On Certain McKay Numbers of Symmetric Groups

Abstract

For primes $\ell$ and nonnegative integers $a$, we study the partition functions $$p_\ell(a;n):= \#\{\lambda \vdash n : \text{ord}_\ell(H(\lambda))=a\},$$ where $H(\lambda)$ denotes the product of hook lengths of a partition $\lambda$. 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 $$p_\ell(a;\ell^k n - \delta(a,\ell)) \equiv 0 \pmod{\ell^{k+1}},$$ for $0<a< \ell$, where $\ell = 5, 7, 11$ and $\delta(a,\ell) := (\ell^2 - 1)/24 + a\ell$, which answer a question of Ono.