Characteristic polynomials and some combinatorics for finite Coxeter groups

TL;DR

The paper studies whether the multivariable characteristic polynomial $d(S,\rho)$, defined by $d(S,\rho)=\det[x_0I+\sum_{i=1}^n x_i\rho(s_i)]$, uniquely determines a finite-dimensional representation $\rho$ of a finite Coxeter group $W$. To prove this, it introduces independent signature sequences (ISS) and lower triangular independent signature matrices (ISM) as the central combinatorial framework and establishes their existence for all irreducible finite Coxeter groups, both classical and exceptional. The main result is that $d(S,\rho_1)=d(S,\rho_2)$ implies $\rho_1\cong\rho_2$, yielding a semigroup isomorphism between representations and their characteristic polynomials; the polynomials generate the representation semigroup freely. The work combines structural Coxeter theory, signature-based character arguments, and computational verification (notably for exceptional types), providing explicit constructions and consequences for branching rules and examples.

Abstract

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 $ρ_1 \cong ρ_2$.

Theorems & Definitions (62)

• Theorem 1.1
• Definition 2.1
• Theorem 2.2: H
• Definition 2.3
• Definition 2.4
• Proposition 2.5: H
• Theorem 2.6: H
• Corollary 2.7
• Theorem 3.1: CST
• Definition 3.2