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

August, 2007
Report Number: 
Steven N. Evans

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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