Characteristic polynomials and some combinatorics for finite Coxeter groups

Let $W$ be a finite Coxeter group with Coxeter generating set $S=\{s_1,\ldots,s_n\}$, and $\rho$ be a complex finite dimensional representation of $W$. The characteristic polynomial of $\rho$ is defined as \begin{equation*} d(S,\rho)=\det[x_0I+x_1\rho(s_1)+\cdots+x_n\rho(s_n)], \end{equation*} where $I$ is the identity operator. In this paper, we show the existence of a combinatorics structure within $W$, and thereby prove that for any two complex finite dimensional representations $\rho_1$ and $\rho_2$ of $W$, $d(S,\rho_1)=d(S,\rho_2)$ if and only if $\rho_1 \cong \rho_2$.