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

We consider the asymptotic behavior as $n\to \mathrm{\infty }$ 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 $\{1,2,\mathrm{.}\mathrm{.}\mathrm{.},n\}$ 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 $\{1,2,\mathrm{.}\mathrm{.}\mathrm{.},n\}$ are indexed by partitions ${\lambda }_n$ of $n$. A consequence of the results we establish is that if ${\lambda }_{n,1}\ge {\lambda }_{n,2}\ge \mathrm{.}\mathrm{.}\mathrm{.}\ge 0$ is the partition of $n$ corresponding to ${\rho }_n$, $\mu_{n,1}\ge\mu_{n,2}\ge

…\ge0$isthecorrespondingconjugatepartitionof$n$ (i.e., the Young diagram of ${\mu }_n$ is the transpose of the Young diagram of ${\lambda }_n$), ${\lim }_{n\to \mathrm{\infty }}\frac{{\lambda }_{n,i}}{n}=p_i$ for each $i\ge 1$, and ${\lim }_{n\to \mathrm{\infty }}\frac{{\mu }_{n,j}}{n}=q_j$ 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 random 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.