论文标题
$ \ ell $ core索引的对称组的角色表中的零
Zeros in the Character Tables of Symmetric Groups with an $\ell$-Core Index
论文作者
论文摘要
令$ \ MATHCAL {C} _n = \ left [χ_λ（μ）\右] _ {λ，μ} $是$ s_n的字符表，$ s $λ$ and $λ$和$μ$在$ p（n）上运行的$ n integer分区$n。$ n。 $c_λ（μ）$ in $ \ MATHCAL {C} _n，$，其中$λ$是每个Prime $ \ ell \ geQ 5的$ \ ell $ - cor-core分区的$n。$，$ $ $ \ geq 5，我们证明是$ z _ {\ ell}（n）（n）（n）sim sim sim-simα________的$n。$ σ_ {\ ell}（n+δ_ {\ ell}）P（n）\ gg _ {\ ell}除数函数，$δ_ {\ ell}：=（\ ell^2-1）/24，$和$ 1/α_ {\ ell}> 0 $是dirichlet $ l $ - value的归一化$ l \ left（\ left（\ frac {\ cdot} {\ ell} \ right），\ frac {\ ell-1} {\ ell-1} {2} {2} \ right）。$ for Primes $ \ ell $ \ ell $ and $ n> \ ell^6/24，$ nes $ nes $quy_λ（μ）此外，如果$ z^*_ {\ ell}（n）$是由两个$ \ ell $ -cores索引的零条目的数量，那么对于$ \ ell \ geq 5 $，我们获得了渐近的$ z^*_ {\ ell} _ {\ ell}（\ ell}（n）\ sim sim sim sim al} al al} _ { n+δ_ {\ ell}）^2 \ gg _ {\ ell} n^{\ ell-3}。 $$
Let $\mathcal{C}_n =\left [χ_λ(μ)\right]_{λ, μ}$ be the character table for $S_n,$ where the indices $λ$ and $μ$ run over the $p(n)$ many integer partitions of $n.$ In this note we study $Z_{\ell}(n),$ the number of zero entries $χ_λ(μ)$ in $\mathcal{C}_n,$ where $λ$ is an $\ell$-core partition of $n.$ For every prime $\ell\geq 5,$ we prove an asymptotic formula of the form $$ Z_{\ell}(n)\sim α_{\ell}\cdot σ_{\ell}(n+δ_{\ell})p(n)\gg_{\ell} n^{\frac{\ell-5}{2}}e^{π\sqrt{2n/3}},$$ where $σ_{\ell}(n)$ is a twisted Legendre symbol divisor function, $δ_{\ell}:=(\ell^2-1)/24,$ and $1/α_{\ell}>0$ is a normalization of the Dirichlet $L$-value $L\left(\left(\frac{\cdot}{\ell}\right),\frac{\ell-1}{2}\right).$ For primes $\ell$ and $n>\ell^6/24,$ we show that $χ_λ(μ)=0$ whenever $λ$ and $μ$ are both $\ell$-cores. Furthermore, if $Z^*_{\ell}(n)$ is the number of zero entries indexed by two $\ell$-cores, then for $\ell\geq 5$ we obtain the asymptotic $$ Z^*_{\ell}(n)\sim α_{\ell}^2 \cdot σ_{\ell}( n+δ_{\ell})^2 \gg_{\ell} n^{\ell-3}. $$
