Each permutation $\pi\in S_n$ 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.
The convex hull of the coordinate basis $\{e_1,\cdots,e_n\}$ is a simplex,
$$ \Bigl\{(x_1,\cdots,x_n)\,\Big|\, {0\le x_1,\cdots,x_n\le1 \atop x_1+\cdots+x_n=1}\Bigr\}, $$
which resides in the affine hyperplane defined by $x_1+\cdots+x_n=1$, parellel to the hyperplane defined by the homogeneous equation $x_1+\cdots+x_n=0$ which is the orthogonal complement of the vector $\vec{\mathbf{1}}=(1,\cdots,1)$.
Any symmetry (linear isometry) of the simplex can be extended to all of $\mathbb{R}^n$ (by fiat fixing $\vec{\mathbf{1}}$) and thus acts on the simplex within the affine hyperplane. The action must preserve the corners, the coordinate basis vectors. Such operators are none other than permutation matrices.