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We consider the asymptotic behavior as $n\to\infty$ of the spectra of randommatrices 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 applyingan 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 eigenvalueseither +1 or -1]. Irreducible representations of the symmetric group on$\{1,2,...,n\}$ are indexed by partitions $\lambda_n$ of $n$. A consequence ofthe 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 Youngdiagram 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 spectralmeasure of the resulting random matrix converges in distribution to a randomprobability 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.

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