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

Report Number
736
Authors
Steven N. Evans
Abstract

We consider the asymptotic behavior as $n \rightarrow \infty$ 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 $\{1,2,\ldots,n\}$ 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 $\{1,2,\ldots,n\}$ are indexed by partitions $\lambda_n$ of $n$. A consequence of the results we establish is that if $\lambda_{n,1} \ge \lambda_{n,2} \ge \cdots \ge 0$ is the partition of $n$ corresponding to $\rho_n$, $\mu_{n,1} \ge \mu_{n,2} \ge \cdots \ge 0$ is the corresponding conjugate partition of $n$ (that is, the Young diagram of $\mu_n$ is the transpose of the Young diagram of $\lambda_n$), $\lim_{n \rightarrow \infty} \frac{\lambda_{n,i}}{n} = p_i$ for each $i \ge 1$, and $\lim_{n \rightarrow \infty} \frac{\mu_{n,j}}{n} = q_j$ for each $j \ge 1$, then the spectral measure of the resulting random matrix converges in distribution to a random probability measure that is Gaussian with mean $\theta Z$ and variance $1 - \theta^2$, where $\theta$ is the constant $\sum_i p_i^2 - \sum_j q_j^2$ and $Z$ is a standard Gaussian random variable.

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