A Relationship Between Character Values Of Wreath Products And The Symmetric Group

A relation between certain irreducible character values of the hyperoctahedral group $B_n$ ($\mathbb{Z}/2\mathbb{Z} \wr S_n$) and the symmetric group $S_{2n}$ was proved by F. L\"ubeck and D. Prasad in 2021. Their proof is algebraic in nature and uses Lie theory. Using combinatorial methods, R. Adin and Y. Roichman proved a similar relation between certain character values of $G\wr S_n$ and $S_{rn}$, where $G$ is an abelian group of order $r$ (generalizing the result of L\"ubeck-Prasad). Using their result, we prove yet another relation between certain irreducible character values of $G\wr S_n$ and $S_{rn}$, where $G$ is an abelian group of order $r$.