Coxeter matrix of Dyck path

Question

I am trying to understand Gjergji Zaimi's answer in What are the periodic Dyck paths?. In the third paragraph he claims that

Next, we define the matrix $X_D$ similarly to the Cartan matrix except we put ones in coordinates $(i,i+c_i−1)$ and zeros everywhere else. So $X_D$ essentially consists of just the "righmost" 1's in the Cartan matrix. The matrix $Y_D$ is defined as the matrix with −1 's in positions $(u,u+1),(u,u+2),…,(u,v)$ as $(u,v)$ ranges through all the valleys, and zeros everywhere else. Finally let $A_n$ be the matrix with 1's in entries $(i,i+1)$ for i=1,…,n−1 . One can check the following explicit form for the Coxeter matrix of a Dyck path: $\phi_D=A_n+Y_D−X^T_D$.

My question is: why is $\phi_D=A_n+Y_D−X^T_D$?
So far, I tried to compute it straightforwardly, but that didn't work.

Answer 1

I will keep the notation and terminology from that question. First we can compute the inverse of the Cartan matrix: The matrix $C_D^{-1}$ has entries $$ (C_D)^{-1}_{(i,j)}= \begin{cases} 1, & \text{if }\, i=j \\ 1, & \text{if } (i,j)\, \text{ is a valley of } C_D\\ -1, & \text{if }\, j=i+1 \\ 0, & \text{otherwise} \\ \end{cases} $$ You can check this explicitly by multiplying this matrix with $C_D$ and ensuring that you get the identity matrix. Thus you can write $C_D^{-1}=I_n-A_n+V_n$ where $V_n$ denotes the matrix that has an entry $1$ at the locations of the valleys of $C_D$ and $0$ everywhere else. So far we have $$\phi_D=-(I_n-A_n+V_n)C_D^{T}.$$ The matrix $A_nC_D^T$ is easy to describe: Remove the first row from $C_D^T$ and add row of zeros at the bottom. Thus $(I_n-A_n)C_D^T$ is the matrix with $-1$'s on the entries right above the diagonal, and $1$'s on the entries $(i+c_i-1, i)$. This gives us $(I_n-A_n)C_D^T=X_D^T-A_n.$

Next, the matrix $V_nC_D^T$ can also be checked to have the following entries: $1$'s in positions $(i,i+1), (i,i+2),\dots, (i,j)$ whenever $(i,j)$ is a valley and $0$ everywhere else. This gives us $V_nC_D^T=-Y_D$.