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Representations of Coxeter groups over fusion rings and hyperplane complements

Edmund Heng, Luis Paris

Abstract

We study faithful realisations of Coxeter groups over fusion rings and study Vinberg systems associated to them. We show that they induce embeddings of hyperplane complements, which provide geometrical realisations of certain types of strong admissible (LCM) homomorphisms between Artin--Tits groups.

Representations of Coxeter groups over fusion rings and hyperplane complements

Abstract

We study faithful realisations of Coxeter groups over fusion rings and study Vinberg systems associated to them. We show that they induce embeddings of hyperplane complements, which provide geometrical realisations of certain types of strong admissible (LCM) homomorphisms between Artin--Tits groups.

Paper Structure

This paper contains 22 sections, 26 theorems, 74 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Hyperplane systems and Coxeter groups
  3. Fusion rings
  4. Main results
  5. Preliminaries
  6. Coxeter systems and realisations
  7. Realisations over real numbers
  8. Realisations over fusion rings
  9. Fusion rings
  10. Frobenius--Perron dimension
  11. Geometric realisations over fusion rings
  12. Main examples
  13. From geometric realisations over fusion rings to realisations over integers
  14. Vinberg systems and embeddings of hyperplane complements
  15. Contragredient representations from fusion rings
  16. ...and 7 more sections

Key Result

theorem 1.1

The set of hyperplanes $\mathcal{A}$ can be obtained from $\check{\mathcal{A}}$ as follows: Moreover, $T^\circ \subseteq \check{T}^\circ$, and so the inclusion $\Theta\subseteq \check{\Theta}$ restricts to an embedding of hyperplane complements

Figures (1)

  • Figure 1: The hyperplane complement $T^\circ \setminus \bigcup_{H \in \mathcal{A}} H = \Theta \setminus \bigcup_{H \in \mathcal{A}} H$ for $\mathbb W\cong D_5$, with hyperplanes $\check{H}_? \in \check{\mathcal{A}}$ associated to $\check{\mathbb W} \cong S_5$ indicated. The 10 hyperplanes for $\check{\mathbb W}$ (labelled by the positive roots) intersect in pairs to give the 5 hyperplanes (lines) for $\mathbb W$ in $\Theta$.

Theorems & Definitions (86)

  • theorem 1.1: =\ref{['thm:hyperplanesystem']}
  • theorem 1.2: =\ref{['thm:hyperplanecompembedding']}
  • definition 2.1
  • definition 2.2
  • remark 2.3: Non-commutative subtlety in matrix representations
  • proposition 2.4: see e.g. bjorner_anders_brenti_2010
  • definition 2.5
  • remark 2.6
  • definition 3.1
  • remark 3.2
  • ...and 76 more