Characteristic polynomials and some combinatorics for finite Coxeter groups

Abstract

Let $W$ be a finite Coxeter group with Coxeter generating set $S=\{s_1,\dots ,s_n\}$, and $\ddot{I}$ be a complex finite dimensional representation of $W$. The characteristic polynomial of $\ddot{I}$ is defined as \begin{equation*} d(S,ρ)=\det[x_0I+x_1ρ(s_1)+\cdots+x_nρ(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 $\ddot{I}{}_1$ and $\ddot{I}{}_2$ of $W$, $d(S,\ddot{I}{}_1)=d(S,\ddot{I}{}_2)$ if and only if $\ddot{I}{}_1\cong \ddot{I}{}_2$.

29pages