On representations of the triplet group and some of its extensions

TL;DR

This paper investigates various representations of the triplet group and its extensions, proving irreducibility, faithfulness, and classifying local representations, thereby advancing understanding of their algebraic structure and extensions.

Contribution

It introduces new representations of the triplet group, classifies local representations, and explores their extensions to virtual and welded triplet groups, providing a comprehensive analysis of their properties.

Findings

Proved irreducibility of the classical Tits representation.

Constructed a new representation : L_n bb a0Aut(a0F_n) and analyzed its properties.

Classified complex homogeneous 2-local representations of L_n for n a3 3.

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 $\Theta: L_n \to \mathrm{GL}_{n-1}(\mathbb{C})$ over the complex field $\mathbb{C}$ and constructing a new representation $\mu: L_n \to \mathrm{Aut}(\mathbb{F}_n)$, where $\mathbb{F}_n$ is the free group of rank $n$. For the representation $\mu$, 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 $\mu$. 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 $WL_n$.