Spectra of random linear combinations of matrices defined via representations and Coxeter generators of the symmetric group

Abstract

We consider the asymptotic behavior as $n\to\infty$ of the spectra of random matrices of the form \[\frac{1}{\sqrt{n-1}}\sum_{k=1}^{n-1}Z_{nk}ρ_n ((k,k+1)),\] where for each $n$ the random variables $Z_{nk}$ are i.i.d. standard Gaussian and the matrices $ρ_n((k,k+1))$ are obtained by applying an irreducible unitary representation $ρ_n$ of the symmetric group on $\{1,2,...,n\}$ to the transposition $(k,k+1)$ that interchanges $k$ and $k+1$ [thus, $ρ_n((k,k+1))$ is both unitary and self-adjoint, with all eigenvalues either +1 or -1]. Irreducible representations of the symmetric group on $\{1,2,...,n\}$ are indexed by partitions $λ_n$ of $n$. A consequence of the results we establish is that if $λ_{n,1}\geλ_{n,2}\ge...\ge0$ is the partition of $n$ corresponding to $ρ_n$, $μ_{n,1}\geμ_{n,2}\ge >...\ge0$ is the corresponding conjugate partition of $n$ (i.e., the Young diagram of $μ_n$ is the transpose of the Young diagram of $λ_n$), $\lim_{n\to\infty}\frac{λ_{n,i}}{n}=p_i$ for each $i\ge1$, and $\lim_{n\to\infty}\frac{μ_{n,j}}{n}=q_j$ for each $j\ge1$, then the spectral measure of the resulting random matrix converges in distribution to a random probability measure that is Gaussian with random mean $θZ$ and variance $1-θ^2$, where $θ$ is the constant $\sum_ip_i^2-\sum_jq_j^2$ and $Z$ is a standard Gaussian random variable.