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

TL;DR

This paper studies the asymptotic spectral distribution of certain random matrices derived from symmetric group representations, revealing a Gaussian limit with random mean and variance depending on the representation's partition structure.

Contribution

It establishes the limiting spectral measure for matrices constructed from symmetric group representations, connecting representation theory with random matrix asymptotics.

Findings

Spectral measure converges to a Gaussian distribution with random mean and variance.

Limit depends on the asymptotic shape of the partitions defining the representations.

Results link representation theory of symmetric groups to spectral properties of random matrices.

Abstract

We consider the asymptotic behavior as $n\to\infty$ of the spectra of random matrices of the form \[\frac{1}{\sqrt{n-1}}\sum_{k=1}^{n-1}Z_{nk}\rho_n ((k,k+1)),\] where for each $n$ the random variables $Z_{nk}$ are i.i.d. standard Gaussian and the matrices $\rho_n((k,k+1))$ are obtained by applying an irreducible unitary representation $\rho_n$ of the symmetric group on $\{1,2,...,n\}$ to the transposition $(k,k+1)$ that interchanges $k$ and $k+1$ [thus, $\rho_n((k,k+1))$ 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 $\lambda_n$ of $n$. A consequence of the results we establish is that if $\lambda_{n,1}\ge\lambda_{n,2}\ge...\ge0$ is the partition of $n$ corresponding to $\rho_n$, $\mu_{n,1}\ge\mu_{n,2}\ge >...\ge0$ is the corresponding conjugate partition of $n$ (i.e., the Young diagram of $\mu_n$ is the transpose of the Young diagram of $\lambda_n$), $\lim_{n\to\infty}\frac{\lambda_{n,i}}{n}=p_i$ for each $i\ge1$, and $\lim_{n\to\infty}\frac{\mu_{n,j}}{n}=q_j$ 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 $\theta Z$ and variance $1-\theta^2$, where $\theta$ is the constant $\sum_ip_i^2-\sum_jq_j^2$ and $Z$ is a standard Gaussian random variable.