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Coxeter groups and the proper joint spectrums of their faithful representations

Shoumin Liu, Zhaohuan Peng, Xumin Wang

TL;DR

The paper tackles the problem of determining a Coxeter group with finite bonds from the proper joint spectrum of a faithful representation. It develops a purely algebraic framework, starting with the dihedral case $W(\mathrm I_2(n))$ to compute irreducible spectra via determinants $\det(-I+x_1\rho(s_1)+x_2\rho(s_2))$, and then derives a faithful-representation criterion in terms of kernel intersections of irreducible constituents. The main result shows that, for any Coxeter group with finite bonds, the proper joint spectrum $U$ of a faithful representation uniquely determines the orders $m_{ij}=\mathrm{ord}(s_i s_j)$, by analyzing the restricted spectra on generator pairs and their irreducible components. This establishes a close link between generating relations, representation theory, and geometric spectra, and motivates further exploration of connections to Hecke algebras and Kazhdan-Lusztig polynomials.

Abstract

In this paper, we analyze the faithful representations of the dihedral groups, and prove that the Coxeter groups can be determined by the proper joint spectrum of their faithful representations.

Coxeter groups and the proper joint spectrums of their faithful representations

TL;DR

The paper tackles the problem of determining a Coxeter group with finite bonds from the proper joint spectrum of a faithful representation. It develops a purely algebraic framework, starting with the dihedral case to compute irreducible spectra via determinants , and then derives a faithful-representation criterion in terms of kernel intersections of irreducible constituents. The main result shows that, for any Coxeter group with finite bonds, the proper joint spectrum of a faithful representation uniquely determines the orders , by analyzing the restricted spectra on generator pairs and their irreducible components. This establishes a close link between generating relations, representation theory, and geometric spectra, and motivates further exploration of connections to Hecke algebras and Kazhdan-Lusztig polynomials.

Abstract

In this paper, we analyze the faithful representations of the dihedral groups, and prove that the Coxeter groups can be determined by the proper joint spectrum of their faithful representations.

Paper Structure

This paper contains 5 sections, 5 theorems, 16 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Some basic notions
  3. Proper joint spectrum of an irreducible representation of $W({\rm I}_{2}(n))$
  4. The faithfulness for a representation $\rho$ of $W({\rm I}_{2}(n))$
  5. Main theorem

Key Result

Lemma 3.1

For the finite dihedral group $W({\rm I}_{2}(n))$, its proper joint spectrums for irreducible representations can be presented in the following three tables. Table $1$ is for the case $n=2$; the Table $2$ is for the case $n>2$, $2\nmid n$, and $1\leq k\leq \frac{n-1}{2}$; the Table $3$ is for the ca

Theorems & Definitions (11)

  • Definition 2.1
  • Definition 2.2
  • Lemma 3.1
  • Theorem 4.1
  • proof
  • Corollary 4.2
  • Theorem 4.3
  • proof
  • Theorem 5.1
  • proof
  • ...and 1 more