Each permutaion $\pi\in S_n$ corresponds to a permutation matrix $P_\pi$ with $(0,1)$ entries. To be more specific, $P_\pi = [e_{\pi(1)},\ldots,e_{\pi(n)}]$, where $e_1,\ldots,e_n$ are standard basis of $\mathbb{R}^n$.
Note that $P_\pi \mathbb{1}=\mathbb{1}$, where $\mathbb{1}$ is a vector with all $1$ elements. Then we can block diagonalize $P_\pi$ for $\pi\in S_n$ simultaneously. See this problem about block diagonalization. There is an orthogonal change of basis $V$ such that $$V^t P_\pi V = \begin{bmatrix} \hat{P_\pi} & 0\\ 0 & 1 \end{bmatrix} $$ for all $\pi\in S_n$. I'm reading a paper, which says that $\{\hat{P}_\pi,\pi\in S_n\}$ can be interpreted as the symmetry group of a regular simplex in $\mathbb{R}^{n-1}$. I need more details about why it is true. Thanks for reading! Any comments are appreciated.
