On Certain McKay Numbers of Symmetric Groups

Abstract

For primes $\ell$ and nonnegative integers $a$, we study the partition functions $$p_{\ell }(a;n):=\#\{\lambda ⊢n:{\text{ord}}_{\ell }(H(\lambda ))=a\},$$ 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 $$p_{\ell }(a;{\ell }^{k}n-\delta (a,\ell ))\equiv 0(\mathrm{mod}{\ell }^{k+1}),$$ 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.

arXiv comment15 pages, updated with referee comments. Ramanujan J (2023)