Consider $S_{n}$ the symmetric group and for each $\sigma\in S_{n}$ let $U_{\sigma}$ be its $n\times n$ permutation matrix. Let $A$ be an Hermitian $n\times n$ matrix. I'm interested in computing the average $$ \mathbb{E}(A):=\sum_{\sigma \in S_{n}}{w(\sigma) U_{\sigma} A U_{\sigma}^{*}} $$

where the $w(\sigma)$ are some positive weight adding up to one.

For instance some natural weights are the ones coming from the Ewens's probability distribution of parameter $\theta>0$ on $S_{n}$ defined as

$$ w(\sigma)=\frac{\theta^{K(\sigma)}}{\theta(\theta+1)\ldots(\theta+n-1)} $$

where $K(\sigma)$ is the number of disjoint cycles of $\sigma$. The case of $\theta=1$ is simply the uniform distribution on $S_{n}$. For the case $\theta=1$, it is known that $$ \mathbb{E}(A)=\alpha \frac{ee^{T}}{n} + \Bigg(\frac{\mathrm{Tr}(A)-\alpha}{n-1}\Bigg)\Bigg(I_{n}-\frac{ee^{T}}{n}\Bigg) $$ where $e$ is the vector $e=(1,1,\ldots,1)$ and $\alpha=\frac{e^{T}Ae}{n}$.

My question are:

• Is there anything known about the averages for the more general case of $\theta>0$.

• Are there known asymptotics results as $n\to\infty$?

Consider $S_{n}$ the symmetric group and for each $\sigma\in S_{n}$ let $U_{\sigma}$ be its $n\times n$ permutation matrix. Let $A$ be an Hermitian $n\times n$ matrix. I'm interested in computing the average $$ \mathbb{E}(A):=\sum_{\sigma \in S_{n}}{w(\sigma) U_{\sigma} A U_{\sigma}^{*}} $$

where the $w(\sigma)$ are some positive weight adding up to one.

For instance some natural weights are the ones coming from the Ewens's probability distribution of parameter $\theta>0$ on $S_{n}$ defined as

$$ w(\sigma)=\frac{\theta^{K(\sigma)}}{\theta(\theta+1)\ldots(\theta+n-1)} $$

where $K(\sigma)$ is the number of disjoint cycles of $\sigma$. The case of $\theta=1$ is simply the uniform distribution on $S_{n}$. For the case $\theta=1$, it is known that $$ \mathbb{E}(A)=\alpha \frac{ee^{T}}{n} + \Bigg(\frac{\mathrm{Tr}(A)-\alpha}{n-1}\Bigg)\Bigg(I_{n}-\frac{ee^{T}}{n}\Bigg) $$ where $e$ is the vector $e^{T}=(1,1,\ldots,1)$ and $\alpha=\frac{e^{T}Ae}{n}$.

My question are:

• Is there anything known about the averages for the more general case of $\theta>0$.

• Are there known asymptotics results as $n\to\infty$?