Symmetric group: equation between standard representation and natural matrix representation

Question

I was reading this article and didn't understand their claim. Let $$A = \begin{pmatrix} 1 & 0& 0 &\cdots & 0 \\ -1& 1 & 0 &\cdots & 0\\ 0& -1& 1 &\cdots & 0\\ 0& 0& -1& \cdots & 0\\ \vdots &\vdots &\vdots & \ddots & 1\\ 0 & 0 & 0 &\cdots& -1 \\ \end{pmatrix},$$ so that $A^TA = C$, the Cartan matrix associated to the symmetric group. Let $\pi$ be a permutation, which we identify with the matrix expressed as 1's and 0's as usual. Let $P$ be the matrix representation of $\pi$ given by the action on the simple roots, in the coordinates defined by those simple roots. Then the paper simply says "We make the observation that $AP = \pi A$". I don't understand what happened there.

Answer 1

In your notation, $A=(a_1,a_2,\cdots,a_{n-1})$, where $a_i=e_i-e_{i+1}$ are the simple roots and viewed as column vectors. The matrix $P=(p_{ij})$ is defined by $\pi(a_i)=\sum_j p_{ji}a_j$.

Therefore $$\pi A=(\pi(a_1),\cdots,\pi(a_{n-1}))=(\sum_j p_{j1}a_j,\cdots,\sum_j p_{j(n-1)}a_j)=(a_1,\cdots,a_{n-1})P=AP$$.