The standard representation of symmetry group

Question

The standard representation of $S_n$ is usually realized by restriction to the subspace with zero sum of coordinates of the permutation representation of $n$ basis vectors. But I think the action of $S_n$ on vertices of a Regular $(n-1)$-simplex in $n-1$ dimensional space centered at the origin also gives the standard representation.( for example, $S_3$ acts on the triangle in $\mathbb R^2$) How to show this rigorously?

Answer 1

Seems like you are interested in the simplex representation of the Potts model in statistical physics.

The paper "Functional RG approach to the Potts model" seems to have an explicit construction of the vectors from the origin to the vertices of the simplex.