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On representations of the triplet group and some of its extensions

Mohamad N. Nasser, Nafaa Chbili, Khaled Qazaqzeh

TL;DR

This work investigates representations of the triplet group $L_n$ and its extensions $VL_n$ and $WL_n$, focusing on linear and automorphism-based realizations. It proves the irreducibility of the classical Tits representation $\Theta$ over $\mathbb{C}$, constructs a new representation $\mu: L_n \to \mathrm{Aut}(\mathbb{F}_n)$ with an explicit matrix form via $\mu'$, and establishes faithfulness only in small ranks. It fully classifies complex homogeneous $2$-local representations of $L_n$ (and non-homogeneous cases for $L_3$), showing these correspond to specializations of $\mu'$ and detailing their faithfulness and reducibility. The paper then extends representations to the virtual and welded triplet groups, identifying standard extension mechanisms and classifying nontrivial homogeneous $2$-local representations of $VL_n$ and $WL_n$, including conditions for faithfulness and irreducibility. Overall, the results advance understanding of how $L_n$-representations behave under extensions to $VL_n$ and $WL_n$, and they pose open questions about further extensions beyond the studied framework.

Abstract

In this paper, we study the representations of the triplet group $L_n$, where $n$ is a positive integer, and its extensions to the virtual and welded triplet groups $VL_n$ and $WL_n$, respectively. We first introduce $L_n$, its extensions, and its pure subgroup. We then investigate several representations, proving the irreducibility of the classical Tits representation $Θ: L_n \to \mathrm{GL}_{n-1}(\mathbb{C})$ over the complex field $\mathbb{C}$ and constructing a new representation $μ: L_n \to \mathrm{Aut}(\mathbb{F}_n)$, where $\mathbb{F}_n$ is the free group of rank $n$. For the representation $μ$, we determine its matrix form, faithfulness, and irreducibility. We also classify all complex homogeneous $2$-local representations of $L_n$ for $n \ge 3$ and all non-homogeneous $2$-local representations of $L_3$, establishing connections with the complex specialization of the representation $μ$. Finally, we examine extensions of $L_n$ representations to $VL_n$ and $WL_n$, proving their existence, classifying non-trivial complex homogeneous $2$-local representations, and analyzing their faithfulness and irreducibility. The paper concludes with an open question regarding further extension of representation of $L_n$ to $VL_n$ and $WL_n$.

On representations of the triplet group and some of its extensions

TL;DR

This work investigates representations of the triplet group and its extensions and , focusing on linear and automorphism-based realizations. It proves the irreducibility of the classical Tits representation over , constructs a new representation with an explicit matrix form via , and establishes faithfulness only in small ranks. It fully classifies complex homogeneous -local representations of (and non-homogeneous cases for ), showing these correspond to specializations of and detailing their faithfulness and reducibility. The paper then extends representations to the virtual and welded triplet groups, identifying standard extension mechanisms and classifying nontrivial homogeneous -local representations of and , including conditions for faithfulness and irreducibility. Overall, the results advance understanding of how -representations behave under extensions to and , and they pose open questions about further extensions beyond the studied framework.

Abstract

In this paper, we study the representations of the triplet group , where is a positive integer, and its extensions to the virtual and welded triplet groups and , respectively. We first introduce , its extensions, and its pure subgroup. We then investigate several representations, proving the irreducibility of the classical Tits representation over the complex field and constructing a new representation , where is the free group of rank . For the representation , we determine its matrix form, faithfulness, and irreducibility. We also classify all complex homogeneous -local representations of for and all non-homogeneous -local representations of , establishing connections with the complex specialization of the representation . Finally, we examine extensions of representations to and , proving their existence, classifying non-trivial complex homogeneous -local representations, and analyzing their faithfulness and irreducibility. The paper concludes with an open question regarding further extension of representation of to and .
Paper Structure (8 sections, 15 theorems, 85 equations, 10 figures)

This paper contains 8 sections, 15 theorems, 85 equations, 10 figures.

Table of Contents

  1. Introduction
  2. The Triplet Group and Some of Its Extensions
  3. Representations of the Triplet Group $L_n$ and Their Properties
  4. The Tits Representation of $L_n$
  5. Constructing a New Representation of $L_n$
  6. The Complex Homogeneous $2$-Local Representations of $L_n$
  7. The Complex Non-Homogeneous $2$-Local Representations of $L_3$
  8. Extensions of Representations of $L_n$ to $VL_n$ and $WL_n$

Key Result

Lemma 10

Let $\Theta : L_n \longrightarrow \mathrm{GL}_{n-1}(\mathbb{C})$ be the Tits representation defined in Definition DefTits, and let $U \subseteq \mathbb{C}^{\,n-1}$ be an invariant subspace under $\Theta$. Let $\{e_1,e_2,\ldots,e_{n-1}\}$ denote the standard basis of $\mathbb{C}^{\,n-1}$. If there ex

Figures (10)

  • Figure 1: The braid group generator $\sigma_i$ and its inverse $\sigma_i^{-1}$.
  • Figure 2: The relation $\sigma_{i}\sigma_{i+1}\sigma_{i}=\sigma_{+1}\sigma_{i}\sigma_{i+1}$.
  • Figure 3: The generator $\ell_i.$
  • Figure 4: The relation $\ell_i^2 =1$.
  • Figure 5: The relation $\ell_i \ell_{i+1} \ell_i = \ell_{i+1} \ell_i \ell_{i+1}$.
  • ...and 5 more figures

Theorems & Definitions (39)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Definition 7
  • Definition 8
  • Definition 9
  • Lemma 10
  • ...and 29 more