{"title":"具有$\\ell$核心索引的对称群的字符表中的零","authors":"Eleanor Mcspirit, K. Ono","doi":"10.4153/S0008439522000443","DOIUrl":null,"url":null,"abstract":"Abstract Let \n$\\mathcal {C}_n =\\left [\\chi _{\\lambda }(\\mu )\\right ]_{\\lambda , \\mu }$\n be the character table for \n$S_n,$\n where the indices \n$\\lambda $\n and \n$\\mu $\n run over the \n$p(n)$\n many integer partitions of \n$n.$\n In this note, we study \n$Z_{\\ell }(n),$\n the number of zero entries \n$\\chi _{\\lambda }(\\mu )$\n in \n$\\mathcal {C}_n,$\n where \n$\\lambda $\n is an \n$\\ell $\n -core partition of \n$n.$\n For every prime \n$\\ell \\geq 5,$\n we prove an asymptotic formula of the form \n$$ \\begin{align*}Z_{\\ell}(n)\\sim \\alpha_{\\ell}\\cdot \\sigma_{\\ell}(n+\\delta_{\\ell})p(n)\\gg_{\\ell} n^{\\frac{\\ell-5}{2}}e^{\\pi\\sqrt{2n/3}}, \\end{align*} $$\n where \n$\\sigma _{\\ell }(n)$\n is a twisted Legendre symbol divisor function, \n$\\delta _{\\ell }:=(\\ell ^2-1)/24,$\n and \n$1/\\alpha _{\\ell }>0$\n is a normalization of the Dirichlet L-value \n$L\\left (\\left ( \\frac {\\cdot }{\\ell } \\right ),\\frac {\\ell -1}{2}\\right ).$\n For primes \n$\\ell $\n and \n$n>\\ell ^6/24,$\n we show that \n$\\chi _{\\lambda }(\\mu )=0$\n whenever \n$\\lambda $\n and \n$\\mu $\n are both \n$\\ell $\n -cores. Furthermore, if \n$Z^*_{\\ell }(n)$\n is the number of zero entries indexed by two \n$\\ell $\n -cores, then, for \n$\\ell \\geq 5$\n , we obtain the asymptotic \n$$ \\begin{align*}Z^*_{\\ell}(n)\\sim \\alpha_{\\ell}^2 \\cdot \\sigma_{\\ell}( n+\\delta_{\\ell})^2 \\gg_{\\ell} n^{\\ell-3}. \\end{align*} $$","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"3","resultStr":"{\"title\":\"Zeros in the character tables of symmetric groups with an \\n$\\\\ell $\\n -core index\",\"authors\":\"Eleanor Mcspirit, K. Ono\",\"doi\":\"10.4153/S0008439522000443\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Abstract Let \\n$\\\\mathcal {C}_n =\\\\left [\\\\chi _{\\\\lambda }(\\\\mu )\\\\right ]_{\\\\lambda , \\\\mu }$\\n be the character table for \\n$S_n,$\\n where the indices \\n$\\\\lambda $\\n and \\n$\\\\mu $\\n run over the \\n$p(n)$\\n many integer partitions of \\n$n.$\\n In this note, we study \\n$Z_{\\\\ell }(n),$\\n the number of zero entries \\n$\\\\chi _{\\\\lambda }(\\\\mu )$\\n in \\n$\\\\mathcal {C}_n,$\\n where \\n$\\\\lambda $\\n is an \\n$\\\\ell $\\n -core partition of \\n$n.$\\n For every prime \\n$\\\\ell \\\\geq 5,$\\n we prove an asymptotic formula of the form \\n$$ \\\\begin{align*}Z_{\\\\ell}(n)\\\\sim \\\\alpha_{\\\\ell}\\\\cdot \\\\sigma_{\\\\ell}(n+\\\\delta_{\\\\ell})p(n)\\\\gg_{\\\\ell} n^{\\\\frac{\\\\ell-5}{2}}e^{\\\\pi\\\\sqrt{2n/3}}, \\\\end{align*} $$\\n where \\n$\\\\sigma _{\\\\ell }(n)$\\n is a twisted Legendre symbol divisor function, \\n$\\\\delta _{\\\\ell }:=(\\\\ell ^2-1)/24,$\\n and \\n$1/\\\\alpha _{\\\\ell }>0$\\n is a normalization of the Dirichlet L-value \\n$L\\\\left (\\\\left ( \\\\frac {\\\\cdot }{\\\\ell } \\\\right ),\\\\frac {\\\\ell -1}{2}\\\\right ).$\\n For primes \\n$\\\\ell $\\n and \\n$n>\\\\ell ^6/24,$\\n we show that \\n$\\\\chi _{\\\\lambda }(\\\\mu )=0$\\n whenever \\n$\\\\lambda $\\n and \\n$\\\\mu $\\n are both \\n$\\\\ell $\\n -cores. Furthermore, if \\n$Z^*_{\\\\ell }(n)$\\n is the number of zero entries indexed by two \\n$\\\\ell $\\n -cores, then, for \\n$\\\\ell \\\\geq 5$\\n , we obtain the asymptotic \\n$$ \\\\begin{align*}Z^*_{\\\\ell}(n)\\\\sim \\\\alpha_{\\\\ell}^2 \\\\cdot \\\\sigma_{\\\\ell}( n+\\\\delta_{\\\\ell})^2 \\\\gg_{\\\\ell} n^{\\\\ell-3}. \\\\end{align*} $$\",\"PeriodicalId\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2022-05-19\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"3\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.4153/S0008439522000443\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.4153/S0008439522000443","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
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
$$ \begin{align*}Z^*_{\ell}(n)\sim \alpha_{\ell}^2 \cdot \sigma_{\ell}( n+\delta_{\ell})^2 \gg_{\ell} n^{\ell-3}. \end{align*} $$