论文标题
在某些McKay数量的对称组中
On Certain McKay Numbers of Symmetric Groups
论文作者
论文摘要
对于Primes $ \ ell $和非负整数$ a $，我们研究分区函数$$ p_ \ ell（a; n）：= \＃\ {λ\ vdash n：\ text {ord} _ \ ell（h（λ）= a \}，$ h（up）$ h（λ）$ DENITION $ deNITION $ n.这些分区值是在对称组的表示理论中，以麦凯数字$ m_ \ ell（\ text {ord} _ \ ell（n！） - a; s_n）$。我们确定$ p_ \ ell（a; n）$的生成函数在$ p_ \ ell（0; n）$方面和特定d'Arcais多项式的专业化。对于$ \ ell = 2 $和$ 3 $，我们给出了$ p_ \ ell（a; n）$的确切公式，并证明这些值几乎对所有$ n $均为零。对于较大的Primes $ \ ell $，$ p_ \ ell（a; n）$对于足够大的$ n $是阳性。尽管存在这种积极性，但我们证明$ p_ \ ell（a; n）$几乎总是可以用$ m $除整体$ m $。此外，通过这些结果，我们证明了几个Ramanujan型的一致性。这些包括一致性$$ p_ \ ell（a; \ ell^k n -δ（a，\ ell））\ equiv 0 \ pmod {\ ell^{k + 1}}，$$ $ 0 for $ 0 <a <a <\ ell $，$ \ ell $ \ ell = 5，7，7，7，7，7，11 $ and $ eell） Ono。
For primes $\ell$ and nonnegative integers $a$, we study the partition functions $$p_\ell(a;n):= \#\{λ\vdash n : \text{ord}_\ell(H(λ))=a\},$$ where $H(λ)$ denotes the product of hook lengths of a partition $λ$. These partition values arise as the McKay numbers $m_\ell(\text{ord}_\ell(n!) - a; S_n)$ in the representation theory of the symmetric group. We determine the generating functions for $p_\ell(a;n)$ in terms of $p_\ell(0;n)$ and specializations of specific D'Arcais polynomials. For $\ell = 2$ and $3$, we give an exact formula for the $p_\ell(a;n)$ and prove that these values are zero for almost all $n$. For larger primes $\ell$, the $p_\ell(a;n)$ are positive for sufficiently large $n$. Despite this positivity, we prove that $p_\ell(a;n)$ is almost always divisible by $m$ for any integer $m$. Furthermore, with these results we prove several Ramanujan-type congruences. These include the congruences $$p_\ell(a;\ell^k n - δ(a,\ell)) \equiv 0 \pmod{\ell^{k+1}},$$ for $0<a< \ell$, where $\ell = 5, 7, 11$ and $δ(a,\ell) := (\ell^2 - 1)/24 + a\ell$, which answer a question of Ono.
