Characteristic polynomials and some combinatorics for finite Coxeter groups

2025/04/28 by Shoumin Liu, Yu-Xiang Wang, Liu, Shoumin +1
Mathematics · #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Markov Chains and Monte Carlo Methods

paper · pdf · doi:10.48550/arxiv.2504.19468

Abstract

Let W be a finite Coxeter group with Coxeter generating set S=\s₁,…,s_n\, and ρ be a complex finite dimensional representation of W. The characteristic polynomial of ρ is defined as d(S,ρ)=det[x₀I+x₁ρ(s₁)+⋯+x_nρ(s_n)], 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 ρ₁ and ρ₂ of W, d(S,ρ₁)=d(S,ρ₂) if and only if ρ₁ ≅ ρ₂.

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