Descents and flag major index on conjugacy classes of colored permutation groups without short cycles

Abstract: We consider the descent and flag major index statistics on the colored permutation groups, which are wreath products of the form $\mathfrak{S}{n,r}=\mathbb{Z}_r\wr \mathfrak{S}_n$. We show that the $k$-th moments of these statistics on $\mathfrak{S}{n,r}$ will coincide with the corresponding moments on all conjugacy classes without cycles of lengths $1,2,\ldots,2k$. Using this, we establish the asymptotic normality of the descent and flag major index statistics on conjugacy classes of $\mathfrak{S}_{n,r}$ with sufficiently long cycles. Our results generalize prior work of Fulman involving the descent and major index statistics on the symmetric group $\mathfrak{S}_n$. Our methods involve an intricate extension of Fulman's work on $\mathfrak{S}_n$ combined with the theory of the degree for a colored permutation statistic, as introduced by Campion Loth, Levet, Liu, Sundaram, and Yin.