Eigenvalue Fluctuations of Symmetric Group Permutation Representations on k-tuples and k-subsets

Let the term $k$-representation refer to the permutation representations of the symmetric group ${\mathfrak{S}}_n$ on $k$-tuples and $k$-subsets as well as the $S^{(n-k,1^k)}$ irreducible representation of ${\mathfrak{S}}_n$. Endow ${\mathfrak{S}}_n$ with the Ewens distribution and let $\alpha$ and $\beta$ be linearly independent irrational numbers over $\mathbb{Q}$. Then for fixed $k > 1$weshowthatas$n \to \infty$, the normalized count of the number of eigenangles in a fixed interval $(\alpha ,\beta )$ of a $k$-representation evaluated at a random element $\sigma \in {\mathfrak{S}}_n$ converges weakly to a compactly supported distribution. In particular, we compute the limiting moments and moreover provide an explicit formula for the limiting density when $k=2$ and the Ewens parameter $\theta =1$ (uniform probability measure). This is in contrast to the $k=1$ case where it has been shown previously that the distribution is asymptotically Gaussian.