Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Coxeter planes as fixed points of Verlinde fusion rings

Max Boyle, Edmund Heng

TL;DR

The paper addresses folding ADE Coxeter planes through a hypergroup framework derived from Verlinde fusion rings. It develops an action of the even part of the Verlinde fusion ring on the ADE lattice, producing a fixed-point subspace identified with the Coxeter plane via a decomposition into irreducible $Z_+$-modules and regular elements. A key step is showing that the eigenstructure of the operator $Delta_1$ yields eigenvectors corresponding to the Coxeter plane, with an explicit Frobenius-Perron eigenvalue $FP(Delta_1)=2\cos(\pi/h)$. This approach extends folding methods beyond crystallographic types and links ADE classification with Verlinde fusion theory, contributing to the interplay between Coxeter theory, fusion rings, and hypergroups in representation-theoretic contexts.

Abstract

For the Coxeter groups of ADE type, we provide a construction of their Coxeter planes as fixed points of actions of hypergroups associated to Verlinde fusion rings. This builds upon the well-known ADE classification of $\mathbb{Z}_+$-modules over these fusion rings.

Coxeter planes as fixed points of Verlinde fusion rings

TL;DR

The paper addresses folding ADE Coxeter planes through a hypergroup framework derived from Verlinde fusion rings. It develops an action of the even part of the Verlinde fusion ring on the ADE lattice, producing a fixed-point subspace identified with the Coxeter plane via a decomposition into irreducible -modules and regular elements. A key step is showing that the eigenstructure of the operator yields eigenvectors corresponding to the Coxeter plane, with an explicit Frobenius-Perron eigenvalue . This approach extends folding methods beyond crystallographic types and links ADE classification with Verlinde fusion theory, contributing to the interplay between Coxeter theory, fusion rings, and hypergroups in representation-theoretic contexts.

Abstract

For the Coxeter groups of ADE type, we provide a construction of their Coxeter planes as fixed points of actions of hypergroups associated to Verlinde fusion rings. This builds upon the well-known ADE classification of -modules over these fusion rings.
Paper Structure (13 sections, 14 theorems, 42 equations, 1 figure)

This paper contains 13 sections, 14 theorems, 42 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries on Coxeter groups
  3. Coxeter groups and Coxeter elements
  4. Geometric representations and Coxeter planes
  5. Fusion Rings and hypergroup actions
  6. Fusion rings and non-negative Z-modules
  7. Hypergroups and fusion rings
  8. Action of hypergroups and non-negative Z-modules
  9. Verlinde fusion ring and its Z-modules
  10. Chebyshev polynomials
  11. The Fusion Ring $R_n$
  12. Irreducible non-negative Z-Modules of Verlinde fusion rings.
  13. Main theorem

Key Result

Theorem 1

Let $\Gamma$ be an ADE Dynkin diagram (see fig:CoxDyn) and let $V_\Gamma$ be the associated geometric representation of the Coxeter group $W$. There exists a hypergroup $\mathcal{H}_\Gamma$ acting on $V_\Gamma$ such that the subspace of fixed points of $\mathcal{H}_\Gamma$ is equal to the Coxeter pl

Figures (1)

  • Figure 1: Coxeter--Dynkin diagrams classifying the irreducible finite Coxeter groups. The ADE Dynkin diagrams consist of the $A_n$ and $D_n$ families, and the three exceptional types $E_6, E_7$ and $E_8$.

Theorems & Definitions (55)

  • Theorem 1: =\ref{['thm:main-theorem']}
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4: MR1581693
  • Definition 2.5
  • Proposition 2.6: See e.g. casselman_coxeter_elements
  • Definition 2.7
  • Definition 2.8
  • Definition 2.9
  • ...and 45 more