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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2504.19468 |
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Table of Contents:
- Let $W$ be a finite Coxeter group with Coxeter generating set $S=\{s_1,\ldots,s_n\}$, and $ρ$ be a complex finite dimensional representation of $W$. The characteristic polynomial of $ρ$ 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 $ρ_1$ and $ρ_2$ of $W$, $d(S,ρ_1)=d(S,ρ_2)$ if and only if $ρ_1 \cong ρ_2$.