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The characteristic polynomials of imprimitive groups and affine Coxeter groups

Chenyue Feng, Shoumin Liu, Xumin Wang

TL;DR

This work develops a generator-aware framework for characteristic polynomials of finite and affine complex groups, showing that for imprimitive groups $G(r,1,n)$ and their products, as well as for irreducible affine Coxeter groups $\tilde{W}$ and the imprimitive groups $G(r,p,n)$, the characteristic polynomial $d(S,\rho)$ uniquely determines the representation $\rho$ up to isomorphism. Central to the method is the construction of a canonical word reduction via echelon form and admissible transformations, which allows comparing characters through word signatures and then deducing full representation data from $d(S,\rho)$. A general semidirect-product abelian-subgroup framework (Theorem 4.4) underpins the affine case, enabling analogous conclusions by combining finite-part conjugacy data with a translation lattice. The results provide a bridge between algebraic geometry and representation theory by linking determinant-based invariants to the complete representation theory of these groups, with concrete implications for $G(r,p,n)$ and affine extensions.

Abstract

In this paper, we will seek appropriate generators to define the characteristic polynomials of $G(r,1,n)$, and prove that for every finite dimensional representation of $G(r,1,n)$, the characteristic polynomial of $G(r,1,n)$ determines the character of this representation. Furthermore, the same conclusion holds for affine Coxeter groups \(\widetilde{W}\) and $G(r,p,n)$.

The characteristic polynomials of imprimitive groups and affine Coxeter groups

TL;DR

This work develops a generator-aware framework for characteristic polynomials of finite and affine complex groups, showing that for imprimitive groups and their products, as well as for irreducible affine Coxeter groups and the imprimitive groups , the characteristic polynomial uniquely determines the representation up to isomorphism. Central to the method is the construction of a canonical word reduction via echelon form and admissible transformations, which allows comparing characters through word signatures and then deducing full representation data from . A general semidirect-product abelian-subgroup framework (Theorem 4.4) underpins the affine case, enabling analogous conclusions by combining finite-part conjugacy data with a translation lattice. The results provide a bridge between algebraic geometry and representation theory by linking determinant-based invariants to the complete representation theory of these groups, with concrete implications for and affine extensions.

Abstract

In this paper, we will seek appropriate generators to define the characteristic polynomials of , and prove that for every finite dimensional representation of , the characteristic polynomial of determines the character of this representation. Furthermore, the same conclusion holds for affine Coxeter groups and .
Paper Structure (8 sections, 20 theorems, 81 equations)

This paper contains 8 sections, 20 theorems, 81 equations.

Table of Contents

  1. Introduction
  2. Signatures of words in $G(r,1,p)$
  3. Characteristic polynomials for $G(r,1,n)$ and $G(r_1,1,n_1)\times G(r_2,1,n_2)$
  4. Cases for $G(r,1,n)$
  5. Characteristic polynomials for $G(r_1,1,n_1)\times G(r_2,1,n_2)$
  6. Characteristic polynomials for affine Coxeter groups and $G(r,p,n)$
  7. Characteristic polynomials for Affine coxeter groups
  8. The subgroup $G(r,p,n)$ of $G(r,1,n)$

Key Result

Theorem 1.1

If $G = \{1, g_1, \cdots, g_n\}$ is a finite group, then Moreover, each $f_{\pi}(z)$ is an irreducible polynomial of degree $d_{\pi}$.

Theorems & Definitions (52)

  • Theorem 1.1
  • Definition 1.2
  • Definition 2.1
  • Proposition 2.2
  • proof
  • Definition 2.3
  • Theorem 2.4
  • proof
  • Definition 2.5
  • Definition 2.6
  • ...and 42 more