UC Berkeley

For each k ≥ 1, we study the eigenvalue distributions of permutation representations of the symmetric group 𝔖_n on k-tuples and k-subsets as well as the S^{(n-k,1,...,1)} irreducible

representation of 𝔖_n (we will call these k-representations).

First, we consider the fluctuations of eigenvalue statistics. Endow 𝔖_n with the uniform measure and let α and θ be linearly independent irrational numbers over ℚ. Then we show that as n → ∞, the scaled count of the number of eigenangles in a fixed interval (α, θ) of a k-representation evaluated at a random element σ ∈ 𝔖_n converges weakly to a compactly supported distribution. In particular, we compute the limiting moments and moreover provide a formula for the limiting density in the case k = 2.

Next, we study the scaled eigenvalue point process of the k-representations. Here, we endow 𝔖_n with the Ewens measure of parameter θ for which the uniform distribution corresponds to the special case θ = 1. The eigenvalue point processes exhibit drastically different behavior depending on whether one “zooms in” at rational angles or irrational angles. We obtain the limiting eigenvalue point process at rational angles as well as angles of irrationality measure 2 for all k-representations. A power series representation for the limiting gap probability is also given.

Finally, we study the distribution of entries of a random permutation matrix under a “randomized basis.” This means that we conjugate the random permutation matrix by a random orthogonal matrix drawn from Haar measure. Then it is shown that under certain conditions, the linear combination of entries of a random permutation matrix under a “randomized basis” converges to a sum of independent variables sY + Z where Y is Poisson distributed, Z is normally distributed, and s is a constant.