On representations of the triplet group and some of its extensions

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 $WL_n$.