{ll}n^{\frac{\ell-5}{2}}}}e^{\pi\sqrt{2n/3}}}}的n渐近公式,\ end{align*}$$$其中$\sigma{\eell}(n)$是一个扭曲的勒让德符号除数函数,$\delta{\el}:=(\ell^2-1)/24,$和$1/\alpha{\ell}>0$是Dirichlet L-值$L\left(\left)(\frac{\cdot}{\ell}\right),\frac{\ell-1}{2}\right.)的归一化$对于素数$\ell$和$n>\ell^6/24,我们证明了$\chi{\lambda}(\mu)=0$,只要$\lambda$和$\mu$都是$\ell$-核。此外,如果$Z^*{ell}(n)$是由两个$\ell-核索引的零条目的数量,那么,对于$\ell\geq5$,我们获得了渐近的$$$\boot{align*}Z^*{ell}(n)\sim\alpha_{ell}^2 \cdot\sigma_{ell}(n+\delta_{ell}})^2 3}。\结束{align*}$$
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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 $$ \begin{align*}Z^*_{\ell}(n)\sim \alpha_{\ell}^2 \cdot \sigma_{\ell}( n+\delta_{\ell})^2 \gg_{\ell} n^{\ell-3}. \end{align*} $$
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